Merge pull request #1 from AndreasFMueller/master

update

217 files changed, 13427 insertions, 0 deletions

diff --git a/vorlesungen/slides/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc
new file mode 100644
index 0000000..a6bb320
--- /dev/null
+++ b/vorlesungen/slides/0/Makefile.inc
@@ -0,0 +1,17 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 20920 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+chapter0 =								\
+	../slides/0/was.tex						\
+	../slides/0/intro.tex						\
+	../slides/0/resourcen.tex					\
+	../slides/0/latextipps.tex					\
+	../slides/0/nextsteps.tex					\
+	../slides/0/themen1.tex						\
+	../slides/0/themen2.tex						\
+	../slides/0/themen3.tex						\
+	../slides/0/chapter.tex
+
diff --git a/vorlesungen/slides/0/chapter.tex b/vorlesungen/slides/0/chapter.tex
new file mode 100644
index 0000000..6e09557
--- /dev/null
+++ b/vorlesungen/slides/0/chapter.tex
@@ -0,0 +1,13 @@
+%
+% chapter.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{0/intro.tex}
+\folie{0/was.tex}
+\folie{0/resourcen.tex}
+\folie{0/latextipps.tex}
+\folie{0/themen1.tex}
+\folie{0/themen2.tex}
+\folie{0/themen3.tex}
+\folie{0/nextsteps.tex}
diff --git a/vorlesungen/slides/0/intro.tex b/vorlesungen/slides/0/intro.tex
new file mode 100644
index 0000000..acda6d1
--- /dev/null
+++ b/vorlesungen/slides/0/intro.tex
@@ -0,0 +1,98 @@
+%
+% intro.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\r{4}
+
+\def\rad#1{
+\begin{scope}[rotate=#1]
+\fill[color=blue!20] (0,0) -- (-60:\r) arc (-60:60:\r) -- cycle;
+\fill[color=darkgreen!20] (0,0) -- (60:\r) arc (60:180:\r) -- cycle;
+\fill[color=orange!20] (0,0) -- (180:\r) arc (180:300:\r) -- cycle;
+
+\node[color=darkgreen] at (120:3.7) [rotate={#1+30}] {Algebra};
+\node[color=orange] at (240:3.7) [rotate={#1+150}] {Analysis};
+\node[color=blue] at (0:3.7) [rotate={#1-90}] {Zerlegung};
+\end{scope}
+}
+
+\begin{frame}
+\frametitle{Intro --- Matrizen}
+
+\vspace{-25pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\only<1-8>{
+	\rad{-30}
+	\only<2->{ \node at (90:3.0) {Rechenregeln $A^2+A+I=0$}; }
+	\only<3->{ \node at (90:2.5) {Polynome $\chi_A(A)=0$, $m_A(A)=0$}; }
+	\only<4->{ \node at (90:2.0) {Projektion: $P^2=P$}; }
+	\only<5->{ \node at (90:1.5) {nilpotent: $N^k=0$}; }
+}
+
+\only<9-14>{
+	\rad{90}
+	\only<10->{ \node at (90:2.7) {Eigenbasis: $A=\sum \lambda_k P_k$}; }
+	\only<11->{ \node at (90:2.2) {Invariante Räume:
+		$AV\subset V, AV^\perp\subset V^\perp$}; }
+}
+
+\only<15-22>{
+	\rad{210}
+	\only<16->{ \node at (90:3.3) {Symmetrien}; }
+	\only<17->{ \node at (90:2.8) {Skalarprodukt erhalten:
+		$\operatorname{SO}(n)$}; }
+	\only<18->{ \node at (90:2.3) {Konstant $\Rightarrow$ Ableitung $=0$}; }
+	\only<19->{ \node at (90:1.5) {$\displaystyle \exp(A)
+		= \sum_{k=0}^\infty \frac{A^k}{k!}$};
+	}
+}
+
+\fill[color=red!20] (0,0) circle[radius=1.0];
+\node at (0,0.25) {Matrizen};
+\node at (0,-0.25) {$M_{m\times n}(\Bbbk)$};
+
+\uncover<6->{
+	\node[color=darkgreen] at (4.3,3.4) [right] {Algebra};
+	\node at (4.3,2.2) [right] {\begin{minipage}{5cm}
+	\begin{itemize}
+	\item<6-> Algebraische Strukturen
+	\item<7-> Polynome, Teilbarkeit
+	\item<8-> Minimalpolynom
+	\end{itemize}
+	\end{minipage}};
+}
+
+\uncover<12->{
+	\node[color=blue] at (4.3,0.8) [right] {Zerlegung};
+	\node at (4.3,-0.4) [right] {\begin{minipage}{5cm}
+	\begin{itemize}
+	\item<12-> Eigenvektoren, -räume
+	\item<13-> Projektionen, Drehungen
+	\item<14-> Invariante Unterräume
+	\end{itemize}
+	\end{minipage}};
+}
+
+\uncover<20->{
+	\node[color=orange] at (4.3,-1.8) [right] {Analysis};
+	\node at (4.3,-3.0) [right] {\begin{minipage}{6cm}
+	\begin{itemize}
+	\item<20-> Symmetrien
+	\item<21-> Matrix-DGL
+	\item<22-> Matrix-Potenzreihen
+	\end{itemize}
+	\end{minipage}};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/0/latextipps.tex b/vorlesungen/slides/0/latextipps.tex
new file mode 100644
index 0000000..09d7c89
--- /dev/null
+++ b/vorlesungen/slides/0/latextipps.tex
@@ -0,0 +1,16 @@
+%
+% latextipps.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{\LaTeX-Tipps/Anforderungen}
+\begin{enumerate}
+\item<1-> Formeln sind Bestandteil von Sätzen, dürfen nicht alleine stehen.
+\item<2-> Über die Platzierung von Abbildungen/Tabellen entscheidet das System
+(mit Verweisen arbeiten).
+\item<3-> Neuer Absatz: Leerzeile (nicht \texttt{\textbackslash\textbackslash})
+\item<4-> Jeden Satz auf einer neuen Zeile beginnen (GIT)
+\item<5-> Bilder PDF (PNG/JPG mindestens 300 dpi)
+\end{enumerate}
+\end{frame}
diff --git a/vorlesungen/slides/0/nextsteps.tex b/vorlesungen/slides/0/nextsteps.tex
new file mode 100644
index 0000000..cb9a07e
--- /dev/null
+++ b/vorlesungen/slides/0/nextsteps.tex
@@ -0,0 +1,21 @@
+%
+% nextsteps.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Nächste Schritte}
+
+\begin{enumerate}
+\item<2->
+Thema wählen, Teams bilden, Thema wird festgelegt zu Beginn der Woche 2
+\item<3->
+Grundlagen studieren (Skript, Wikipedia, Bücher)
+\item<4->
+Eigenes Seminarthema vertiefen 
+\item <5->
+Plan für Seminararbeit und Vortrag
+\end{enumerate}
+
+\end{frame}
diff --git a/vorlesungen/slides/0/resourcen.tex b/vorlesungen/slides/0/resourcen.tex
new file mode 100644
index 0000000..02a3fb8
--- /dev/null
+++ b/vorlesungen/slides/0/resourcen.tex
@@ -0,0 +1,37 @@
+%
+% resourcen.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Resourcen}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Moodle Modul MathSem}
+\begin{enumerate}
+\item<2-> Skript
+\begin{itemize}
+\item<3-> Aktuellste Version in Github
+\item<4-> regelmässige Updates in Moodle: \texttt{buch.pdf}
+\end{itemize}
+\item<5-> Informationen zur Planung: Kurztests, Vorträge
+\item<6-> Anleitung für die Seminararbeit
+\item<7-> Aufgabenstellungen
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Weitere Quellen}
+\begin{enumerate}
+\item<9-> Zusätzliche Literaturhinweise in der Aufgabenbeschreibung im Moodle
+\item<10-> Bibliothek
+\item<11-> Google
+\item<12-> Google Scholar
+\item<13-> Paper ist nicht öffentlich zugänglich? $\rightarrow$ kann via
+Bibliothek organisiert werden
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/0/themen1.tex b/vorlesungen/slides/0/themen1.tex
new file mode 100644
index 0000000..756e037
--- /dev/null
+++ b/vorlesungen/slides/0/themen1.tex
@@ -0,0 +1,27 @@
+%
+% themen1.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{Seminararbeitsthemen I}
+\begin{enumerate}[<+->]
+\item
+Verkehrsnetze und Verkehrsfluss
+\item
+Mittelwert von Matrizen
+\item
+Pascal-Matrizen
+\item 
+Stirling-Matrizen
+\item
+Vandermonde-Matrix
+\item
+Probabilistische Matrix-Produkt-Kontrolle
+\item
+Der Satz von Furrer-Hungerbühler-Jantschgi
+\item
+Clifford-Algebren
+\end{enumerate}
+\end{frame}
+
diff --git a/vorlesungen/slides/0/themen2.tex b/vorlesungen/slides/0/themen2.tex
new file mode 100644
index 0000000..1fbdab3
--- /dev/null
+++ b/vorlesungen/slides/0/themen2.tex
@@ -0,0 +1,27 @@
+%
+% themen2.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Seminararbeitsthemen II}
+\begin{enumerate}[<+->]
+\setcounter{enumi}{7}
+\item
+Schnelle Matrixmultiplikation
+\item
+Parkettierungen mit Dominosteinen zählen
+\item
+Punktgruppen und Kristallographie
+\item
+Symmetriegruppen und Machine Learning
+\item
+Floyd-Warshall-Algorithmus
+\item
+Laser
+\item
+Munkres-Algorithmus
+\end{enumerate}
+\end{frame}
+
diff --git a/vorlesungen/slides/0/themen3.tex b/vorlesungen/slides/0/themen3.tex
new file mode 100644
index 0000000..5a6bdf5
--- /dev/null
+++ b/vorlesungen/slides/0/themen3.tex
@@ -0,0 +1,26 @@
+%
+% themen3.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Seminararbeitsthemen III}
+\begin{enumerate}[<+->]
+\setcounter{enumi}{14}
+\item
+Iwasawa-Zerlegung
+\item 
+Reed-Solomon Code
+\item
+Pauli- und Dirac-Matrizen
+\item
+Klassifikation der Lie-Gruppen
+\item
+Iterierte Funktionsschemata
+\item
+QR-Codes
+\item
+McEliece-Kryptosystem
+\end{enumerate}
+\end{frame}
diff --git a/vorlesungen/slides/0/was.tex b/vorlesungen/slides/0/was.tex
new file mode 100644
index 0000000..685ee22
--- /dev/null
+++ b/vorlesungen/slides/0/was.tex
@@ -0,0 +1,50 @@
+%
+% was.tex -- was wird erwartet
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Was wird erwartet}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Seminararbeit}
+\begin{itemize}
+\item<3-> Ihr Thema: gestalten Sie es!
+\item<4-> Eine spannende Story erzählen
+\item<5-> Immer an den Leser denken: Ihre Kollegen
+\item<6-> So lang wie nötig, so kurz wie möglich
+\item<7-> Bewertet durch Seminarleiter
+\end{itemize}
+\end{block}}
+\vspace{-5pt}
+\uncover<14->{%
+\begin{block}{Hilfe}
+\begin{itemize}
+\item Einführungsvorlesungen
+\item \texttt{andreas.mueller@ost.ch}
+\item \texttt{roy.seitz@ost.ch}
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Seminarvortrag}
+\begin{itemize}
+\item<8-> Vortrag $\ne$ Arbeit
+\item<9-> So lang wie nötig, so kurz wie möglich
+\item<10-> Konzentration auf das Wesentliche
+\item<11-> $>30\,\text{min}$ ist fast sicher zu lang
+\item<12-> Bewertet durch die Seminarteilnehmer
+\end{itemize}
+\end{block}}
+\vspace{-5pt}
+\uncover<13->{%
+\begin{block}{3 Kurztests}
+Ziel: Sie befassen sich auch mit den Themen
+ausserhalb ihrer eigenen Seminararbeit
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/Makefile.inc b/vorlesungen/slides/1/Makefile.inc
new file mode 100644
index 0000000..38b47b3
--- /dev/null
+++ b/vorlesungen/slides/1/Makefile.inc
@@ -0,0 +1,23 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 20920 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+chapter1 =								\
+	../slides/1/zahlensysteme.tex					\
+	../slides/1/peano.tex						\
+	../slides/1/ganz.tex						\
+	../slides/1/bruch.tex						\
+	../slides/1/ring.tex						\
+	../slides/1/schwierigkeiten.tex					\
+	../slides/1/strukturen.tex					\
+	../slides/1/j.tex						\
+	../slides/1/vektorraum.tex					\
+	../slides/1/matrixalgebra.tex					\
+	../slides/1/algebrastruktur.tex					\
+	../slides/1/speziell.tex					\
+	../slides/1/dreieck.tex						\
+	../slides/1/hadamard.tex					\
+	../slides/1/chapter.tex
+
diff --git a/vorlesungen/slides/1/algebrastruktur.tex b/vorlesungen/slides/1/algebrastruktur.tex
new file mode 100644
index 0000000..fd474eb
--- /dev/null
+++ b/vorlesungen/slides/1/algebrastruktur.tex
@@ -0,0 +1,93 @@
+%
+% algebrastruktur.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\begin{frame}[t]
+\frametitle{Algebra über $\Bbbk$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\pgfmathparse{atan(7/4)}
+\xdef\a{\pgfmathresult}
+\uncover<2->{
+	\fill[color=red!40,opacity=0.5]
+		({-4-2.5},{2+1.0})
+		--
+		({-2.5},{-3-1.0})
+		--
+		({2.5},{-3-1.0})
+		--
+		({-4+2.5},{2+1.0})
+		-- cycle;
+}
+
+\uncover<4->{
+	\fill[color=blue!40,opacity=0.5]
+		({4-2.5},{2+1.0})
+		--
+		({-2.5},{-3-1.0})
+		--
+		({2.5},{-3-1.0})
+		--
+		({4+2.5},{2+1.0})
+		-- cycle;
+}
+
+\uncover<6->{
+	\fill[color=darkgreen!40,opacity=0.5]
+		({-4-2.5},{2+1.0})
+		-- 
+		({-4-2.5+2*(4/7)},{2-1})
+		-- 
+		({+4+2.5-2*(4/7)},{2-1})
+		-- 
+		({+4+2.5},{2+1})
+		--
+		cycle;
+}
+
+\node at ({-3-0.5},2) {Skalarmultiplikation};
+
+\node at (3.5,2.2) {Multiplikation};
+\node at (3.5,1.8) {\tiny Monoid};
+
+\node at (0,-2.8) {Addition};
+\node at (0,-3.2) {\tiny Gruppe};
+
+\uncover<4->{
+	\node[color=blue] at (4.8,-0.5) [rotate=\a] {Ring\strut};
+}
+
+\uncover<2->{
+	\node[color=red] at (-4.8,-0.5) [rotate=-\a] {Vektorraum\strut};
+}
+
+\uncover<6->{
+	\node[color=darkgreen] at (0,2.6) {$(\lambda a)b=\lambda(ab)$};
+}
+
+\uncover<3->{
+	\node[color=red] at (-2.5,-0.5) {$\displaystyle
+	\begin{aligned}
+	\lambda(a+b)&=\lambda a + \lambda b\\
+	(\lambda+\mu)a&=\lambda a +\mu a
+	\end{aligned}$};
+}
+
+\uncover<5->{
+	\node[color=blue] at (2.5,-0.5) {$\displaystyle
+	\begin{aligned}
+	a(b+c)&=ab+ac\\
+	(a+b)c&=ac+bc
+	\end{aligned}$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/1/bruch.tex b/vorlesungen/slides/1/bruch.tex
new file mode 100644
index 0000000..65521a2
--- /dev/null
+++ b/vorlesungen/slides/1/bruch.tex
@@ -0,0 +1,73 @@
+%
+% bruch.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Brüche}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-8pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Division}
+Nicht für alle $a,b\in\mathbb{Z}$ hat die Gleichung
+\[
+ax=b
+\uncover<2->{
+\;\Rightarrow\;
+x=\frac{b}{a}}
+\]
+eine Lösung in $\mathbb{Z}$\uncover<2->{, nämlich wenn $b\nmid a$}
+\end{block}
+\uncover<3->{%
+\begin{block}{Brüche}
+Idee: $\displaystyle\frac{b}{a} = (b,a)$
+\begin{enumerate}
+\item<4-> $(b,a)\in\mathbb{Z}\times\mathbb{Z}$
+\item<5-> Äquivalenzrelation
+\[
+(b,a)\sim (d,c)
+\ifthenelse{\boolean{presentation}}{
+\only<5>{
+\Leftrightarrow
+\text{``
+$\displaystyle
+\frac{b}{a}=\frac{d}{c}
+$
+''}
+}}{}
+\only<6->{
+\Leftrightarrow
+bc=ad
+}
+\]
+\end{enumerate}
+\vspace{-15pt}
+\uncover<7->{%
+$\Rightarrow$ alle Quotienten
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<9->{%
+\begin{block}{Gruppe}
+$\mathbb{Q}^* = \mathbb{Q}\setminus\{0\}$ ist eine multiplikative Gruppe:
+\begin{enumerate}
+\item<10-> Neutrales Element: $1\in \mathbb{Q}^*$
+\item<11-> Inverses Element $q=\frac{b}{a}\in\mathbb{Q}
+\Rightarrow
+q^{-1}=\frac{a}{b}\in\mathbb{Q}$
+\end{enumerate}
+\end{block}
+}
+\uncover<8->{%
+\begin{block}{Rationale Zahlen}
+Alle Brüche, gleiche Werte zusammengefasst:
+\[
+\mathbb{Q} = \mathbb{Z}\times\mathbb{Z}/\sim
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/chapter.tex b/vorlesungen/slides/1/chapter.tex
new file mode 100644
index 0000000..7bdda34
--- /dev/null
+++ b/vorlesungen/slides/1/chapter.tex
@@ -0,0 +1,19 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{1/zahlensysteme.tex}
+\folie{1/peano.tex}
+\folie{1/ganz.tex}
+\folie{1/bruch.tex}
+\folie{1/ring.tex}
+\folie{1/schwierigkeiten.tex}
+\folie{1/strukturen.tex}
+\folie{1/j.tex}
+\folie{1/vektorraum.tex}
+\folie{1/matrixalgebra.tex}
+\folie{1/algebrastruktur.tex}
+\folie{1/speziell.tex}
+\folie{1/dreieck.tex}
+\folie{1/hadamard.tex}
diff --git a/vorlesungen/slides/1/dreieck.tex b/vorlesungen/slides/1/dreieck.tex
new file mode 100644
index 0000000..3797e4b
--- /dev/null
+++ b/vorlesungen/slides/1/dreieck.tex
@@ -0,0 +1,69 @@
+%
+% dreieck.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Dreiecksmatrizen}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.31\textwidth}
+\begin{block}{Dreiecksmatrix}
+\begin{align*}
+R&=
+\begin{pmatrix}
+*&*&*&\dots&*\\
+0&*&*&\dots&*\\
+0&0&*&\dots&*\\
+\vdots&\vdots&\vdots&\ddots&\vdots\\
+0&0&0&\dots&*
+\end{pmatrix}
+\\
+U&=
+\begin{pmatrix}
+1&*&*&\dots&*\\
+0&1&*&\dots&*\\
+0&0&1&\dots&*\\
+\vdots&\vdots&\vdots&\ddots&\vdots\\
+0&0&0&\dots&1
+\end{pmatrix}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.31\textwidth}
+\uncover<2->{%
+\begin{block}{Nilpotente Matrix}
+\[
+N=
+\begin{pmatrix}
+0&*&*&\dots&*\\
+0&0&*&\dots&*\\
+0&0&0&\dots&*\\
+\vdots&\vdots&\vdots&\ddots&\vdots\\
+0&0&0&\dots&0
+\end{pmatrix}
+\]
+\uncover<3->{%
+$\Rightarrow N^n=0$
+}
+\end{block}}
+\end{column}
+\begin{column}{0.31\textwidth}
+\uncover<4->{%
+\begin{block}{Jordan-Matrix}
+\[
+J_\lambda=\begin{pmatrix}
+\lambda&1&0&\dots&0\\
+0&\lambda&1&\dots&0\\
+0&0&\lambda&\dots&0\\
+\vdots&\vdots&\vdots&\ddots&\vdots\\
+0&0&0&\dots&\lambda
+\end{pmatrix}
+\]
+\uncover<5->{%
+$\Rightarrow J_\lambda -\lambda I$ ist nilpotent
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/ganz.tex b/vorlesungen/slides/1/ganz.tex
new file mode 100644
index 0000000..7930826
--- /dev/null
+++ b/vorlesungen/slides/1/ganz.tex
@@ -0,0 +1,106 @@
+%
+% ganz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Ganze Zahlen: Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{block}{Subtrahieren}
+Nicht für alle $a,b\in \mathbb{N}$ hat die 
+Gleichung
+\[
+a+x=b
+\uncover<2->{
+\quad
+\Rightarrow
+\quad
+x=b-a}
+\]
+eine Lösung in $\mathbb{N}$\uncover<2->{, nämlich wenn $a>b$}%
+\end{block}
+\uncover<3->{%
+\begin{block}{Ganze Zahlen = Paare}
+Idee: $b-a = (b,a)$
+\begin{enumerate}
+\item<4-> $(b,a)=\mathbb{N}\times\mathbb{N}$
+\item<5-> Äquivalenzrelation
+\[
+(b,a)\sim (d,c)
+\ifthenelse{\boolean{presentation}}{
+\only<6>{\Leftrightarrow
+\text{``\strut}
+b-a=c-d
+\text{\strut''}}}{}
+\only<7->{
+\Leftrightarrow
+b+d=c+a}
+\]
+\end{enumerate}
+\vspace{-10pt}
+\uncover<8->{%
+Ganze Zahlen:
+\(
+\mathbb{Z}
+=
+\mathbb{N}\times\mathbb{N}/\sim
+\)}
+\\
+\uncover<9->{%
+$z\in\mathbb{Z}$, $z=\mathstrut$ Paare $(u,v)$ mit 
+``gleicher Differenz''}
+\uncover<10->{%
+$\Rightarrow$ alle Differenzen in $\mathbb{Z}$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\uncover<11->{%
+\begin{block}{Gruppe}
+Monoid $\ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}{}\only<12->{G}$ mit inversem Element
+\[
+a\in \ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}{}\only<12->{G}
+\Rightarrow
+\ifthenelse{\boolean{presentation}}{\only<11>{-a\in\mathbb{Z}}}{}\only<12->{a^{-1}\in G}
+\text{ mit }
+\ifthenelse{\boolean{presentation}}{
+\only<11>{
+a+(-a)=0
+}}{}
+\only<12->{
+\left\{
+\begin{aligned}
+aa^{-1}&=e
+\\
+a^{-1}a&=e
+\end{aligned}
+\right.
+}
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<13->{%
+\begin{block}{Abelsche Gruppe}
+Verknüpfung ist kommutativ:
+\[
+a+b=b+a
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<14->{%
+\begin{block}{Beispiele}
+\begin{itemize}
+\item<15-> Brüche, reelle Zahlen
+\item<16-> invertierbare Matrizen: $\operatorname{GL}_n(\mathbb{R})$
+\item<17-> Drehmatrizen: $\operatorname{SO}(n)$
+\item<18-> Matrizen mit Determinante $1$: $\operatorname{SL}_n(\mathbb R)$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/hadamard.tex b/vorlesungen/slides/1/hadamard.tex
new file mode 100644
index 0000000..5cb692a
--- /dev/null
+++ b/vorlesungen/slides/1/hadamard.tex
@@ -0,0 +1,51 @@
+%
+% hadamard.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Hadamard-Algebra}
+\begin{block}{Alternatives Produkt: Hadamard-Produkt}
+\[
+\begin{pmatrix}
+a_{11}&\dots&a_{1n}\\
+\vdots&\ddots&\vdots\\
+a_{m1}&\dots&a_{mn}\\
+\end{pmatrix}
+\odot
+\begin{pmatrix}
+b_{11}&\dots&b_{1n}\\
+\vdots&\ddots&\vdots\\
+b_{m1}&\dots&b_{mn}\\
+\end{pmatrix}
+=
+\begin{pmatrix}
+a_{11}b_{11}&\dots&a_{1n}b_{1n}\\
+\vdots&\ddots&\vdots\\
+a_{m1}b_{m1}&\dots&a_{mn}b_{mn}\\
+\end{pmatrix}
+\]
+\end{block}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.58\textwidth}
+\uncover<2->{%
+\begin{block}{Algebra}
+\begin{itemize}
+\item<3-> $M_{mn}(\Bbbk)$ ist eine Algebra mit
+$\odot$ als Produkt
+\item<4-> Neutrales Element $U$: Matrix aus lauter Einsen
+\item<5-> Anwendung: Wahrscheinlichkeitsmatrizen
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.38\textwidth}
+\uncover<6->{%
+\begin{block}{Nicht so interessant}
+Die Hadamard-Algebra ist kommutativ 
+\uncover<7->{$\Rightarrow$ 
+kann ``keine'' interessanten algebraischen Relationen darstellen}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/j.tex b/vorlesungen/slides/1/j.tex
new file mode 100644
index 0000000..132f1d0
--- /dev/null
+++ b/vorlesungen/slides/1/j.tex
@@ -0,0 +1,63 @@
+%
+% j.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Beispiele}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Imaginäre Einheit $i$}
+Gibt es eine Zahl $i$ mit $i^2=-1$?
+\end{block}
+\uncover<2->{%
+\begin{block}{Matrixlösung}
+Die Matrix
+\[
+J
+=
+\begin{pmatrix}0&-1\\1&0\end{pmatrix}
+\]
+erfüllt
+\[
+J^2
+=
+%\begin{pmatrix}0&-1\\1&0\end{pmatrix}
+%\begin{pmatrix}0&-1\\1&0\end{pmatrix}
+%=
+\begin{pmatrix}-1&0\\0&-1\end{pmatrix}
+=
+-I
+\]
+$\Rightarrow$ $J$ ist eine Matrixdarstellung von $i$
+
+Drehmatrix mit Winkel $90^\circ$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Quadratwurzel $\sqrt{2}$}
+Gibt es eine Zahl $\sqrt{2}$ derart, dass $(\sqrt{2})^2=2$?
+\end{block}}
+\uncover<4->{%
+\begin{block}{Matrixlösung}
+%\setlength{\abovedisplayskip}{5pt}
+%\setlength{\belowdisplayskip}{5pt}
+Die Matrix
+\[
+W
+=
+\begin{pmatrix}0&2\\1&0\end{pmatrix}
+\]
+erfüllt
+\[
+W^2
+=
+\begin{pmatrix}2&0\\0&2\end{pmatrix} = 2I
+\]
+$\Rightarrow$ $W$ ist eine Matrixdarstellung von $\sqrt{2}$
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/matrixalgebra.tex b/vorlesungen/slides/1/matrixalgebra.tex
new file mode 100644
index 0000000..a3c3a76
--- /dev/null
+++ b/vorlesungen/slides/1/matrixalgebra.tex
@@ -0,0 +1,77 @@
+%
+% matrixalgebra.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\newtcbox{\myboxA}{blank,boxsep=0mm,
+clip upper,minipage,
+width=31.0mm,height=17.0mm,nobeforeafter,
+borderline={0.0pt}{0.0pt}{white},
+}
+\definecolor{magenta}{rgb}{0.8,0.2,0.8}
+
+\begin{frame}[t]
+\frametitle{Matrix-Algebra}
+\vspace{-10pt}
+\[
+\begin{pmatrix}
+a_{11}&\dots &a_{1n}\\
+\vdots&\ddots&\vdots\\
+a_{m1}&\dots &a_{mn}
+\end{pmatrix}
++
+\begin{pmatrix}
+b_{11}&\dots &b_{1n}\\
+\vdots&\ddots&\vdots\\
+b_{m1}&\dots &b_{mn}
+\end{pmatrix}
+=
+\begin{pmatrix}
+a_{11}+b_{11}&\dots &a_{1n}+b_{1n}\\
+\vdots&\ddots&\vdots\\
+a_{m1}+b_{m1}&\dots &a_{mn}+b_{mn}
+\end{pmatrix}
+\]
+\[
+\lambda
+\begin{pmatrix}
+a_{11}&\dots &a_{1n}\\
+\vdots&\ddots&\vdots\\
+a_{m1}&\dots &a_{mn}
+\end{pmatrix}
+=
+\begin{pmatrix}
+\lambda a_{11}&\dots &\lambda a_{1n}\\
+\vdots&\ddots&\vdots\\
+\lambda a_{m1}&\dots &\lambda a_{mn}
+\end{pmatrix}
+\]
+\uncover<2->{%
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}[xshift=-4.5cm]
+\node at (1.5,1.53) {$\left(\myboxA{}\right)$};
+\draw[color=red,line width=3pt] (0,2) -- (3,2);
+\draw (0,0) rectangle (3,3);
+\end{scope}
+\node at (-0.75,1.5) {$\mathstrut\cdot\mathstrut$};
+\begin{scope}[xshift=0cm]
+\node at (1.5,1.53) {$\left(\myboxA{}\right)$};
+\draw[color=blue,line width=3pt] (2.7,0) -- (2.7,3);
+\draw (0,0) rectangle (3,3);
+\end{scope}
+\node at (3.75,1.5) {$\mathstrut=\mathstrut$};
+\begin{scope}[xshift=4.5cm]
+\node at (1.5,1.53) {$\left(\myboxA{}\right)$};
+\draw[color=gray,line width=1pt] (2.7,0) -- (2.7,3);
+\draw[color=gray,line width=1pt] (0,2) -- (3,2);
+\fill[color=magenta] (2.7,2) circle[radius=0.12];
+\draw (0,0) rectangle (3,3);
+\end{scope}
+\end{tikzpicture}
+\end{center}}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/1/peano.tex b/vorlesungen/slides/1/peano.tex
new file mode 100644
index 0000000..219c853
--- /dev/null
+++ b/vorlesungen/slides/1/peano.tex
@@ -0,0 +1,72 @@
+%
+% peano.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Natürliche Zahlen\uncover<2->{: Peano}}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zählen}
+Mit den natürlichen Zahlen zählt man: 
+\[
+\mathbb{N}
+=
+\left\{
+\begin{minipage}{5cm}
+\raggedright
+Äquivalenzklassen von gleich mächtigen
+endlichen Mengen
+\end{minipage}
+\right\}
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Peano-Axiome}
+\begin{enumerate}
+\item<3-> $0\in\mathbb{N}$
+\item<4-> $n\in\mathbb{N}\Rightarrow \text{Nachfolger }n'\in\mathbb{N}$
+\item<5-> $0$ ist nicht Nachfolger
+\item<6-> $n,m\in\mathbb{N}\wedge n'=m'\Rightarrow n=m$
+\item<7-> $X\subset \mathbb{N}\wedge 0\in X\wedge \forall n\in X(n'\in X)
+\Rightarrow
+\mathbb{N}=X
+$
+\end{enumerate}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Monoid}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+Menge $\only<8-10>{\mathbb{N}}\only<11->{M}$ mit einer
+zweistelligen Verknüpfung $a\only<8-10>{+}\only<11->{*}b$
+\begin{enumerate}
+\item<9-> Assoziativ: $a,b,c\in M$
+\[
+(a\only<8-10>{+}\only<11->{*}b)\only<8-10>{+}\only<11->{*}c=a\only<8-10>{+}\only<11->{*}(b\only<8-10>{+}\only<11->{*}c)
+\]
+\item<10-> Neutrales Element: $\only<8-10>{0}\only<11->{e}\in M$
+\[
+\only<8-10>{0+}\only<11->{e*} a
+=
+a \only<8-10>{+0}\only<11->{*e}
+\]
+\end{enumerate}
+\end{block}}%
+\vspace{-15pt}
+\uncover<12->{%
+\begin{block}{Axiom 5 = Vollständige Induktion}
+$X=\{n\in\mathbb{N}\;|\; \text{$P(n)$ ist wahr}\}$
+\begin{enumerate}
+\item<13-> Verankerung: $0\in X$
+\item<14-> Induktionsannahme: $n\in X$
+\item<15-> Induktionsschritt: $n'\in X$
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/ring.tex b/vorlesungen/slides/1/ring.tex
new file mode 100644
index 0000000..9641975
--- /dev/null
+++ b/vorlesungen/slides/1/ring.tex
@@ -0,0 +1,58 @@
+%
+% ring.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Ring\only<15->{/Körper}}
+\vspace{-10pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Addition und Multiplikation}
+$\mathbb{Z}$ und $\mathbb{Q}$
+haben zwei Verknüpfungen:
+\begin{enumerate}
+\item<2-> Addition
+\[
+a,b\in R\Rightarrow a+b\in R
+\]
+\item<3-> Multiplikation
+\[
+a,b\in R\Rightarrow a\cdot b=ab\in R
+\]
+\end{enumerate}
+\vspace{-5pt}
+\uncover<4->{%
+Gilt auch für
+\begin{itemize}
+\item<5-> Polynome
+\item<6-> $M_{n}(\mathbb{R})$
+\item<7-> $\mathbb{R}^3$ mit Vektorprodukt
+\end{itemize}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Definition}
+Ein Ring\only<15->{/{\color{red}Körper}} ist eine Menge $R$ mit zwei
+Verknüpfungen $+$ und $\cdot$:
+\begin{enumerate}
+\item<9->
+$R$ mit $+$ ist eine abelsche Gruppe
+\item<10->
+$R$ mit $\cdot$ ist ein Monoid\only<15->{/{\color{red}eine Gruppe}}
+\item<11->
+Verträglichkeit: Distributivgesetz
+\begin{align*}
+\uncover<12->{a(b+c)&=ab+bc}
+\\
+\uncover<13->{(a+b)c&=ac+bc}
+\end{align*}
+\uncover<14->{(Ausmultiplizieren)}
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/schwierigkeiten.tex b/vorlesungen/slides/1/schwierigkeiten.tex
new file mode 100644
index 0000000..fb22e58
--- /dev/null
+++ b/vorlesungen/slides/1/schwierigkeiten.tex
@@ -0,0 +1,90 @@
+%
+% schwierigkeiten.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Schwierigkeiten}
+\vspace{-15pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Nullteiler}
+Elemente $a,b$ mit $ab=0$
+$\Rightarrow$ nicht invertierbar
+\begin{itemize}
+\item<3-> Projektionen
+\[
+\begin{pmatrix}
+1&0\\0&0
+\end{pmatrix}
+\begin{pmatrix}
+0&0\\0&1
+\end{pmatrix}
+=
+0
+\]
+\item<4-> Nilpotente Matrizen
+\[
+\begin{pmatrix}
+0&1&0\\
+0&0&1\\
+0&0&0
+\end{pmatrix}^3
+=0
+\]
+\item<5->
+In $\mathbb{Z}/15\mathbb{Z}$ (modulo 15):
+\[
+3\cdot 5 = 15 \equiv 0\mod 15
+\]
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Invertierbarkeit}
+\begin{itemize}
+\item<7->
+$7\in\mathbb{Z}$, aber $7^{-1}\not\in\mathbb{Z}$, $7^{-1}\in\mathbb{Q}$
+\item<8->
+$A$ regulär heisst nicht $A^{-1}\in M_n(\mathbb{Z})$
+\[
+A=\begin{pmatrix}
+1&-1\\
+1&1
+\end{pmatrix}
+\;\Rightarrow\;
+A^{-1}
+=
+\begin{pmatrix}
+\frac12&\frac12\\
+-\frac12&\frac12
+\end{pmatrix}
+\]
+\item<9->
+$A\in\operatorname{SL}_n(\mathbb{Z})$ invertierbar in
+$M_n(\mathbb{Z})$:
+\[
+A=
+\begin{pmatrix}
+5&4\\4&3
+\end{pmatrix}
+\;
+\Rightarrow
+\;
+A^{-1}=
+\begin{pmatrix}
+-3&4\\4&-5
+\end{pmatrix}
+\]
+\end{itemize}
+\uncover<10->{%
+Invertierbarkeit erreichen durch ``vergrössern'' des Ringes
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/speziell.tex b/vorlesungen/slides/1/speziell.tex
new file mode 100644
index 0000000..5b93da6
--- /dev/null
+++ b/vorlesungen/slides/1/speziell.tex
@@ -0,0 +1,46 @@
+%
+% speziell.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\frametitle{Diagonalmatrizen}
+\begin{block}{Einheitsmatrix}
+\[
+I=\begin{pmatrix}
+1&0&\dots&0\\
+0&1&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+0&0&\dots&1
+\end{pmatrix}
+\]
+Neutrales Element der Matrixmultiplikation:
+\[
+AI=IA=A
+\]
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<2->{%
+\begin{block}{Diagonalmatrix}
+\[
+\operatorname{diag}(\lambda_1,\lambda_2,\dots,\lambda_n)
+=
+\begin{pmatrix}
+\lambda_1&0&\dots&0\\
+0&\lambda_2&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+0&0&\dots&\lambda_n
+\end{pmatrix}
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Hadamard-Algebra}
+Die Algebra der Diagonalmatrizen ist die Hadamard-Algebra
+(siehe später)
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/strukturen.tex b/vorlesungen/slides/1/strukturen.tex
new file mode 100644
index 0000000..a5fc09a
--- /dev/null
+++ b/vorlesungen/slides/1/strukturen.tex
@@ -0,0 +1,35 @@
+%
+% strukturen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Strukturen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.42\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/10-vektorenmatrizen/images/strukturen.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.54\textwidth}
+\begin{itemize}[<+->]
+\item Gruppen: Drehungen, Symmetrien
+\item Vektorraum: Geometrie
+\item Ring (mit Eins)
+\item Algebra: Vektorraum und Ring
+\item Algebra mit Eins: Vektorraum und Ring mit Eins
+\item Körper
+\end{itemize}
+\uncover<7->{%
+\begin{block}{Matrizen}
+Jede beliebige Struktur lässt sich mit Matrizen darstellen:
+\begin{itemize}
+\item<8-> Permutationsmatrizen
+\item<9-> Wahrscheinlichkeitsmatrizen
+\item<10-> Wurzeln
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/vektorraum.tex b/vorlesungen/slides/1/vektorraum.tex
new file mode 100644
index 0000000..2566085
--- /dev/null
+++ b/vorlesungen/slides/1/vektorraum.tex
@@ -0,0 +1,54 @@
+%
+% vektorraum.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Vektorraum}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Operationen}
+Addition:
+\[
+\begin{pmatrix}a_1\\\vdots\\a_n \end{pmatrix}
++
+\begin{pmatrix}b_1\\\vdots\\b_n \end{pmatrix}
+=
+\begin{pmatrix}a_1+b_1\\\vdots\\a_n+b_n \end{pmatrix}
+\]
+Skalarmultiplikation:
+\[
+\lambda\begin{pmatrix}a_1\\\vdots\\a_n \end{pmatrix}
+=
+\begin{pmatrix}\lambda a_1\\\vdots\\\lambda a_n \end{pmatrix}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Additive Gruppe}
+$\mathbb{R}^n$ ist eine Gruppe bezüglich der Addition
+mit 
+\[
+0=\begin{pmatrix}0\\\vdots\\0\end{pmatrix},
+\qquad
+-a
+=
+-\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}
+=
+\begin{pmatrix}-a_1\\\vdots\\-a_n\end{pmatrix}
+\]
+\end{block}}
+\vspace{-5pt}
+\uncover<3->{%
+\begin{block}{Skalarmultiplikation}
+Distributivgesetz
+\begin{align*}
+(\lambda+\mu)a&=\lambda a + \mu a\\
+\lambda (a+b)&=\lambda a + \lambda b
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/1/zahlensysteme.tex b/vorlesungen/slides/1/zahlensysteme.tex
new file mode 100644
index 0000000..9131cc6
--- /dev/null
+++ b/vorlesungen/slides/1/zahlensysteme.tex
@@ -0,0 +1,46 @@
+%
+% zahlensysteme.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Zahlensysteme}
+\begin{center}
+\begin{tabular}{|>{$}c<{$}|p{7cm}|p{3cm}|}
+\hline
+\text{Zahlenmenge}&\text{Eigenschaften}&\text{Struktur}
+\\
+\hline
+\mathbb{N}
+&\phantom{}\raggedright\uncover<2->{Addition, neutrales Element $0$}
+&\phantom{}\uncover<2->{Monoid}
+\\
+\mathbb{Z}
+&\phantom{}\raggedright\uncover<3->{Addition, neutrales Element $0$,
+inverses Element der Addition}
+&\phantom{}\uncover<3->{Gruppe}
+\\
+\mathbb{Z}
+&\phantom{}\raggedright\uncover<4->{zusätzlich: Multiplikation, neutrales Element $1$}
+&\phantom{}\uncover<4->{Ring}
+\\
+\mathbb{Q}
+&\phantom{}\raggedright\uncover<5->{Addition und Multiplikation mit Inversen}
+&\phantom{}\uncover<5->{Körper}
+\\
+\mathbb{R}
+&\phantom{}\raggedright\uncover<6->{zusätzlich: Ordnungsrelation, Vollständigkeit}
+&\phantom{}\uncover<6->{Körper mit Ordnung}
+\\
+\mathbb{C}
+&\phantom{}\raggedright\uncover<7->{zusätzlich: Alle Wurzeln}
+&\phantom{}\uncover<7->{algebraisch abgeschlossener Körper}
+\\
+\uncover<8->{\mathbb{H}}
+&\phantom{}\raggedright\uncover<8->{höhere Dimension, nichtkommutativ}
+&\phantom{}\uncover<8->{Schiefkörper}
+\\
+\hline
+\end{tabular}
+\end{center}
+\end{frame}
diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc
new file mode 100644
index 0000000..c857fec
--- /dev/null
+++ b/vorlesungen/slides/2/Makefile.inc
@@ -0,0 +1,21 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter2 =								\
+	../slides/2/norm.tex						\
+	../slides/2/skalarprodukt.tex					\
+	../slides/2/cauchyschwarz.tex					\
+	../slides/2/polarformel.tex					\
+	../slides/2/funktionenraum.tex					\
+	../slides/2/operatornorm.tex					\
+	../slides/2/linearformnormen.tex				\
+	../slides/2/funktionenalgebra.tex				\
+	../slides/2/frobeniusnorm.tex					\
+	../slides/2/frobeniusanwendung.tex				\
+	../slides/2/quotient.tex					\
+	../slides/2/quotientv.tex					\
+	../slides/2/chapter.tex
+
diff --git a/vorlesungen/slides/2/cauchyschwarz.tex b/vorlesungen/slides/2/cauchyschwarz.tex
new file mode 100644
index 0000000..a24ada8
--- /dev/null
+++ b/vorlesungen/slides/2/cauchyschwarz.tex
@@ -0,0 +1,94 @@
+%
+% cauchyschwarz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Cauchy-Schwarz-Ungleichung}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Cauchy-Schwarz)}
+$\langle\;,\;\rangle$ eine positiv definite, hermitesche Sesquilinearform
+\[
+{\color{darkgreen}
+|\operatorname{Re}\langle u,v\rangle|
+\le
+|\langle u,v\rangle|
+\le
+\|u\|_2\cdot \|v\|_2
+}
+\]
+Gleichheit genau dann, wenn $u$ und $v$ linear abhängig sind
+\end{block}
+\begin{block}{Dreiecksungleichung}
+\vspace{-12pt}
+\begin{align*}
+\|u+v\|_2^2
+&=
+\|u\|_2^2 + 2\operatorname{Re}\langle u,v\rangle + \|v\|_2^2
+\\
+&\le
+\|u\|_2^2 + 2{\color{darkgreen}|\langle u,v\rangle|} + \|v\|_2^2
+\\
+&\le
+\|u\|_2^2 + 2{\color{darkgreen}\|u\|_2\cdot \|v\|_2} + \|v\|_2^2
+\\
+&=(\|u\|_2 + \|v\|_2)^2
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{proof}[Beweis]
+Die quadratische Funktion
+\begin{align*}
+Q(t)
+&=
+\langle u+tv,u+tv\rangle \ge 0
+\\
+\uncover<3->{
+Q(t)
+&=
+\|u\|_2^2 + 2t\operatorname{Re}\langle u,v\rangle + t^2\|v\|_2^2}
+\end{align*}
+\uncover<4->{hat ihr Minimum bei}%
+\begin{align*}
+\uncover<5->{
+t&=
+-\operatorname{Re}\langle u,v\rangle/\|v\|_2^2}
+\intertext{\uncover<6->{mit Wert}}
+\uncover<7->{
+Q(t)
+&=
+\|u\|_2^2
+-2\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<7->{
+&\qquad + \operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<8->{
+0
+&\le
+\|u\|_2^2-\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<9->{
+\operatorname{Re}\langle u,v\rangle^2
+&\le
+\|u\|_2^2\cdot\|v\|_2^2}
+\\
+\uncover<10->{
+\operatorname{Re}\langle u,v\rangle
+&\le
+\|u\|_2\cdot\|v\|_2}
+\qedhere
+\end{align*}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex
new file mode 100644
index 0000000..49e656a
--- /dev/null
+++ b/vorlesungen/slides/2/chapter.tex
@@ -0,0 +1,17 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{2/norm.tex}
+\folie{2/skalarprodukt.tex}
+\folie{2/cauchyschwarz.tex}
+\folie{2/polarformel.tex}
+\folie{2/funktionenraum.tex}
+\folie{2/operatornorm.tex}
+\folie{2/linearformnormen.tex}
+\folie{2/funktionenalgebra.tex}
+\folie{2/frobeniusnorm.tex}
+\folie{2/frobeniusanwendung.tex}
+\folie{2/quotient.tex}
+\folie{2/quotientv.tex}
diff --git a/vorlesungen/slides/2/frobeniusanwendung.tex b/vorlesungen/slides/2/frobeniusanwendung.tex
new file mode 100644
index 0000000..277d600
--- /dev/null
+++ b/vorlesungen/slides/2/frobeniusanwendung.tex
@@ -0,0 +1,80 @@
+%
+% frobeniusanwendung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Anwendung der Frobenius-Norm}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung nach $X\in M_{m\times n}(\mathbb{R})$}
+Die Ableitung $Df=\partial f/\partial X$ der Funktion
+$f\colon M_{m\times n}(\mathbb{R})\to \mathbb{R}$ ist die Matrix
+mit Einträgen
+\begin{align*}
+\biggl(
+\frac{\partial f}{\partial X}
+\biggr)_{ij}
+&=
+\frac{\partial f}{\partial x_{ij}}
+=
+D_{ij}f
+\end{align*}
+\end{block}
+\uncover<2->{%
+\begin{block}{Richtungsableitung}
+\uncover<5->{Die Matrix $Df$ ist ein Gradient:}
+\begin{align*}
+\frac{\partial}{\partial t}f(X+tY)\bigg|_{t=0}
+&=\uncover<3->{
+\sum_{i,j}
+D_{ij} f(X) \cdot y_{ij}}
+\\
+&\uncover<4->{=
+\langle D_{ij}f(X), Y\rangle_F}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Quadratische Minimalprobleme}
+$A=A^t,B,X\in M_n(\mathbb{R})$, Minimum von
+\begin{align*}
+f(X)&=\langle X,AX\rangle_F + \langle B,X\rangle_F
+\intertext{\uncover<7->{Folgerungen:}}
+\uncover<8->{
+\langle X,AY\rangle_F&=\langle AX,Y\rangle_F
+}
+\\
+\uncover<9->{
+D\langle B,\mathstrut\cdot\mathstrut\rangle_F
+&=
+B
+}
+\\
+\uncover<10->{
+D_X\langle X, AY\rangle_F
+&=AY
+}
+\\
+\uncover<11->{
+D_Y\langle X, AY\rangle_F
+&=AX
+}
+\\
+\uncover<12->{
+Df &= 2AX + B
+}
+\intertext{\uncover<13->{Minimum:}}
+\uncover<14->{
+X&=-\frac12 A^{-1}B
+}
+\end{align*}
+\uncover<15->{(Kalman-Filter)}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/frobeniusnorm.tex b/vorlesungen/slides/2/frobeniusnorm.tex
new file mode 100644
index 0000000..461005a
--- /dev/null
+++ b/vorlesungen/slides/2/frobeniusnorm.tex
@@ -0,0 +1,96 @@
+%
+% frobeniusnorm.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Frobenius-Norm}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Skalarprodukt}
+$A,B\in M_{m\times n}(\mathbb{C})$
+\begin{align*}
+\langle A,B\rangle_F
+&\uncover<2->{=
+\sum_{i,j} \overline{a}_{ik}b_{ik}}
+\uncover<3->{=
+\operatorname{Spur} A^*B}
+\\
+\uncover<4->{
+\|A\|_F^2
+&=
+\langle A,A\rangle}
+\uncover<5->{=
+\sum_{i,k} |a_{ik}|^2}
+\end{align*}
+\uncover<6->{%
+$\Rightarrow M_{m\times n}(\mathbb{C})$ ist ein normierter Raum}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Singulärwertzerlegung}
+\vspace{-12pt}
+\begin{align*}
+\uncover<13->{
+A
+&=
+U\Sigma V^*}
+\\
+\uncover<14->{
+A^*A
+&=
+V\Sigma^*U^*U\Sigma V^*}
+\uncover<15->{=
+V\Sigma^*\Sigma V^*}
+\\
+\uncover<16->{%
+\operatorname{Spur}{A^*A}
+&=
+\operatorname{Spur}V\Sigma^*\Sigma V^*}
+\\
+\uncover<17->{%
+&=
+\operatorname{Spur}V^*V\Sigma^*\Sigma}
+\\
+\uncover<18->{%
+&=
+\operatorname{Spur}\Sigma^*\Sigma}
+\uncover<19->{=
+\sum_{i} |\sigma_i|^2}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<7->{%
+\begin{block}{Produkt}
+\vspace{-10pt}
+\begin{align*}
+\|AB\|_F
+\uncover<8->{=
+\sum_{i,j}
+\biggl|
+\sum_{k}
+a_{ik}b_{kj}
+\biggr|^2}
+&\uncover<9->{\le
+\sum_{i,j}
+\biggl(
+\sum_k  |a_{ik}|^2
+\biggr)
+\biggl(
+\sum_l  |b_{lj}|^2
+\biggr)}
+\\
+\uncover<10->{
+&=
+\sum_{i,k} |a_{ik}|^2
+\sum_{l,j} |b_{lj}|^2}
+\uncover<11->{=
+\|A\|_F\cdot \|B\|_F}
+\end{align*}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex
new file mode 100644
index 0000000..9116be4
--- /dev/null
+++ b/vorlesungen/slides/2/funktionenalgebra.tex
@@ -0,0 +1,88 @@
+%
+% funktionenalgebra.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Funktionenalgebra}
+\vspace{-17pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Algebra $C([0,1])$}
+Funktionenraum
+\[
+C([0,1])
+=
+\{f\colon[0,1]\to\mathbb{C}\;|\;\text{$f$ stetig}\}
+\]
+mit Supremum-Norm\uncover<2->{ und punktweisem Produkt
+\[
+(f\cdot g)(x)
+=
+f(x)\cdot g(x)
+\]}
+\end{block}
+\vspace{-8pt}
+\uncover<3->{%
+\begin{block}{Algebranorm}
+\vspace{-12pt}
+\begin{align*}
+\|f\cdot g\|_\infty
+&=
+\sup_{x\in[0,1]} |f(x)g(x)|
+\\
+\uncover<4->{
+&\le
+\sup_{x\in[0,1]}|f(x)|
+\sup_{y\in[0,1]}|g(y)|
+}
+\\
+\uncover<5->{
+&=
+\|f\|_\infty \cdot \|g\|_\infty
+}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Faltungs-Algebra $L^2([0,1])$}
+Funktionenraum
+\[
+L^2=\{f\colon \mathbb{R}\to\mathbb{C}\;|\;\text{$f$ $1$-periodisch}\}
+\]
+mit $L^2$-Skalarprodukt\uncover<7->{ und Faltungsprodukt
+\[
+f*g(x)
+=
+\int_0^1
+\underbrace{f(x-t)}_{(=\gamma_x\check{f})(t)} g(t)\,dx
+\]}
+\end{block}}
+\vspace{-21pt}
+\uncover<8->{%
+\begin{block}{Norm}
+\vspace{-12pt}
+\begin{align*}
+\|f*g\|_2^2
+&\uncover<9->{=\int_0^1 |
+\langle \gamma_x\check{f},g\rangle
+|^2\,dx}
+\\
+\uncover<10->{
+&\le
+\int_0^1
+\|\gamma_t\check{f}\|_2^2
+\|g\|_2^2
+\,dx}
+\\
+\uncover<11->{
+&=\|f\|_2^2\cdot \|g\|_2^2
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/funktionenraum.tex b/vorlesungen/slides/2/funktionenraum.tex
new file mode 100644
index 0000000..f7733cc
--- /dev/null
+++ b/vorlesungen/slides/2/funktionenraum.tex
@@ -0,0 +1,70 @@
+%
+% funktionenraum.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Funktionenraum}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Supremum-Norm}
+Vektorraum
+\[
+C([a,b])
+=
+\{f\colon[a,b]\to\mathbb{R}\;|\; \text{$f$ stetig}\}
+\]
+\only<2->{wird Banachraum }%
+mit der Norm
+\(\displaystyle
+\|f\|
+=
+\|f\|_{\infty}
+=
+\sup_{x\in[a,b]} |f(x)|
+\)
+\end{block}
+\uncover<3->{%
+\begin{block}{$L^1$-Norm}
+Vektorraum
+\[
+L^1([a,b])
+=
+\{f\colon[a,b]\;|\;\text{$f$ integrierbar}\}
+\]
+\only<4->{wird Banachraum }%
+mit der Norm
+\[
+\|f\|_1
+=
+\int_a^b |f(x)|\,dx
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{$L^2$-Norm}
+Vektorraum
+\[
+L^2([a,b])
+=
+\{f\colon[a,b]\to\mathbb{R}\;|\; \|f\|_2^2<\infty\}
+\]
+mit Skalarprodukt
+\begin{align*}
+\langle f,g\rangle
+&=
+\int_a^b \overline{f}(x)g(x)\,dx
+\\
+\|f\|_2^2
+&=
+\int_a^b |f(x)|^2\,dx
+\end{align*}
+\uncover<6->{ist ein Banachraum}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/images/Makefile b/vorlesungen/slides/2/images/Makefile
new file mode 100644
index 0000000..8bce5c9
--- /dev/null
+++ b/vorlesungen/slides/2/images/Makefile
@@ -0,0 +1,32 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	quotient1.jpg quotient2.jpg quotient1.pdf quotient2.pdf
+
+quotient1.png:	quotient1.pov quotient.inc
+	povray +A0.1 +W1920 +H1080 -Oquotient1.png quotient1.pov
+
+quotient1.jpg:	quotient1.png Makefile
+	convert -extract 1360x1040+330+20 quotient1.png \
+		-density 300 -units PixelsPerInch quotient1.jpg
+
+quotient2.png:	quotient2.pov quotient.inc
+	povray +A0.1 +W1920 +H1080 -Oquotient2.png quotient2.pov
+
+quotient2.jpg:	quotient2.png Makefile
+	convert -extract 1360x1040+330+20 quotient2.png \
+		-density 300 -units PixelsPerInch quotient2.jpg
+
+quotient:	quotient.ini quotient.inc quotient.pov
+	rm -rf quotient
+	mkdir quotient
+	povray +A0.1 -Oquotient/0.png -W1920 -H1080 quotient.ini
+
+quotient1.pdf:	quotient1.tex quotient1.jpg
+	pdflatex quotient1.tex
+
+quotient2.pdf:	quotient2.tex quotient2.jpg
+	pdflatex quotient2.tex
+
diff --git a/vorlesungen/slides/2/images/quotient.inc b/vorlesungen/slides/2/images/quotient.inc
new file mode 100644
index 0000000..3fa49d1
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient.inc
@@ -0,0 +1,186 @@
+//
+// quotient.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.035;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+	location <8, 15, -50>
+	look_at <0.4, 0.2, 0.4>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-4, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro kasten()
+	box { <-0.5,-0.5,-0.5>, <1.5,1,1.5> }
+#end
+
+
+arrow(<-0.6,0,0>, <1.6,0,0>, at, White)
+arrow(<0,0,-0.6>, <0,0,1.6>, at, White)
+arrow(<0,-0.6,0>, <0,1.2,0>, at, White)
+
+#declare U = <-1,3,-0.5>;
+#declare V1 = <1,0.2,0>;
+#declare V2 = <0,0.2,1>;
+
+#macro gerade(richtung, farbe)
+	intersection {
+		kasten()
+		cylinder { -U + richtung, U + richtung, at }
+		pigment {
+			color farbe
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#declare A = <0.8, -0.2, 0>;
+#declare B = <0.2, 0.8, 0>;
+
+#macro ebene(vektor1, vektor2)
+#declare n = vcross(vektor1,vektor2);
+
+
+intersection {
+	kasten()
+	plane { n, 0.005 }
+	plane { -n, 0.005 }
+	pigment {
+		color rgbf<0.8,0.8,1,0.7>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+intersection {
+	kasten()
+	union {
+		#declare Xstep = 0.45;
+		#declare X = -5 * Xstep;
+		#while (X < 5.5 * Xstep)
+			cylinder { X*vektor1 - 5*vektor2, X*vektor1 + 5*vektor2, at/2 }
+			#declare X = X + Xstep;
+		#end
+		#declare Ystep = 0.45;
+		#declare Y = -5 * Ystep;
+		#while (Y < 5.5 * Ystep)
+			cylinder { -5*vektor1 + Y*vektor2, 5*vektor1 + Y*vektor2, at/2 }
+			#declare Y = Y + Ystep;
+		#end
+	}
+	pigment {
+		color rgb<0.9,0.9,1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+
+gerade(O, Red)
+
+#declare gruen = rgb<0.2,0.4,0.2>;
+#declare blau = rgb<0,0.4,0.8>;
+#declare rot = rgb<1,0.4,0.0>;
+
+#macro repraesentanten(vektor1, vektor2) 
+
+#declare d1 = A.x*vektor1 + A.y*vektor2;
+#declare d2 = B.x*vektor1 + B.y*vektor2;
+
+arrow(0, d1 + d2, at, rot)
+gerade(d1 + d2, rot)
+
+gerade(d1, blau)
+arrow(O, d1, at, blau)
+cylinder { d1, d1 + d2, 0.6 * at
+	pigment {
+		color gruen
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+gerade(d2, gruen)
+arrow(O, d2, at, gruen)
+cylinder { d2, d1 + d2, 0.6 * at
+	pigment {
+		color blau
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+#end
+
+#macro vektorraum(s)
+#declare b1 = V1 + s * 0.03 * U;
+#declare b2 = V2 + s * 0.03 * U;
+
+ebene(b1, b2)
+repraesentanten(b1, b2)
+#end
+
diff --git a/vorlesungen/slides/2/images/quotient.ini b/vorlesungen/slides/2/images/quotient.ini
new file mode 100644
index 0000000..f62b21a
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient.ini
@@ -0,0 +1,7 @@
+Input_File_Name="quotient.pov"
+Initial_Frame=0
+Final_Frame=100
+Initial_Clock=-1
+Final_Clock=1
+Cyclic_Animation=off
+Pause_when_Done=off
diff --git a/vorlesungen/slides/2/images/quotient1.jpg b/vorlesungen/slides/2/images/quotient1.jpg
new file mode 100644
index 0000000..aeb713e
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.jpg
diff --git a/vorlesungen/slides/2/images/quotient1.pov b/vorlesungen/slides/2/images/quotient1.pov
new file mode 100644
index 0000000..60bab7f
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.pov
@@ -0,0 +1,8 @@
+//
+// quotient1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "quotient.inc"
+
+vektorraum(-1)
diff --git a/vorlesungen/slides/2/images/quotient1.tex b/vorlesungen/slides/2/images/quotient1.tex
new file mode 100644
index 0000000..30d82d2
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.tex
@@ -0,0 +1,29 @@
+%
+% quotient1.tex -- Vektorraumquotient
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.7,0,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\node at (0,0) {\includegraphics[width=8cm]{quotient1.jpg}};
+
+\node[color=blue] at (0.7,-1.3) {$v$};
+\node[color=darkgreen] at (-1.0,0.1) {$w$};
+\node[color=orange] at (2.5,0.1) {$v+w$};
+\node[color=darkred] at (-2.1,-0.9) {$0$};
+\node[color=darkred] at (-3.1,2.4) {$U$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/vorlesungen/slides/2/images/quotient2.jpg b/vorlesungen/slides/2/images/quotient2.jpg
new file mode 100644
index 0000000..345cf22
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.jpg
diff --git a/vorlesungen/slides/2/images/quotient2.pov b/vorlesungen/slides/2/images/quotient2.pov
new file mode 100644
index 0000000..771425d
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.pov
@@ -0,0 +1,8 @@
+//
+// quotient2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "quotient.inc"
+
+vektorraum(1)
diff --git a/vorlesungen/slides/2/images/quotient2.tex b/vorlesungen/slides/2/images/quotient2.tex
new file mode 100644
index 0000000..607fd03
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.tex
@@ -0,0 +1,29 @@
+%
+% quotient2.tex -- Vektorraumquotient
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.7,0,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\node at (0,0) {\includegraphics[width=8cm]{quotient2.jpg}};
+
+\node[color=blue] at (0.57,-0.94) {$v$};
+\node[color=darkgreen] at (-1.15,0.65) {$w$};
+\node[color=orange] at (2.15,1) {$v+w$};
+\node[color=darkred] at (-2.1,-0.9) {$0$};
+\node[color=darkred] at (-3.1,2.4) {$U$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/vorlesungen/slides/2/linearformnormen.tex b/vorlesungen/slides/2/linearformnormen.tex
new file mode 100644
index 0000000..8993f66
--- /dev/null
+++ b/vorlesungen/slides/2/linearformnormen.tex
@@ -0,0 +1,76 @@
+%
+% linearformnormen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Linearformen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Linearformen $\varphi\colon L^1\to\mathbb{R}$}
+Beispiel: $g\in C([a,b])$
+\[
+\varphi(f)
+=
+\int_a^b g(x)f(x)\,dx
+\]
+\uncover<2->{%
+erfüllt
+\begin{align*}
+|\varphi(f)|
+&=
+\biggl|\int_a^b g(x)f(x)\,dx\biggr|
+\\
+\uncover<3->{
+&\le \|g\|_\infty\cdot \|f\|_1
+}
+\end{align*}}
+\uncover<4->{%
+und hat daher die Operatornorm
+\[
+\|\varphi\|_{C([a,b])^*}
+=
+\|g\|_\infty
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Linearformen $\varphi\colon L^2\to\mathbb{R}$}
+\uncover<5->{%
+Darstellungssatz von Riesz: $\exists g\in L^2$
+\[
+\varphi(f) = \langle g,f\rangle
+\]}
+\uncover<6->{%
+erfüllt Cauchy-Schwarz}
+\begin{align*}
+\uncover<7->{
+|\varphi(f)|
+&=
+|\langle g,f\rangle|}
+\\
+\uncover<8->{
+&\le
+\|g\|_2 \cdot \|f\|_2
+}
+\end{align*}
+\uncover<9->{%
+und hat daher die Operatornorm
+\[
+\|\varphi\|_{L^2([a,b])^*}
+= \|g\|_2
+\]}
+\end{block}
+\end{column}
+\end{columns}
+
+\vspace{8pt}
+{\usebeamercolor[fg]{title}
+\uncover<10->{%
+$\Rightarrow$
+Operatornorm hängt von den Vektorraumnormen ab}
+}
+\end{frame}
diff --git a/vorlesungen/slides/2/norm.tex b/vorlesungen/slides/2/norm.tex
new file mode 100644
index 0000000..35d2513
--- /dev/null
+++ b/vorlesungen/slides/2/norm.tex
@@ -0,0 +1,58 @@
+%
+% norm.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Norm}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Wozu}
+Ziel: Konvergenz von Folgen, Grenzwert in einem Vektorraum
+\end{block}
+\uncover<7->{%
+\begin{block}{Cauchy-Folge}
+Eine Folge $(x_n)_{n\in\mathbb{N}}$ von Vektoren in $V$ heisst
+{\em Cauchy-Folge},
+wenn es für alle $\varepsilon >0$ ein $N$ gibt mit
+\[
+\|x_n-x_m\| < \varepsilon\; \forall n,m>N
+\]
+\end{block}}
+\vspace{-8pt}
+\uncover<8->{%
+\begin{block}{Grenzwert}
+$x\in V$ heisst Grenzwert der Folge $x_n$, wenn es für alle $\varepsilon>0$
+ein $N$ gibt mit
+\[
+\| x-x_n\| < \varepsilon \;\forall n>N
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Definition}
+$V$ ein $\mathbb{R}$-Vektorraum.
+Eine Funktion
+\[
+\|\cdot\| \colon V \to \mathbb{R}_{\ge 0} : v \mapsto \|v\|
+\]
+heisst eine {\em Norm}, wenn
+\begin{itemize}
+\item<3-> $\| v \|>0$ für $v\ne 0$
+\item<4-> $\|\lambda v\| = |\lambda|\cdot\|v\|$
+\item<5-> $\| u + v \| \le \|u\| + \|v\|$ (Dreiecksungleichung)
+\end{itemize}
+\uncover<6->{%
+Ein Vektorraum mit einer Norm heisst {\em normierter Raum}}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Banach-Raum}
+Normierter Raum, in dem jede Cauchy-Folge einen Grenwzert hat
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/operatornorm.tex b/vorlesungen/slides/2/operatornorm.tex
new file mode 100644
index 0000000..d20461a
--- /dev/null
+++ b/vorlesungen/slides/2/operatornorm.tex
@@ -0,0 +1,59 @@
+%
+% operatorname.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Operatornorm}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Lineare Operatoren}
+$A\colon U\to V$ lineare Abbildung mit $U$, $V$ normiert
+\end{block}}
+\uncover<3->{%
+\begin{block}{Operatornorm}
+eines linearen Operators $A$:
+\[
+\|A\| 
+=
+\sup_{\|x\|_U\le 1} \|Ax\|_V
+\]
+\uncover<4->{$\Rightarrow \|Ax\| \le \| A \|\cdot \|x\|$}
+\end{block}}
+\uncover<5->{%
+\begin{block}{Stetigkeit}
+Wenn $\|A\|<\infty$, dann ist $A$ stetig, d.~h.
+\[
+\lim_{n\to\infty} Ax_n
+=
+A\lim_{n\to\infty} x_n
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Algebranorm}
+$A$ ein normierter Raum, der auch ein Algebra ist.
+Dann heisst $A$ eine normierte Algebra, wenn
+\[
+\| ab\| \le \| a\|\cdot \|b\|
+\quad\forall a,b\in A
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Operatoralgebra}
+$U$ ein normierter Raum, dann ist die Algebra der linearen Operatoren
+$A\colon U\to U$ mit der Operatornorm eine normierte Algebra
+\end{block}}
+\uncover<8->{%
+\begin{block}{Banach-Algebra}
+Ein Banach-Raum, der auch eine normierte Algebra ist
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/polarformel.tex b/vorlesungen/slides/2/polarformel.tex
new file mode 100644
index 0000000..ebdbf81
--- /dev/null
+++ b/vorlesungen/slides/2/polarformel.tex
@@ -0,0 +1,113 @@
+%
+% polarformel.tex
+%
+% (c) 2021 Prod Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkcolor}{rgb}{0,0.6,0}
+\def\yone{-2.1}
+\def\ytwo{-3.55}
+\def\ythree{-5.0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Polarformel}
+\vspace{-5pt}
+\begin{block}{Aufgabe}
+$\langle x,y\rangle$ aus Werten von $\|\cdot\|_2$ rekonstruieren:
+
+\end{block}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\node at (0,0) {$
+\begin{aligned}
+\uncover<2->{
+\|x+ty\|_2^2
+&=
+\|x\|_2^2
++t\langle x,y\rangle
++\overline{t}\langle y,x\rangle
++ \|y\|_2^2}
+\\
+\uncover<3->{
+&=
+\|x\|_2^2
++t\langle x,y\rangle
++\overline{t\langle x,y\rangle}
++ \|y\|_2^2}
+\\
+\uncover<4->{
+&=
+\|x\|_2^2
++2\operatorname{Re}(t\langle x,y\rangle)
++ \|y\|_2^2}
+\end{aligned}$};
+
+\uncover<5->{
+	\draw[->] (-1,-0.9) -- (-3.3,{\yone+0.25});
+	\node at (-3.5,\yone) {$
+	\|x\pm y\|_2^2
+	=
+	\|x\|_2^2
+	\pm2\operatorname{Re}\langle x,y\rangle
+	+
+	\|y\|_2^2
+	$};
+}
+
+\uncover<8->{
+	\draw[->] (1,-0.9) -- (3.3,{\yone+0.25});
+	\node at (3.5,\yone) {$
+	\|x\pm iy\|_2^2
+	=
+	\|x\|_2^2
+	\pm2i\operatorname{Im}\langle x,y\rangle
+	+
+	\|y\|_2^2
+	$};
+}
+
+\uncover<6->{
+	\draw[->] (-3.5,{\yone-0.2}) -- (-3.5,{\ytwo+0.2});
+	\node at (-3.5,\ytwo) {$\operatorname{Re}\langle x,y\rangle
+	=
+	\frac12\bigl(
+	\|x+y\|_2^2-\|x-y\|_2^2
+	\bigr)
+	$};
+}
+
+\uncover<9->{
+	\draw[->] (3.5,{\yone-0.2}) -- (3.5,{\ytwo+0.2});
+	\node at (3.5,\ytwo) {$
+	\operatorname{Im}\langle x,y\rangle
+	=
+	\frac1{2i}\bigl(
+	\|x+iy\|_2^2-\|x-iy\|_2^2
+	\bigr)
+	$};
+}
+
+\uncover<7->{
+	\draw[->] (-3.3,{\ytwo-0.25}) -- (-1.5,{\ythree+0.25});
+	\node at (0,\ythree) {$
+	\langle x,y\rangle
+	=
+	\frac12\bigl(
+	\|x+y\|_2^2-\|x-y\|_2^2
+	\uncover<10->{
+		+
+		\|x+iy\|_2^2-\|x-iy\|_2^2
+	}
+	\bigr)$};
+}
+
+\uncover<10->{
+	\draw[->] (3.3,{\ytwo-0.25}) -- (1.5,{\ythree+0.25});
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/quotient.tex b/vorlesungen/slides/2/quotient.tex
new file mode 100644
index 0000000..24b0523
--- /dev/null
+++ b/vorlesungen/slides/2/quotient.tex
@@ -0,0 +1,110 @@
+%
+% quotient.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.7,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\s{0.3}
+\def\punkt#1#2{({#1-3*#2},{8*#2})}
+\def\gerade#1{
+\draw[darkgreen,line width=1.4pt]
+	\punkt{#1}{1}
+	--
+	\punkt{#1}{-1};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quotientenraum}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Einen Unterraum ``ignorieren''}
+{\usebeamercolor[fg]{title}Gegeben:} $U\subset V$ ein Unterraum
+\\
+{\usebeamercolor[fg]{title}Gesucht:} Eine Projektion auf einen Vektorraum,
+in dem die Richtungen in $U$ zu $0$ gemacht werden
+\end{block}
+\uncover<2->{%
+\begin{block}{Projektion}
+In $V$ Klassen bilden:
+\[
+\pi
+\colon
+v\mapsto
+\llbracket v\rrbracket
+=
+v+U
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<3->{%
+\begin{block}{Quotientenraum}
+\vspace{-12pt}
+\begin{align*}
+V/U
+&=
+\{ v+U\;|\; v\in V \}
+\\
+\uncover<4->{\pi(\lambda v)&=\lambda v+U= \lambda \pi(v)}
+\\
+\uncover<5->{\pi(v+w)
+&=
+v+w+U}
+\ifthenelse{\boolean{presentation}}{
+\only<6>{=
+v+U+w+U}}{}
+\uncover<7->{=
+\pi(v) + \pi(w)}
+\phantom{blubb}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (U) at (-3,8);
+\def\t{0.03}
+\begin{scope}
+\clip (-2,-2) rectangle (4,4.8);
+\draw[color=darkred,line width=2pt] (-3,8) -- (1.5,-4);
+\node[color=darkred] at (-1.45,4.6) {$U$};
+\node[color=darkred] at (-0.05,-0.05) [above left] {$0$};
+
+\gerade{2.5}
+
+\ifthenelse{\boolean{presentation}}{
+	\foreach \n in {8,...,25}{
+		\pgfmathparse{(\n-12)*0.04}
+		\xdef\s{\pgfmathresult}
+		\only<\n>{
+			\draw[color=blue,line width=1.2pt]
+				\punkt{-5}{-2*\s} -- \punkt{5}{2*\s};
+			\draw[->,color=blue,line width=2pt]
+				(0,0) -- \punkt{2.5}{\s};
+			\node[color=blue] at \punkt{2.5}{\s}
+				[above right] {$v'$};
+		}
+	}
+}{
+	\xdef\s{0.35}
+	\draw[color=blue,line width=1.2pt]
+		\punkt{-5}{-2*\s} -- \punkt{5}{2*\s};
+	\draw[->,color=blue,line width=2pt] (0,0) -- \punkt{2.5}{\s};
+	\node[color=blue] at \punkt{2.5}{\s} [above right] {$v'$};
+}
+
+\draw[->,color=darkgreen,line width=1.4pt] (0,0) -- \punkt{2.5}{0.1};
+
+\node[color=darkgreen] at \punkt{2.5}{0.1} [above right] {$v$};
+
+\end{scope}
+\draw[->] (-2,0) -- (4,0) coordinate[label={$x$}];
+\draw[->] (0,-2) -- (0,5) coordinate[label={right:$x$}];
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/quotientv.tex b/vorlesungen/slides/2/quotientv.tex
new file mode 100644
index 0000000..dc01f21
--- /dev/null
+++ b/vorlesungen/slides/2/quotientv.tex
@@ -0,0 +1,62 @@
+%
+% quotientv.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.7,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Quotient}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.33\textwidth}
+\begin{block}{Repräsentanten}
+Jeder Unterraum $W\subset V$ mit 
+$W\cap U = \{0\}$
+kann als Menge von Repräsentanten
+für
+\begin{align*}
+V/U
+&=
+\{v+U\;|\;v \in V\}
+\\
+&\simeq W
+\end{align*}
+dienen.
+\end{block}
+\uncover<3->{%
+\begin{block}{Orthogonalraum}
+Mit Skalarprodukt ist
+$W=U^\perp$ eine bevorzugte Wahl
+\end{block}}
+\end{column}
+\begin{column}{0.66\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\only<1>{
+	\node at (0,0)
+	{\includegraphics[width=8.5cm]{../slides/2/images/quotient1.jpg}};
+	\node[color=darkgreen] at (-0.5,0.3) {$v$};
+	\node[color=blue] at (0.7,-1.4) {$w$};
+	\node[color=orange] at (2.7,0.1) {$v+w$};
+	\fill[color=white,opacity=0.5] (3.7,1.0) circle[radius=0.25];
+	\node at (3.7,1.0) {$W$};
+}
+\only<2->{
+	\node at (0,0)
+	{\includegraphics[width=8.5cm]{../slides/2/images/quotient2.jpg}};
+	\node[color=darkgreen] at (-0.75,0.95) {$v$};
+	\node[color=blue] at (0.6,-1.05) {$w$};
+	\node[color=orange] at (2.36,1.05) {$v+w$};
+	\fill[color=white,opacity=0.5] (3.7,2.9) circle[radius=0.25];
+	\node at (3.7,2.9) {$W$};
+}
+\node[color=darkred] at (-3.3,2.6) {$U$};
+\node[color=darkred] at (-2.25,-1.0) {$0$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex
new file mode 100644
index 0000000..99d8a73
--- /dev/null
+++ b/vorlesungen/slides/2/skalarprodukt.tex
@@ -0,0 +1,96 @@
+%
+% skalarprodukt.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Skalarprodukt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Positiv definite, symmetrische Bilinearform}
+$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$
+\begin{itemize}
+\item<2->
+Bilinear:
+\begin{align*}
+\langle \alpha u+\beta v,w\rangle
+&=
+\alpha\langle u,w\rangle
++
+\beta\langle v,w\rangle
+\\
+\langle u,\alpha v+\beta w\rangle
+&=
+\alpha\langle u,v\rangle
++
+\beta\langle u,w\rangle
+\end{align*}
+\item<3->
+Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$
+\item<4->
+$\langle x,x\rangle >0 \quad\forall x\ne 0$
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Positive definite, hermitesche Sesquilinearform}
+$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$
+\begin{itemize}
+\item<6->
+Sesquilinear:
+\begin{align*}
+\langle \alpha u+\beta v,w\rangle
+&=
+\overline{\alpha}\langle u,w\rangle
++
+\overline{\beta}\langle v,w\rangle
+\\
+\langle u,\alpha v+\beta w\rangle
+&=
+\alpha\langle u,v\rangle
++
+\beta\langle u,w\rangle
+\end{align*}
+\item<7->
+Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$
+\item<8->
+$\langle x,x\rangle >0 \quad\forall x\ne 0$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.28\textwidth}
+\uncover<9->{%
+\begin{block}{$2$-Norm}
+$\|v\|_2^2 = \langle v,v\rangle$
+\\
+$\|v\|_2 = \sqrt{\langle v,v\rangle}$
+\end{block}}
+\end{column}
+\begin{column}{0.78\textwidth}
+\uncover<10->{%
+\begin{itemize}
+\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$
+\item<12-> $\| \lambda v \|_2
+=
+\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut}
+=
+\sqrt{\overline{\lambda}\lambda\langle v,v\rangle}
+=
+|\lambda|\cdot \|v\|_2$
+\item<13->
+\raisebox{-8pt}{
+$\begin{aligned}
+\|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2
+\\
+(\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2
+\end{aligned}$}
+\end{itemize}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc
new file mode 100644
index 0000000..442bd15
--- /dev/null
+++ b/vorlesungen/slides/3/Makefile.inc
@@ -0,0 +1,37 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter3 =								\
+	../slides/3/motivation.tex					\
+	../slides/3/inverse.tex						\
+	../slides/3/polynome.tex					\
+	../slides/3/division.tex					\
+	../slides/3/division2.tex					\
+	../slides/3/ringstruktur.tex					\
+	../slides/3/teilbarkeit.tex					\
+	../slides/3/ideal.tex						\
+	../slides/3/nichthauptideal.tex					\
+	../slides/3/nichthauptideal2.tex				\
+	../slides/3/idealverband.tex					\
+	../slides/3/maximalideal.tex					\
+	../slides/3/quotientenring.tex					\
+	../slides/3/faktorisierung.tex					\
+	../slides/3/faktorzerlegung.tex					\
+	../slides/3/einsetzen.tex					\
+	../slides/3/maximalergrad.tex					\
+	../slides/3/minimalbeispiel.tex					\
+	../slides/3/fibonacci.tex					\
+	../slides/3/minimalpolynom.tex					\
+	../slides/3/drehmatrix.tex					\
+	../slides/3/drehfaktorisierung.tex				\
+	../slides/3/operatoren.tex					\
+	../slides/3/adjunktion.tex					\
+	../slides/3/adjalgebra.tex					\
+	../slides/3/wurzel2.tex						\
+	../slides/3/phi.tex						\
+	../slides/3/multiplikation.tex					\
+	../slides/3/chapter.tex
+
diff --git a/vorlesungen/slides/3/adjalgebra.tex b/vorlesungen/slides/3/adjalgebra.tex
new file mode 100644
index 0000000..e65b621
--- /dev/null
+++ b/vorlesungen/slides/3/adjalgebra.tex
@@ -0,0 +1,43 @@
+%
+% adjalgebra.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Adjunktion einer Nullstelle, abstrakt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom.
+
+\uncover<2->{%
+\begin{block}{Existenz}
+Es gibt ein ``Objekt'' $\alpha$ mit
+\(
+m(\alpha) = 0
+\)
+\end{block}}
+
+\uncover<3->{%
+\begin{block}{Körpererweiterung}
+Der kleinste Körper, der $\Bbbk$ und $\alpha$ enthält ist
+\[
+\Bbbk(\alpha)
+=
+\left
+\{ p(\alpha)
+\;\left|\;
+\begin{minipage}{8cm}\raggedright
+$p\in\Bbbk[X]$ ein Polynom vom Grad
+$\deg p<\deg m$
+\end{minipage}
+\right.
+\right\}
+\]
+\uncover<4->{Das Polynom $m$ definiert, wie mit $\alpha$ gerechnet werden
+muss:
+\[
+\alpha^n = -m_0-m_1\alpha-m_2\alpha^2 - \dots - m_{n-1}\alpha^{n-1}
+\]}
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/adjunktion.tex b/vorlesungen/slides/3/adjunktion.tex
new file mode 100644
index 0000000..a974a76
--- /dev/null
+++ b/vorlesungen/slides/3/adjunktion.tex
@@ -0,0 +1,35 @@
+%
+% adjunktion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Adjunktion einer Nullstelle von $m(X)$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom.
+\uncover<2->{%
+\[
+X^n = -m_{n-1}X^{n-1} - \dots - m_1X - m_0
+\]
+}%
+\uncover<3->{%
+Nullstelle $W$ als Operator betrachten:
+\[
+W = \begin{pmatrix}
+     0&     0&     0&\dots &     0&   -m_0\\
+     1&     0&     0&\dots &     0&   -m_1\\
+     0&     1&     0&\dots &     0&   -m_2\\
+     0&     0&     1&\dots &     0&   -m_3\\
+\vdots&\vdots&\vdots&\ddots&\vdots& \vdots\\
+     0&     0&     0&\dots &     1&-m_{n-1}
+\end{pmatrix}
+\]}
+\uncover<4->{%
+Man kann nachrechnen, dass immer $m(W)=0$.
+}
+\medskip
+
+\uncover<5->{$\Rightarrow \Bbbk(W) = \{p(W)\;|\;p\in\Bbbk[X], \deg p<\deg m\}$
+ist ein Körper, in dem $m(X)$ faktorisiert werden kann $m(X) = (X-W)q(X)$.}
+\end{frame}
diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex
new file mode 100644
index 0000000..3fbc3fd
--- /dev/null
+++ b/vorlesungen/slides/3/chapter.tex
@@ -0,0 +1,33 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{3/motivation.tex}
+\folie{3/inverse.tex}
+\folie{3/polynome.tex}
+\folie{3/division.tex}
+\folie{3/division2.tex}
+\folie{3/ringstruktur.tex}
+\folie{3/teilbarkeit.tex}
+\folie{3/ideal.tex}
+\folie{3/nichthauptideal.tex}
+\folie{3/nichthauptideal2.tex}
+\folie{3/maximalideal.tex}
+\folie{3/idealverband.tex}
+\folie{3/quotientenring.tex}
+\folie{3/faktorisierung.tex}
+\folie{3/faktorzerlegung.tex}
+\folie{3/einsetzen.tex}
+\folie{3/maximalergrad.tex}
+\folie{3/minimalbeispiel.tex}
+\folie{3/fibonacci.tex}
+\folie{3/minimalpolynom.tex}
+\folie{3/drehmatrix.tex}
+\folie{3/drehfaktorisierung.tex}
+\folie{3/operatoren.tex}
+\folie{3/adjunktion.tex}
+\folie{3/adjalgebra.tex}
+\folie{3/wurzel2.tex}
+\folie{3/phi.tex}
+\folie{3/multiplikation.tex}
diff --git a/vorlesungen/slides/3/division.tex b/vorlesungen/slides/3/division.tex
new file mode 100644
index 0000000..94df27b
--- /dev/null
+++ b/vorlesungen/slides/3/division.tex
@@ -0,0 +1,32 @@
+%
+% division.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynomdivision}
+\begin{block}{Aufgabe}
+Finde Quotient und Rest für
+$a= X^4- X^3-7X^2+ X+6\in\mathbb{Z}[X]$
+und
+$b= X^2+X+1\in\mathbb{Z}[X]$
+\end{block}
+\uncover<2->{%
+\begin{block}{Lösung}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcrcrcrcr}
+\llap{$($}X^4&-& X^3&-&7X^2&+& X&+&6\rlap{$)$}&\;\mathstrut:\mathstrut&(X^2&+&X&+&1)&=&\uncover<3->{X^2}&\uncover<7->{-&2X}&\uncover<11->{-&6}=q\\
+\uncover<4->{\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}}& &  & & & &   & & & & & &   & &  & & \\ 
+   &\uncover<5->{-&2X^3&-&8X^2}&\uncover<6->{+& X}& & & &   & & & & & &   & &  & & \\
+   &\uncover<8->{\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}}& & & &   & & & & & &   & &  & & \\ 
+   & &    &\uncover<9->{-&6X^2&+&3X}&\uncover<10->{+&6}& &   & & & & & &   & &  & & \\
+   & &    &\uncover<12->{\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}}& &   & & & & & &   & &  & & \\ 
+   & &    & &    & &\uncover<13->{9X&+&12\rlap{$\mathstrut=r$}}& &   & & & & & &   & &  & &
+\end{array}
+\]
+\uncover<14->{
+Funktioniert weil $b$ normiert ist!
+}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/division2.tex b/vorlesungen/slides/3/division2.tex
new file mode 100644
index 0000000..0602598
--- /dev/null
+++ b/vorlesungen/slides/3/division2.tex
@@ -0,0 +1,34 @@
+%
+% division2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Division in $\Bbbk[X]$}
+\vspace{-5pt}
+\begin{block}{Aufgabe}
+Finde Quotienten und Rest der Polynome
+$a(X) = X^4-X^3-7X^2+X+6$
+und
+$b(X) = 2X^2+X+1$
+\end{block}
+\uncover<2->{%
+\begin{block}{Lösung}
+\vspace{-15pt}
+\[
+\arraycolsep=1.4pt
+\renewcommand{\arraystretch}{1.2}
+\begin{array}{rcrcrcrcrcrcrcrcrcrcr}
+\llap{$($}X^4&-&       X^3&-&         7X^2&+&          X&+&           6\rlap{$)$}&\mathstrut\;:\mathstrut&(2X^2&+&X&+&1)&=&\uncover<3->{\frac12X^2}&\uncover<7->{-&\frac34X}&\uncover<11->{-\frac{27}{8}} = q\\
+\uncover<4->{\llap{$-($}X^4&+&\frac12X^3&+&   \frac12X^2\rlap{$)$}}& &           & &            & &    & & & & & &          & &        &             \\
+   &\uncover<5->{-&\frac32X^3&-&\frac{15}2X^2}&\uncover<6->{+&          X}& &            & &    & & & & & &          & &        &             \\
+   &\uncover<8->{\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}}& &            & &    & & & & & &          & &        &             \\
+   & &          &\uncover<9->{-&\frac{27}4X^2&+&\frac{ 7}4X}&\uncover<10->{+&           6}& &    & & & & & &          & &        &             \\
+   & &          &\uncover<12->{\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}}& &    & & & & & &          & &        &             \\
+   & &          & &             & &\uncover<13->{\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}}& &    & & & & & &          & &        &             \\
+\end{array}
+\]
+Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/drehfaktorisierung.tex b/vorlesungen/slides/3/drehfaktorisierung.tex
new file mode 100644
index 0000000..64418d5
--- /dev/null
+++ b/vorlesungen/slides/3/drehfaktorisierung.tex
@@ -0,0 +1,75 @@
+%
+% drehfaktorisierung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{4pt}
+\setlength{\belowdisplayskip}{4pt}
+\frametitle{Faktorisierung von $X^2+X+1$}
+\vspace{-3pt}
+$X^2+X+1$ kann faktorisiert werden, wenn man $i\sqrt{3}$
+hinzufügt:
+\uncover<2->{%
+\[
+\biggl(X+\frac12+\frac{i\sqrt{3}}2\biggr)
+\biggl(X+\frac12-\frac{i\sqrt{3}}2\biggr)
+=
+X^2+X+\frac14
++
+\frac34
+\uncover<3->{=
+X^2+X+1}
+\]}
+\vspace{-10pt}
+\uncover<4->{%
+\begin{block}{Was ist $i\sqrt{3}$?}
+Matrix mit Minimalpolynom $X^2+3$:
+\[
+W=\begin{pmatrix}0&-3\\1&0\end{pmatrix}
+\uncover<5->{%
+\qquad\Rightarrow\qquad
+W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I}
+\uncover<6->{%
+\qquad\Rightarrow\qquad
+W^2+3I=0}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Faktorisierung von $X^2+X+1$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<8->{B_\pm
+&=
+-\frac12I\pm\frac12W}
+&
+&\uncover<10->{\Rightarrow
+&
+(X+B_+)(X+B_-)}
+&\uncover<11->{=
+(X+\frac12I+\frac12W)
+(X+\frac12I-\frac12W)}
+\\
+&\uncover<9->{=
+\smash{
+{\textstyle\begin{pmatrix}-\frac12&-\frac32\\\frac12&-\frac12\end{pmatrix}}
+}}
+&
+&
+&
+&\uncover<12->{=
+X^2+X + \frac14I - \frac14W^2}
+\\
+&
+&
+&%\Rightarrow
+&
+&\uncover<13->{=
+X^2+X + \frac14I + \frac34I}
+\uncover<14->{=
+X^2+X+I}
+\end{align*}
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/drehmatrix.tex b/vorlesungen/slides/3/drehmatrix.tex
new file mode 100644
index 0000000..9e5eb65
--- /dev/null
+++ b/vorlesungen/slides/3/drehmatrix.tex
@@ -0,0 +1,66 @@
+%
+% drehmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Analyse einer Drehung um $120^\circ$}
+$D$ eine Drehung des $\mathbb{R}^3$ um $120^\circ$
+\begin{enumerate}
+\item<2->
+Drehwinkel = $120^\circ\quad\Rightarrow\quad D^3 = I$
+\uncover<3->{
+$\quad\Rightarrow\quad \chi_D(X)=X^3-1$
+}
+\item<4->
+$m_D(X)=X^3-1$
+\item<5->
+$m_D$ ist nicht irreduzibel, weil $m_D(1)=0$:
+$
+m_D(X) = (X-1)(X^2+X+1)
+$
+\item<6->
+Welche Matrix hat $X^2+X+1$ als Minimalpolynom?
+\uncover<7->{%
+\[
+\arraycolsep=1.4pt
+W
+=
+\biggl(\begin{array}{cc}
+-\frac12          & -\frac{\sqrt{3}}2 \\
+ \frac{\sqrt{3}}2 & -\frac12
+\end{array}\biggr)
+\quad\Rightarrow\quad
+W^2+W+I
+=
+\biggl(\begin{array}{cc}
+-\frac12          & -\frac{\sqrt{3}}2 \\
+ \frac{\sqrt{3}}2 & -\frac12
+\end{array}\biggr)
++
+\biggl(\begin{array}{cc}
+-\frac12          & \frac{\sqrt{3}}2 \\
+ -\frac{\sqrt{3}}2 & -\frac12
+\end{array}\biggr)
++
+\biggl(\begin{array}{cc}
+1&0\\0&1
+\end{array}\biggr)
+=0
+\]}
+\item<8-> In einer geeigneten Basis hat $D$ die Form
+\[
+D=\begin{pmatrix}
+1&0&0\\
+0&-\frac12 & -\frac{\sqrt{3}}2 \\
+0&\frac{\sqrt{3}}2 & -\frac12
+\end{pmatrix}
+\uncover<9->{=
+\begin{pmatrix}
+1&0&0\\
+0&\cos 120^\circ & -\sin 120^\circ\\
+0&\sin 120^\circ & \cos 120^\circ
+\end{pmatrix}}
+\]
+\end{enumerate}
+\end{frame}
diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex
new file mode 100644
index 0000000..7f54abb
--- /dev/null
+++ b/vorlesungen/slides/3/einsetzen.tex
@@ -0,0 +1,54 @@
+%
+% einsetzen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Matrix in ein Polynom einsetzen}
+\vspace{-10pt}
+\[
+\begin{array}{rcrcrcrcrcrcr}
+p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\
+\uncover<2->{\bigg\downarrow\hspace*{4pt}}  & &
+\uncover<3->{\bigg\downarrow\hspace*{4pt}}  & &
+\uncover<4->{\bigg\downarrow\hspace*{10pt}} & &     & &
+\uncover<5->{\bigg\downarrow\hspace*{4pt}}  & &
+\uncover<6->{\bigg\downarrow\hspace*{2pt}}  & &
+\uncover<7->{\bigg\downarrow\hspace*{0pt}}  \\
+\uncover<2->{p(A)}&\uncover<3->{=&a_nA^n}&\uncover<4->{+&a_{n-1}A^{n-1}}&\uncover<5->{+&\dots&+&a_2A^2}&\uncover<6->{+&a_1A}&\uncover<7->{+&a_0         I}
+\end{array}
+\]
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Nilpotente Matrizen}
+$p(X) = (X-a)^n$
+\[
+\uncover<9->{p(A) = 0}
+\uncover<10->{
+\quad\Rightarrow\quad
+\text{$A-aI$ ist nilpotent}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Eigenwerte}
+$p(X) = (X-\lambda_1)(X-\lambda_2)$,\\
+$A$ eine $2\times 2$-Matrix
+\[
+\uncover<12->{p(A)=0}
+\uncover<13->{\quad\Rightarrow\quad
+\left\{
+\begin{aligned}
+&\text{$A-\lambda_1I$ ist singulär}\\
+&\text{$A-\lambda_2I$ ist singulär}
+\end{aligned}
+\right.}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/faktorisierung.tex b/vorlesungen/slides/3/faktorisierung.tex
new file mode 100644
index 0000000..b4ea1d5
--- /dev/null
+++ b/vorlesungen/slides/3/faktorisierung.tex
@@ -0,0 +1,47 @@
+%
+% faktorisierung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Faktorisierung}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Primzahlen\strut}
+Eine Zahl $p\in\mathbb{Z}$, $p>1$ heisst Primzahl, wenn sie nicht als Produkt
+$p=ab$ mit $a,b\in\mathbb{Z},a>1, b>1$ geschrieben werden kann.
+\begin{align*}
+\uncover<2->{p&=7}
+\\
+\uncover<3->{2021 &= 43 \cdot 47}
+\\
+\uncover<4->{2048 &= 2^{11}}
+\\
+\uncover<5->{4095667&=2021\cdot 2027}
+\\
+\uncover<6->{p&=43, 47, 1291, 2017, 2027}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Irreduzible Polynome in $\mathbb{Q}[X]$}
+Ein Polynome $p\in\mathbb{Q}[X]$, $\deg p>0$ wenn es nicht als Produkt
+$p=ab$ mit $a,b\in\mathbb{Q}[X]$, $\deg a>0$, $\deg b>0$ geschrieben
+werden kann.
+\begin{align*}
+\uncover<8->{p&=X-9}
+\\
+\uncover<9->{X^2-1&= (X+1)(X-1)}
+\\
+\uncover<10->{X^2-2&\text{\; irreduzibel}}
+\\
+\uncover<11->{X^2-2&=(X-\sqrt{2})(X+\sqrt{2})}
+\end{align*}
+\uncover<12->{%
+aber: $X\pm\sqrt{2}\not\in\mathbb{Q}[X]$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/faktorzerlegung.tex b/vorlesungen/slides/3/faktorzerlegung.tex
new file mode 100644
index 0000000..eb44cf3
--- /dev/null
+++ b/vorlesungen/slides/3/faktorzerlegung.tex
@@ -0,0 +1,62 @@
+%
+% faktorzerlegung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Faktorzerlegung}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{in $\mathbb{Z}$}
+Jede Zahl kann eindeutig in Primfaktoren zerlegt werden:
+\[
+z = p_1^{n_1}\cdot p_2^{n_2} \cdot\dots\cdot p_k^{n_k}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{in $\mathbb{Q}[X]$}
+Jedes Polynom $p\in\mathbb{Q}[X]$
+kann eindeutig faktorisiert werden in irreduzible, normierte Polynome
+\[
+p
+=
+a_n
+p_1^{n_1}
+\cdot
+p_2^{n_2}
+\cdot
+\dots
+\cdot
+p_k^{n_k}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Polynomfaktorisierung hängt vom Koeffizientenring ab}
+Ist $X^2-2$ irreduzibel?
+\vspace{-5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{in $\mathbb{Q}[X]$}
+\[
+X^2-2\quad\text{ist irreduzibel}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{in $\mathbb{R}[X]$}
+\[
+X^2-2 = (X-\sqrt{2})(X+\sqrt{2})
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex
new file mode 100644
index 0000000..3d01020
--- /dev/null
+++ b/vorlesungen/slides/3/fibonacci.tex
@@ -0,0 +1,71 @@
+%
+% fibonacci.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+
+\begin{frame}[t]
+\frametitle{Fibonacci}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{block}{Fibonacci-Rekursion}
+$x_i$ Fibonacci-Zahlen\uncover<2->{, d.~h.~$x_{n+1\mathstrut}=x_{n\mathstrut}+x_{n-1\mathstrut}$}
+\[
+\uncover<3->{
+v_n
+=
+\begin{pmatrix}
+x_{n+1}\\
+x_n
+\end{pmatrix}}
+\uncover<4->{
+\quad\Rightarrow\quad
+v_n = 
+\underbrace{
+\begin{pmatrix}
+1&1\\
+1&0
+\end{pmatrix}
+}_{\displaystyle=\Phi}
+v_{n-1}}
+\uncover<5->{
+\quad\Rightarrow\quad
+v_n
+=
+\Phi^n
+v_0}\uncover<6->{,
+\;
+v_0 = \begin{pmatrix} 1\\0\end{pmatrix}}
+\]
+\end{block}
+\vspace{-5pt}
+\uncover<7->{%
+\begin{block}{Rekursionsformel für $\Phi$}
+\vspace{-12pt}
+\begin{align*}
+v_{n}&=v_{n-1}+v_{n-2}
+&&\uncover<8->{\Rightarrow&
+\Phi^n v_0 &= \Phi^{n-1} v_0 + \Phi^{n-2}v_0}
+&&\uncover<9->{\Rightarrow&
+\Phi^{n-2}(\Phi^2-\Phi-I)v_0&=0}
+\\
+\end{align*}
+\vspace{-22pt}%
+
+\uncover<10->{$\Phi$ ist $\chi_\Phi(X)=m_\Phi(X) = X^2-X-1$, irreduzibel}
+\end{block}}
+
+\uncover<11->{%
+\begin{block}{Faktorisierung}
+\vspace{-12pt}
+\[
+(X-\Phi)(X-(I-\Phi))
+\uncover<12->{=
+X^2-X +\Phi(I-\Phi)}
+\uncover<13->{=
+X^2-X -(\underbrace{\Phi^2-\Phi}_{\displaystyle=I})
+}
+\]
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/ideal.tex b/vorlesungen/slides/3/ideal.tex
new file mode 100644
index 0000000..f7f432e
--- /dev/null
+++ b/vorlesungen/slides/3/ideal.tex
@@ -0,0 +1,63 @@
+%
+% ideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ideal}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Voraussetzungen}
+$R$ ein Ring, $r\in R$
+\end{block}
+\uncover<2->{%
+\begin{block}{Vielfache\uncover<4->{ = Hauptideal}}
+Die Menge aller Elemente, die durch $r$ teilbar sind\uncover<3->{:
+\[
+(r)=rR
+\]}
+\uncover<4->{heisst {\em Hauptideal}}
+\end{block}}
+\uncover<5->{%
+\begin{block}{Ideal}
+$I\subset R$ mit
+\(RI\subset I\), \(I+I\subset I\)
+\end{block}}
+\uncover<6->{%
+\begin{block}{Hauptidealring}
+Jedes Ideal von $R$ ist ein Hauptideal
+\\
+\uncover<7->{{\usebeamercolor[fg]{title}Beispiele:}
+$\mathbb{Z}$,
+$\Bbbk[X]$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Grösster gemeinsamer Teiler}
+$a,b\in R$
+\begin{align*}
+\uncover<9->{(a) + (b)
+&= aR + bR}
+\intertext{\uncover<10->{ist eine Ideal }\uncover<11->{$\Rightarrow$ ein Hauptideal}}
+&\uncover<12->{= cR}\uncover<13->{ = \operatorname{ggT}(a,b)R}
+\end{align*}
+\uncover<14->{Existenz des $\operatorname{ggT}(a,b)$ ist eine
+gemeinsame Eigenschaft}
+\end{block}}
+\uncover<15->{%
+\begin{block}{Allgemein}
+\begin{itemize}
+\item<16->
+Alle euklidischen Ringe sind Hauptidealringe
+\item<17->
+Alle solchen Ringe verwenden den gleichen Algorithmus
+für $\operatorname{ggT}(a,b)$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/idealverband.tex b/vorlesungen/slides/3/idealverband.tex
new file mode 100644
index 0000000..3434868
--- /dev/null
+++ b/vorlesungen/slides/3/idealverband.tex
@@ -0,0 +1,78 @@
+%
+% idealverband.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Idealverband}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {$\mathbb{Z}$};
+
+\uncover<2->{
+\node at (-6,-2) {$2\mathbb{Z}$};
+\node at (-2,-2) {$3\mathbb{Z}$};
+\node at (2,-2) {$5\mathbb{Z}$};
+\node at (6,-2) {$7\mathbb{Z}$};
+\node at (7,-2) {$\dots$};
+}
+
+\uncover<3->{
+\node at (-4,-4) {$6\mathbb{Z}$};
+\node at (-2,-4) {$10\mathbb{Z}$};
+\node at (0,-4) {$15\mathbb{Z}$};
+\node at (2,-4) {$21\mathbb{Z}$};
+\node at (4,-4) {$35\mathbb{Z}$};
+\node at (6,-4) {$\dots$};
+}
+
+\uncover<4->{
+\node at (-2,-6) {$30\mathbb{Z}$};
+\node at (0,-6) {$70\mathbb{Z}$};
+\node at (2,-6) {$105\mathbb{Z}$};
+}
+
+\uncover<5->{
+	\node at (-5,-6) {$\dots$};
+	\node at (5,-6) {$\dots$};
+}
+
+\uncover<2->{
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-6,-2);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-2,-2);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (2,-2);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (6,-2);
+}
+
+\uncover<3->{
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-4,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-2,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (-4,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (0,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (2,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (-2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (0,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (4,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (4,-4);
+}
+
+\uncover<4->{
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-4,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (0,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (-2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (4,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (0,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (4,-4);
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
diff --git a/vorlesungen/slides/3/images/Makefile b/vorlesungen/slides/3/images/Makefile
new file mode 100644
index 0000000..e338fcf
--- /dev/null
+++ b/vorlesungen/slides/3/images/Makefile
@@ -0,0 +1,55 @@
+#
+# Makefile -- build images
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	hauptideal.jpg nichthauptideal.jpg hauptideal2.jpg hauptidealX.jpg \
+	hauptidealR.jpg hauptidealXR.jpg ring.jpg
+
+ring.png:	ring.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Oring.png ring.pov
+ring.jpg:	ring.png
+	convert ring.png -density 300 -units PixelsPerInch ring.jpg
+
+hauptideal.png:	hauptideal.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Ohauptideal.png hauptideal.pov
+hauptideal.jpg:	hauptideal.png
+	convert hauptideal.png -density 300 -units PixelsPerInch \
+		hauptideal.jpg
+
+hauptidealR.png:	hauptidealR.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealR.png hauptidealR.pov
+hauptidealR.jpg:	hauptidealR.png
+	convert hauptidealR.png -density 300 -units PixelsPerInch \
+		hauptidealR.jpg
+
+hauptideal2.png:	hauptideal2.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Ohauptideal2.png hauptideal2.pov
+hauptideal2.jpg:	hauptideal2.png
+	convert hauptideal2.png -density 300 -units PixelsPerInch \
+		hauptideal2.jpg
+
+hauptidealX.png:	hauptidealX.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealX.png hauptidealX.pov
+hauptidealX.jpg:	hauptidealX.png
+	convert hauptidealX.png -density 300 -units PixelsPerInch \
+		hauptidealX.jpg
+
+hauptidealXR.png:	hauptidealXR.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealXR.png hauptidealXR.pov
+hauptidealXR.jpg:	hauptidealXR.png
+	convert hauptidealXR.png -density 300 -units PixelsPerInch \
+		hauptidealXR.jpg
+
+nichthauptideal.png:	nichthauptideal.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Onichthauptideal.png nichthauptideal.pov
+nichthauptideal.jpg:	nichthauptideal.png
+	convert nichthauptideal.png -density 300 -units PixelsPerInch \
+		nichthauptideal.jpg
+
+ideal:	ideal.pov ideal.ini common.inc
+	rm -rf ideal
+	mkdir ideal
+	povray +A0.1 +W1920 +H1080 -Oideal/i.png ideal.ini
+
+
diff --git a/vorlesungen/slides/3/images/common.inc b/vorlesungen/slides/3/images/common.inc
new file mode 100644
index 0000000..36c4e6b
--- /dev/null
+++ b/vorlesungen/slides/3/images/common.inc
@@ -0,0 +1,277 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.40;
+#declare O = <0, 0, 0>;
+#declare at = 0.10;
+
+#declare Xunten = -10;
+#declare Xoben = 10;
+#declare Yunten = -8;
+#declare Yoben = 8;
+#declare Zunten = 0;
+#declare Zoben = 20;
+
+#declare phi0 = 2 * pi * 290 / 360;
+
+camera {
+	location <60 * cos(2*pi*T+phi0), 15, 60 * sin(2*pi*T+phi0) + 10>
+	look_at <0, -2, 10>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-14, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+light_source {
+	<41, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+arrow(< -12.0,   0.0,  0  >, < 12.0,  0.0,  0.0 >, at, Gray)
+arrow(<   0.0,   0.0, -2.0>, <  0.0,  0.0, 22.0 >, at, Gray)
+arrow(<   0.0, -10.0,  0  >, <  0.0, 10.0,  0.0 >, at, Gray)
+
+#macro kasten()
+	box { <-10.5,-8.5,-0.5>, <10.5,8.5,20.5> }
+#end
+
+#declare gruen = rgb<0.2,0.4,0.2>;
+#declare blau  = rgb<0.0,0.4,0.8>;
+#declare rot   = rgb<1.0,0.4,0.0>;
+
+#declare r = 0.4;
+
+#macro Zring()
+	union {
+		#declare X = Xunten;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X, Y, Z>, r }
+
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 1;
+		#end
+		pigment {
+			color rot
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Hauptideal()
+	union {
+		#declare A = Xunten;
+		#while (A <= Xoben + 0.5)
+			#declare B = Zunten;
+			#while (B <= Zoben + 0.5)
+				#declare Y = A + B;
+				#if ((Y >= Yunten - 0.5) & (Y <= Yoben + 0.5))
+					sphere { <A, Y, B>, r }
+				#end
+				#declare B = B + 1;
+			#end
+			#declare A = A + 1;
+		#end
+		pigment {
+			color blau
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro HauptidealR()
+	intersection {
+		kasten()
+		#declare n = vnormalize(< 1, -2, 1 >);
+		plane { n, 0.1 }
+		plane { -n, 0.1 }
+		pigment {
+			color blau
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Ideal2()
+	union {
+		#declare X = Xunten;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X, Y, Z>, r }
+					#declare Z = Z + 2;
+				#end
+				#declare Y = Y + 2;
+			#end
+			#declare X = X + 2;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro IdealX()
+	union {
+		#declare Y = Yunten;
+		#while (Y <= Yoben + 0.5)
+			#declare Z = Zunten;
+			#while (Z <= Zoben + 0.5)
+				sphere { <0, Y, Z>, r }
+				#declare Z = Z + 1;
+			#end
+			#declare Y = Y + 1;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro IdealXR()
+	intersection {
+		kasten()
+		plane { <1,0,0>, 0.1 }
+		plane { <-1,0,0>, 0.1 }
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Nichthauptideal()
+	union {
+		#declare X = Xunten/2;
+		#while (X <= Xoben/2 + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = 0;
+				#while (Z <= Zoben + 0.5)
+					sphere { <2*X,Y,Z>, r }
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 1;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro NichthauptidealKomplement()
+	union {
+		#declare X = Xunten + 1;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X,Y,Z>, r }
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 2;
+		#end
+		pigment {
+			color rot
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+
+
+
+
+
+
diff --git a/vorlesungen/slides/3/images/hauptideal.jpg b/vorlesungen/slides/3/images/hauptideal.jpg
new file mode 100644
index 0000000..769f53c
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal.jpg
diff --git a/vorlesungen/slides/3/images/hauptideal.pov b/vorlesungen/slides/3/images/hauptideal.pov
new file mode 100644
index 0000000..a934e57
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Hauptideal()
+
diff --git a/vorlesungen/slides/3/images/hauptideal2.jpg b/vorlesungen/slides/3/images/hauptideal2.jpg
new file mode 100644
index 0000000..51823f3
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal2.jpg
diff --git a/vorlesungen/slides/3/images/hauptideal2.pov b/vorlesungen/slides/3/images/hauptideal2.pov
new file mode 100644
index 0000000..9da5a1a
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal2.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Ideal2()
+
diff --git a/vorlesungen/slides/3/images/hauptidealR.jpg b/vorlesungen/slides/3/images/hauptidealR.jpg
new file mode 100644
index 0000000..fae5840
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealR.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealR.pov b/vorlesungen/slides/3/images/hauptidealR.pov
new file mode 100644
index 0000000..330e523
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealR.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealR.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+HauptidealR()
+
diff --git a/vorlesungen/slides/3/images/hauptidealX.jpg b/vorlesungen/slides/3/images/hauptidealX.jpg
new file mode 100644
index 0000000..f9b4540
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealX.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealX.pov b/vorlesungen/slides/3/images/hauptidealX.pov
new file mode 100644
index 0000000..d0045f9
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealX.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealX.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+IdealX()
+
diff --git a/vorlesungen/slides/3/images/hauptidealXR.jpg b/vorlesungen/slides/3/images/hauptidealXR.jpg
new file mode 100644
index 0000000..d8906c8
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealXR.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealXR.pov b/vorlesungen/slides/3/images/hauptidealXR.pov
new file mode 100644
index 0000000..5daa3e6
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealXR.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealXR.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+IdealXR()
+
diff --git a/vorlesungen/slides/3/images/ideal.ini b/vorlesungen/slides/3/images/ideal.ini
new file mode 100644
index 0000000..66aa191
--- /dev/null
+++ b/vorlesungen/slides/3/images/ideal.ini
@@ -0,0 +1,7 @@
+Input_File_Name=ideal.pov
+Initial_Frame=0
+Final_Frame=2500
+Initial_Clock=0
+Final_Clock=5
+Cyclic_Animation=off
+Pause_when_Done=off
diff --git a/vorlesungen/slides/3/images/ideal.pov b/vorlesungen/slides/3/images/ideal.pov
new file mode 100644
index 0000000..88afaf7
--- /dev/null
+++ b/vorlesungen/slides/3/images/ideal.pov
@@ -0,0 +1,26 @@
+//
+// ideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = clock;
+#include "common.inc"
+
+#if (T < 1)
+Zring()
+#else
+    #if (T < 2)
+	Hauptideal()
+    #else
+	#if (T < 3)
+	    Ideal2()
+	#else
+	    #if (T < 4)
+		IdealX()
+	    #else
+		Nichthauptideal()
+		NichthauptidealKomplement()
+	    #end
+	#end
+    #end
+#end
diff --git a/vorlesungen/slides/3/images/nichthauptideal.jpg b/vorlesungen/slides/3/images/nichthauptideal.jpg
new file mode 100644
index 0000000..55858d0
--- /dev/null
+++ b/vorlesungen/slides/3/images/nichthauptideal.jpg
diff --git a/vorlesungen/slides/3/images/nichthauptideal.pov b/vorlesungen/slides/3/images/nichthauptideal.pov
new file mode 100644
index 0000000..72a6330
--- /dev/null
+++ b/vorlesungen/slides/3/images/nichthauptideal.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Nichthauptideal()
+NichthauptidealKomplement()
diff --git a/vorlesungen/slides/3/images/ring.jpg b/vorlesungen/slides/3/images/ring.jpg
new file mode 100644
index 0000000..27721b1
--- /dev/null
+++ b/vorlesungen/slides/3/images/ring.jpg
diff --git a/vorlesungen/slides/3/images/ring.pov b/vorlesungen/slides/3/images/ring.pov
new file mode 100644
index 0000000..f854335
--- /dev/null
+++ b/vorlesungen/slides/3/images/ring.pov
@@ -0,0 +1,10 @@
+//
+// ring.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Zring()
+
diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex
new file mode 100644
index 0000000..4ad22d2
--- /dev/null
+++ b/vorlesungen/slides/3/inverse.tex
@@ -0,0 +1,89 @@
+%
+% inverse.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Inverse Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.24\textwidth}
+\begin{block}{Imaginäre Einheit}
+\vspace{-15pt}
+\begin{align*}
+J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix}
+\\
+0&=
+J^2 + I
+\\
+0&=
+J+J^{-1}
+\\
+J^{-1}&=-J
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.25\textwidth}
+\uncover<2->{%
+\begin{block}{Wurzel $\sqrt{2}$}
+\vspace{-15pt}
+\begin{align*}
+W&=\begin{pmatrix}0&2\\1&0\end{pmatrix}
+\\
+0 &= X^2-2
+\\
+0 &= W-2W^{-1}
+\\
+W^{-1}&=\frac12 W
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.41\textwidth}
+\uncover<3->{%
+\begin{block}{Drehmatrix}
+\vspace{-15pt}
+\begin{align*}
+D&=\begin{pmatrix}
+\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\
+\sin \frac{\pi}{1291} &  \cos\frac{\pi}{1291}
+\end{pmatrix}
+\\
+0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\\
+0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{D-2\cos\frac{\pi\mathstrut}{1291\mathstrut}+D^{-1}}
+\\
+D^{-1}
+&= \only<-3>{-D^{1290}\phantom{+\frac{\mathstrut}{\mathstrut}}}%
+\only<4->{-D+2I\cos\frac{\pi\mathstrut}{1291\mathstrut}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-25pt}
+\uncover<5->{
+\begin{block}{3D-Beispiel}
+$p(x) = -x^3-5x^2+5x+1$
+\[
+A=
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}
+\quad\Rightarrow\quad
+A^{-1}
+=
+A^2+5A-5I
+=
+\begin{pmatrix*}[r]
+-1& 1&-1\\
+ 7&-9&10\\
+ 3&-4& 5
+\end{pmatrix*}
+\]
+\end{block}}
+\vspace{-10pt}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/maximalergrad.tex b/vorlesungen/slides/3/maximalergrad.tex
new file mode 100644
index 0000000..d33ddc0
--- /dev/null
+++ b/vorlesungen/slides/3/maximalergrad.tex
@@ -0,0 +1,72 @@
+%
+% maximalergrad.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Jede Matrix hat eine Polynomrelation}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-5pt}
+\begin{block}{Dimension des Matrizenrings}
+Der Ring $M_{n}(\Bbbk)$ ist ein $n^2$-dimensionaler Vektorraum mit
+Basis
+{\tiny
+\begin{align*}
+&\uncover<2->{\begin{pmatrix}
+1&0&\dots&0\\
+0&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<3->{\begin{pmatrix}
+0&1&\dots&0\\
+0&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<4->{\dots}
+&
+&\uncover<5->{\begin{pmatrix}
+0&0&\dots&1\\
+0&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+\\
+&\uncover<6->{\begin{pmatrix}
+0&0&\dots&0\\
+1&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<7->{\begin{pmatrix}
+0&0&\dots&0\\
+0&1&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<8->{\dots}
+&
+&\uncover<9->{\begin{pmatrix}
+0&0&\dots&0\\
+0&0&\dots&1\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+\end{align*}}
+\end{block}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Potenzen von $A$}
+Die $n^2+1$ Matrizen $I,A,A^2,\dots,A^{n^2-1},A^{n^2}$ müssen linear abhängig
+sein:
+\[
+\uncover<11->{
+a_0I+a_1A+a_2A^2+\dots+a_{n^2-1}A^{n^2-1}+a_{n^2}A^{n^2} = 0
+}
+\]
+\uncover<12->{d.~h.~$p(X) = a_0+a_1X+a_2X^2+\dots +a_{n^2-1}X^{n^2-1}+a_{n^2}A^{n^2}\in\Bbbk[X]$ ist ein Polynom mit $p(A)=0$.}
+\end{block}}
+\uncover<13->{%
+$\Rightarrow$ $A$ über die Eigenschaften (Faktorisierung) von $p$ studieren
+}
+\end{frame}
diff --git a/vorlesungen/slides/3/maximalideal.tex b/vorlesungen/slides/3/maximalideal.tex
new file mode 100644
index 0000000..21a945a
--- /dev/null
+++ b/vorlesungen/slides/3/maximalideal.tex
@@ -0,0 +1,64 @@
+%
+% maximalideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Maximale Ideale}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Teilbarkeit}
+$a|b$
+\uncover<2->{$\Rightarrow$
+$b\in aR$}
+\uncover<3->{$\Rightarrow$
+$bR\subset aR$}
+\end{block}
+\uncover<4->{%
+\begin{block}{Nicht mehr teilbar}
+$a\in R$ nicht faktorisierbar
+\\
+\uncover<5->{$\Rightarrow$
+\\
+es gibt kein Ideal zwischen $aR$ und $R$}
+\\
+\uncover<6->{$\Leftrightarrow$
+\\
+$J$ ein Ideal
+$aR \subset J \subset R$, dann ist
+$J=aR$ oder $J=R$}
+\end{block}}
+\uncover<7->{
+\begin{block}{maximales Ideal}
+$I\subset R$ heisst maximal, wenn für jedes Ideal $J$
+mit $I\subset J\subset R$ gilt
+$I=J$ oder $J=R$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{
+\begin{block}{Beispiele}
+\begin{itemize}
+\item Primzahlen $p$ erzeugen maximale Ideale in $\mathbb{Z}$
+\item<9-> Irreduzible Polynome erzeugen maximale Ideale in $\Bbbk[X]$
+\end{itemize}
+\end{block}}
+\uncover<10->{%
+\begin{block}{Körper}
+$M\subset R$ ein maximales Ideal, dann ist
+$R/M$ ein Körper
+\end{block}}
+\uncover<11->{%
+\begin{block}{Beispiel}
+\begin{itemize}
+\item
+$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$
+\item<12->
+$m$ ein irreduzibles Polynom:
+$\Bbbk[X]/ (m)$ ist ein Körper
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/minimalbeispiel.tex b/vorlesungen/slides/3/minimalbeispiel.tex
new file mode 100644
index 0000000..f94cf8d
--- /dev/null
+++ b/vorlesungen/slides/3/minimalbeispiel.tex
@@ -0,0 +1,36 @@
+%
+% minimalbeispiel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Beispiel für $p(A)=0$}
+\begin{block}{Potenzen einer $2\times 2$-Matrix $A$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-10pt}
+\[
+I  ={\tiny\begin{pmatrix}  1 &  0 \\  0 &  1 \end{pmatrix}},\quad
+A  ={\tiny\begin{pmatrix}  3 &  2 \\ -1 & -2 \end{pmatrix}},\quad
+\uncover<2->{A^2={\tiny\begin{pmatrix}  7 &  2 \\ -1 &  2 \end{pmatrix}}}
+\uncover<3->{,\quad A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}}}
+\uncover<4->{,\quad A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 &  2 \end{pmatrix}}}
+\]
+\end{block}
+\vspace{-5pt}
+\uncover<5->{%
+\begin{block}{linear abhängig}
+Bereits die ersten $3$ sind linear abhängig:
+\[
+-4I - A + A^2
+=
+-4\begin{pmatrix}  1 &  0 \\  0 &  1 \end{pmatrix}
+-\begin{pmatrix}  3 &  2 \\ -1 & -2 \end{pmatrix}
++\begin{pmatrix}  7 &  2 \\ -1 &  2 \end{pmatrix}
+= 
+\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
+\]
+\uncover<6->{$p(X) = X^2 - X - 4 \in \mathbb{Q}[X]$ hat die Eigenschaft
+$p(A)=0$}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/minimalpolynom.tex b/vorlesungen/slides/3/minimalpolynom.tex
new file mode 100644
index 0000000..2b36a65
--- /dev/null
+++ b/vorlesungen/slides/3/minimalpolynom.tex
@@ -0,0 +1,30 @@
+%
+% minimalpolynom.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Minimalpolynom}
+\begin{block}{Definition}
+Zu jeder $n\times n$-Matrix $A$
+gibt es ein Polynom $m_A(X)\in\Bbbk[X]$ minimalen Grades $\deg m_A\le n^2$
+derart, dass $m_A(A)=0$.
+\end{block}
+\uncover<2->{%
+\begin{block}{Strategie}
+Das Minimalpolynom ist eine ``Invariante'' der Matrix $A$
+\end{block}}
+\uncover<3->{%
+\begin{block}{Satz von Cayley-Hamilton}
+Für jede $n\times n$-Matrix $A\in M_n(\Bbbk)$ gilt $\chi_A(A)=0$
+\uncover<4->{%
+\[
+\Downarrow
+\]
+Das Minimalpolynom $m_A\in \Bbbk[X]$ ist ein Teiler
+des charakteristischen Polynoms $\chi_A\in \Bbbk[X]$}
+\\
+\uncover<5->{$\Rightarrow $
+Faktorzerlegung on $\chi_A(X)$ ermitteln!}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex
new file mode 100644
index 0000000..048e6a2
--- /dev/null
+++ b/vorlesungen/slides/3/motivation.tex
@@ -0,0 +1,108 @@
+%
+% motivation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Motivation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.24\textwidth}
+\begin{block}{Imaginäre Einheit}
+\vspace{-15pt}
+\begin{align*}
+J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix}
+\\
+p(X) &= X^2 + 1
+\\
+p(J) &= J^2 + I = 0
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.25\textwidth}
+\uncover<2->{%
+\begin{block}{Wurzel $\sqrt{2}$}
+\vspace{-15pt}
+\begin{align*}
+W&=\begin{pmatrix}0&2\\1&0\end{pmatrix}
+\\
+p(X) &= X^2-2
+\\
+p(W) &= W^2-2I=0
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.41\textwidth}
+\uncover<3->{%
+\begin{block}{Drehmatrix}
+\vspace{-15pt}
+\begin{align*}
+D&=\begin{pmatrix}
+\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\
+\sin \frac{\pi}{1291} &  \cos\frac{\pi}{1291}
+\end{pmatrix}
+\\
+p(X)&=
+\ifthenelse{\boolean{presentation}}{\only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{X^2-2X\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\\
+p(D) &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-20pt}
+\uncover<5->{
+\begin{block}{3D-Beispiel}
+$p(x) = -x^3-5x^2+5x+1$
+\[
+\ifthenelse{\boolean{presentation}}{
+\only<5-8>{
+A=
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}}
+\only<6-8>{
+\quad\Rightarrow\quad}}{}
+\uncover<6->{
+-
+\only<-9>{A^3}\only<10->{
+\begin{pmatrix*}[r]
+-169&-35&35\\
+-185&-39&40\\
+ -45&-10&11
+\end{pmatrix*}}
+-5
+\only<-8>{A^2}\only<9->{
+\begin{pmatrix*}[r]
+29&6&-6\\
+32&6&-5\\
+ 8&1& 0
+\end{pmatrix*}}
++5
+\only<-7>{A}\only<8->{
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}}
++
+\only<-6>{I}\only<7->{
+\begin{pmatrix*}[r]
+1&0&0\\
+0&1&0\\
+0&0&1
+\end{pmatrix*}}
+}
+\uncover<11->{=0}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}$\Rightarrow$
+Rechenregeln von Matrizen können durch Polynome ausgedrückt werden}
+}
+\end{frame}
diff --git a/vorlesungen/slides/3/multiplikation.tex b/vorlesungen/slides/3/multiplikation.tex
new file mode 100644
index 0000000..13f4e03
--- /dev/null
+++ b/vorlesungen/slides/3/multiplikation.tex
@@ -0,0 +1,180 @@
+%
+% multiplikation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\N{21}
+\begin{frame}[t,fragile]
+\frametitle{Multiplikation mit $\alpha$ in $\mathbb{Z}(\alpha)$}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+
+\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\sqrt{2})$};
+\node[color=blue] at (3.5,3.2) [above left] {$\sqrt{2}\mathbb{Z}(\sqrt{2})$};
+
+\pgfmathparse{sqrt(2)}
+\xdef\a{\pgfmathresult}
+\pgfmathparse{-int(3.2/\a)}
+\xdef\ymin{\pgfmathresult}
+\pgfmathparse{int(3.2/\a)}
+\xdef\ymax{\pgfmathresult}
+
+\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\sqrt{2}$}];
+
+\def\punkt#1#2#3{
+	({(1-(#3))*(#1)+2*(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1))*\a})
+}
+
+\foreach \x in {-3,...,3}{
+	\draw[color=red,line width=0.5pt]
+		\punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0};
+	\foreach \y in {\ymin,...,\ymax}{
+		\fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08];
+	}
+}
+\foreach \y in {\ymin,...,\ymax}{
+	\draw[color=red,line width=0.5pt]
+		\punkt{-3}{\y}{0} -- \punkt{3}{\y}{0};
+}
+
+
+\def\bildnetz#1{
+	\pgfmathparse{(#1-1)/(\N-1)}
+	\xdef\t{\pgfmathresult}
+	\only<#1>{
+		\uncover<2->{
+			\draw[->,color=blue,line width=1.4pt]
+				(0,\a) -- \punkt{0}{1}{\t};
+			\draw[->,color=blue,line width=1.4pt]
+				(1,0) -- \punkt{1}{0}{\t};
+		}
+		\foreach \x in {-3,...,3}{
+			\draw[color=blue,line width=0.5pt]
+				\punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t};
+			\foreach \y in {\ymin,...,\ymax}{
+				\fill[color=blue]
+				\punkt{\x}{\y}{\t}
+				circle[radius=0.06];
+			}
+		}
+		\foreach \y in {\ymin,...,\ymax}{
+			\draw[color=blue,line width=0.5pt]
+				\punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t};
+		}
+	}
+}
+
+\begin{scope}
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\ifthenelse{\boolean{presentation}}{
+	\foreach \T in {1,...,\N}{
+		\bildnetz{\T}
+	}
+}{
+	\bildnetz{\N}
+}
+\end{scope}
+
+\uncover<\N->{
+\begin{scope}[yshift=-2.5cm]
+\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8);
+\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8);
+\node at (0,0) {$\displaystyle W=\begin{pmatrix}0&2\\1&0\end{pmatrix}$};
+\end{scope}
+}
+
+\node at (0,-3.7) {$\alpha^2 = 2$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+
+\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\varphi)$};
+\node[color=blue] at (3.5,3.2) [above left] {$\varphi\mathbb{Z}(\varphi)$};
+
+\pgfmathparse{(sqrt(5)+1)/2}
+\xdef\a{\pgfmathresult}
+\pgfmathparse{-int(3.3/\a)}
+\xdef\ymin{\pgfmathresult}
+\pgfmathparse{int(3.3/\a)}
+\xdef\ymax{\pgfmathresult}
+\def\punkt#1#2#3{
+	({(1-(#3))*(#1)+(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1+#2))*\a})
+}
+
+\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\varphi$}];
+
+\foreach \x in {-3,...,3}{
+	\draw[color=red,line width=0.5pt]
+		\punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0};
+	\foreach \y in {\ymin,...,\ymax}{
+		\fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08];
+	}
+}
+\foreach \y in {\ymin,...,\ymax}{
+	\draw[color=red,line width=0.5pt]
+		\punkt{-3}{\y}{0} -- \punkt{3}{\y}{0};
+}
+
+\def\bildnetz#1{
+	\pgfmathparse{(#1-1)/(\N-1)}
+	\xdef\t{\pgfmathresult}
+	\only<#1>{
+		\uncover<2->{
+			\draw[->,color=blue,line width=1.4pt]
+				(0,\a) -- \punkt{0}{1}{\t};
+			\draw[->,color=blue,line width=1.4pt]
+				(1,0) -- \punkt{1}{0}{\t};
+		}
+		\foreach \x in {-3,...,3}{
+			\draw[color=blue,line width=0.5pt]
+				\punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t};
+			\foreach \y in {\ymin,...,\ymax}{
+				\fill[color=blue] \punkt{\x}{\y}{\t}
+					circle[radius=0.06];
+			}
+		}
+		\foreach \y in {\ymin,...,\ymax}{
+			\draw[color=blue,line width=0.5pt]
+				\punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t};
+		}
+	}
+}
+
+\begin{scope}
+
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\ifthenelse{\boolean{presentation}}{
+	\foreach \T in {1,...,\N}{
+		\bildnetz{\T}
+	}
+}{
+	\bildnetz{\N}
+}
+\end{scope}
+
+\uncover<\N->{
+\begin{scope}[yshift=-2.5cm]
+\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8);
+\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8);
+\node at (0,0) {$\displaystyle \Phi=\begin{pmatrix}0&1\\1&1\end{pmatrix}$};
+\end{scope}
+}
+
+\node at (0,-3.7) {$\alpha^2 = \alpha + 1$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/nichthauptideal.tex b/vorlesungen/slides/3/nichthauptideal.tex
new file mode 100644
index 0000000..46074b9
--- /dev/null
+++ b/vorlesungen/slides/3/nichthauptideal.tex
@@ -0,0 +1,78 @@
+%
+% nichthauptideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Nicht-Hauptideal in $\mathbb{Z}[X]$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Hauptideal\uncover<2->{ = ``Gerade''}}
+\vspace{-10pt}
+\begin{align*}
+\langle X+1\rangle&=(X+1) = {\color{red}(X+1)\cdot\mathbb{Z}[X]}
+\end{align*}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+\draw[->] (-6.3,0) -- (6.8,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-6.2) -- (0,6.6) coordinate[label={right:$\mathbb{Z}X$}];
+\foreach \x in {-6,...,6}{
+	\fill[color=red] (\x,\x) circle[radius=0.12];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Ideal mit zwei Erzeugenden}
+\vspace{-10pt}
+\begin{align*}
+\uncover<6->{
+{\color{darkgreen}
+\langle 2,X\rangle
+}
+&=}
+\uncover<5->{
+{\color{red}2\cdot\mathbb{Z}[X]}
+}
+\uncover<6->{+}
+\uncover<4->{
+{\color{blue}X\cdot\mathbb{Z}[X]}
+}
+\end{align*}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+\draw[->] (-6.3,0) -- (6.9,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-6.2) -- (0,7.0) coordinate[label={right:$\mathbb{Z}X$}];
+\uncover<6->{
+	\foreach \x in {-6,-4,...,6}{
+		\foreach \y in {-6,...,6}{
+			\fill[color=darkgreen] (\x,\y) circle[radius=0.20];
+		}
+	}
+}
+\uncover<5->{
+	\foreach \x in {-6,-4,...,6}{
+		\foreach \y in {-6,-4,...,6}{
+			\fill[color=red] (\x,\y) circle[radius=0.16];
+		}
+	}
+}
+\uncover<4->{
+	\foreach \y in {-6,...,6}{
+		\fill[color=blue] (0,\y) circle[radius=0.12];
+	}
+}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex
new file mode 100644
index 0000000..e1424ff
--- /dev/null
+++ b/vorlesungen/slides/3/nichthauptideal2.tex
@@ -0,0 +1,95 @@
+%
+% nichthauptideal2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Das Ideal $\langle 2,X\rangle \subset \mathbb{Z}[X]$}
+\vspace{-12pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\c{\clip (-2.8,-2.0) rectangle (2.8,2.0);}
+
+\def\labels{
+	\fill[color=white,opacity=0.5] (1.5,-0.1) circle[radius=0.2];
+	\node at (1.5,-0.1) {$1$};
+	\fill[color=white,opacity=0.5] (-0.9,1.7) circle[radius=0.2];
+	\node at (-0.9,1.7) {$X$};
+	\fill[color=white,opacity=0.5] (0.8,0.7) circle[radius=0.2];
+	\node at (0.8,0.7) {$X^2$};
+}
+
+\only<-3>{
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/ring.jpg}};
+	\end{scope}
+	\node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$};
+\end{scope}
+}
+
+\uncover<2->{
+\begin{scope}[xshift=-3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptideal.jpg}};
+	\end{scope}
+	\node[color=blue] at (-0.2,-1.2) {$(X+1)\cdot\mathbb{Z}[X]$};
+	\labels
+\end{scope}
+}
+
+\uncover<3->{
+\begin{scope}[xshift=-3.0cm,yshift=-1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptideal2.jpg}};
+	\end{scope}
+	\node[color=darkgreen] at (-3.0,-0.8) {$2\cdot\mathbb{Z}[X]$};
+\end{scope}
+
+\begin{scope}[xshift=3.0cm,yshift=-1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptidealX.jpg}};
+	\end{scope}
+	\node[color=darkgreen] at (2.5,-0.8) {$X\cdot\mathbb{Z}[X]$};
+	\labels
+\end{scope}
+}
+
+\uncover<4->{
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/nichthauptideal.jpg}};
+	\end{scope}
+	\node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$};
+	\node[color=darkgreen] at (1.9,-0.4) [right] {$\langle 2,X\rangle$};
+\end{scope}
+}
+
+\draw[color=gray!50] (-6.6,0) -- (6.4,0);
+\draw[color=gray!50] (0,-3.8) -- (0,3.8);
+
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\fill[color=white,opacity=0.5] (1.5,-0.6) circle[radius=0.2];
+	\node at (1.5,-0.6) {$1$};
+	\fill[color=white,opacity=0.5] (-0.4,1.7) circle[radius=0.2];
+	\node at (-0.4,1.7) {$X$};
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex
new file mode 100644
index 0000000..d646353
--- /dev/null
+++ b/vorlesungen/slides/3/operatoren.tex
@@ -0,0 +1,51 @@
+%
+% operatoren.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{$X$ als Operator}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Polynome}
+$a(X)=a_0+a_1X+\dots+a_nX^n$
+\uncover<2->{%
+\[
+a(X)
+=
+\begin{pmatrix}
+a_0\\a_1\\a_2\\a_3\\\vdots\\a_n
+\end{pmatrix}
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<3->{%
+\begin{block}{Multiplikation mit $X$}
+\strut
+\[
+\begin{pmatrix}
+1\\0\\0\\0\\\vdots\\0
+\end{pmatrix}
+\uncover<4->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\1\\0\\0\\\vdots\\0
+\end{pmatrix}}
+\uncover<5->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\0\\1\\0\\\vdots\\0
+\end{pmatrix}}
+\uncover<6->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\0\\0\\1\\\vdots\\0
+\end{pmatrix}}
+\uncover<7->{\overset{\cdot X}{\mapsto}\dots}
+\uncover<8->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\0\\0\\0\\\vdots\\1
+\end{pmatrix}}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/phi.tex b/vorlesungen/slides/3/phi.tex
new file mode 100644
index 0000000..ee0814c
--- /dev/null
+++ b/vorlesungen/slides/3/phi.tex
@@ -0,0 +1,85 @@
+%
+% phi.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\mathbb{Q}(\varphi)=\mathbb{Q}[X]/(X^2-X-1)$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Der Ring $\mathbb{Z}(\varphi)$}
+$\mathbb{Z}(\varphi)$ als Teilrung:
+{\color{blue}
+\[
+R=\{a+b\varphi\;|\; a,b\in\mathbb{Z}\}
+\]}%
+\uncover<2->{$\varphi\not\in\mathbb{Q}$}\uncover<3->{
+$\Rightarrow$
+$1$ und $\varphi$ sind inkommensurabel}\uncover<4->{
+$\Rightarrow$
+$R$ dicht in $\mathbb{R}$}
+\end{block}
+\uncover<5->{%
+\begin{block}{Algebraische Konstruktion}
+\uncover<8->{%
+Das Polynom $X^2-X-1$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$}
+\[
+\uncover<8->{\mathbb{Q}[X]/(X^2-X-1)
+=}
+{\color{red}\{a+b\varphi\;|\;a,b\in\mathbb{Z}\}}
+\]\uncover<7->{%
+mit der Rechenregel: $X^2=X+1$}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Körper}
+$\mathbb{Q}(\varphi) = \mathbb{Q}[X]/(X^2+X+1)$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+\begin{scope}
+\pgfmathparse{(sqrt(5)-1))/2}
+\xdef\gphi{\pgfmathresult}
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\foreach \x in {-10,...,10}{
+	\pgfmathparse{int(\x/\gphi)-10}
+	\xdef\s{\pgfmathresult}
+	\pgfmathparse{int(\x/\gphi)+10}
+	\xdef\t{\pgfmathresult}
+	\foreach \y in {\s,...,\t}{
+		\uncover<4->{
+			\fill[color=blue] ({\x-\y*\gphi},0)
+				circle[radius=0.05];
+		}
+		\uncover<6->{
+			\draw[color=blue,line width=0.1pt] 
+				({\x-\y*\gphi-3.2},3.2) 
+				--
+				({\x-\y*\gphi+3.2},-3.2);
+		}
+	}
+}
+\end{scope}
+
+\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
+
+\uncover<5->{
+	\draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}];
+
+	\foreach \x in {-3,...,3}{
+		\foreach \y in {-5,...,5}{
+			\fill[color=red]
+				({\x},{\y*\gphi}) circle[radius=0.08];
+		}
+	}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/polynome.tex b/vorlesungen/slides/3/polynome.tex
new file mode 100644
index 0000000..d7179a0
--- /dev/null
+++ b/vorlesungen/slides/3/polynome.tex
@@ -0,0 +1,29 @@
+%
+% polynome.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynome}
+$R$ ein Ring, z.~B.~$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$
+
+\begin{definition}
+Polynome in $X$ mit Koeffizienten in $R$:
+\[
+R[X]
+=
+\{
+a(X)\;|\;
+a(X) = a_nX^n+a_{n-1}X^{n-1} + \dots a_2X^2+a_1X + a_0, a_k\in R
+\}
+\]
+\end{definition}
+
+\begin{itemize}
+\item<2-> {\em Grad} des Polynoms: $\deg a(X) = \deg a = n$
+\item<3-> $\deg 0 = -\infty$
+\item<4-> {\em normiertes Polynom}: $a_n=1$
+\end{itemize}
+
+
+\end{frame}
diff --git a/vorlesungen/slides/3/quotientenring.tex b/vorlesungen/slides/3/quotientenring.tex
new file mode 100644
index 0000000..4aa9e43
--- /dev/null
+++ b/vorlesungen/slides/3/quotientenring.tex
@@ -0,0 +1,59 @@
+%
+% Quotientenring.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quotientenring}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Quotientenring}
+$I\subset R$ ein Ideal
+\\
+\uncover<2->{
+$R/I$ hat eine Ringstruktur:
+\begin{align*}
+\uncover<3->{\pi(s)&=s+I}
+\\
+\uncover<4->{\pi(s)\pi(r)&= (s+I)(r+I)}\\
+            &\uncover<5->{= sr +\underbrace{sI + rI}_{\subset RI\subset I} + II = sr+I}
+\\
+\uncover<6->{\pi(s)+\pi(r)&= (s+I)+(r+I)}\\
+	     &\uncover<7->{=s+r+I=\pi(s+r)}
+\end{align*}}
+\end{block}
+\vspace{-15pt}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Beispiele}
+\begin{itemize}
+\item
+$\mathbb{Z}/(n)=\mathbb{Z}/n\mathbb{Z}$,
+$\mathbb{F}_p=\mathbb{Z}/(p)=\mathbb{Z}/p\mathbb{Z}$
+\item<8->
+$p\in\Bbbk[X]$
+$\Rightarrow$
+$\Bbbk[X]/(p)$ ist ein Ring
+\end{itemize}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Algebraideal}
+$I\subset A$
+\begin{itemize}
+\item<10->
+$I$ ein Unterraum von $A$ als Vektorraum
+\item<11->
+$I$ ein Ideal von $A$ als Ring
+\end{itemize}
+\end{block}}
+\uncover<12->{%
+\begin{block}{Quotientenalgebra}
+$A/I$ ist eine Algebra
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/ringstruktur.tex b/vorlesungen/slides/3/ringstruktur.tex
new file mode 100644
index 0000000..d653300
--- /dev/null
+++ b/vorlesungen/slides/3/ringstruktur.tex
@@ -0,0 +1,50 @@
+%
+% ringstruktur.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Ringstruktur}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.46\textwidth}
+\begin{block}{Ring}
+Menge $R$ mit zwei zweistelligen Verknüfpungen $+$ und $\cdot$
+mit
+\begin{enumerate}
+\item<3->
+$R$ ist abelsche Gruppe bezüglich $+$
+\item<5->
+$R\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$
+\item<7->
+Für alle $a,b,c\in R$
+\begin{align*}
+a(b+c) &= ab+ac
+\\
+(a+b)c &= ac+bc
+\end{align*}
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<2->{%
+\begin{block}{Polynomring}
+$R$ ein Ring, $R[X]$ ``erbt'' Addition und Multiplikation mit
+\begin{enumerate}
+\item<4->
+$R[X]$ ist abelsche Gruppe bezüglich $+$
+\item<6->
+$R[X]\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$
+\item<8->
+Für alle $a,b,c\in R[X]$
+\begin{align*}
+a(b+c) &= ab+ac
+\\
+(a+b)c &= ac+bc
+\end{align*}
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/teilbarkeit.tex b/vorlesungen/slides/3/teilbarkeit.tex
new file mode 100644
index 0000000..a5ea9b9
--- /dev/null
+++ b/vorlesungen/slides/3/teilbarkeit.tex
@@ -0,0 +1,47 @@
+%
+% teilbarkeit.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Teilen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Teilen in $\mathbb{Z}$}
+Zu zwei Zahlen $a,b\in \mathbb{Z}$, \only<3->{{\color<3-4>{red}$a>b$}, }gibt es
+immer \only<3->{{\color<3-4>{red}genau}} ein Paar $q,r\in\mathbb{Z}$ derart, dass
+\begin{align*}
+a&=bq+r
+\\
+\uncover<3->{{\color<3-4>{red}r}&{\color<3-4>{red}< b}}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Teilen in $\mathbb{Q}[X]$}
+Zu zwei Polynomen $a,b\in\mathbb{Q}[X]$, \only<4->{{\color<4>{red}$\deg a > \deg b$},}
+gibt es 
+immer \only<4->{{\color<4>{red}bis auf eine Einheit genau }}%
+ein Paar $q,r\in\mathbb{Q}[X]$ derart, dass
+\begin{align*}
+a&=bq+r
+\\
+\uncover<4->{{\color<4>{red}\deg r}&{\color<4>{red}< \deg b}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<5->{%
+\begin{block}{Allgemein: euklidischer Ring}
+Nullteilerfreier Ring $R$ mit einer Funktion
+$d\colon R\setminus{0}\to\mathbb{N}$ mit
+\begin{itemize}
+\item Für $x,y\in R$ gilt $d(xy) \ge d(x)$.
+\item Für $x,y\in R$ gibt es $q,r\in R$ derart
+$x=qy +r$ mit $d(y)>d(r)$
+\end{itemize}
+Euklidische Ringe haben ähnliche Eigenschaften wie Polynomringe
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex
new file mode 100644
index 0000000..d20bfc4
--- /dev/null
+++ b/vorlesungen/slides/3/wurzel2.tex
@@ -0,0 +1,83 @@
+%
+% wurzel2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\mathbb{Z}(\sqrt{2})\only<7->{ = \mathbb{Z}[X]/(X^2-2)}$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Der Ring $\mathbb{Z}(\sqrt{2})$}
+$\mathbb{Z}(\sqrt{2})$ als Teilring:
+{\color{blue}
+\[
+R=\{ a+b\sqrt{2}\;|\; a,b\in\mathbb{Z} \} \subset \mathbb{R}
+\]}%
+\uncover<2->{$\sqrt{2}\not\in\mathbb{Q}$}\uncover<3->{
+$\Rightarrow$
+$1$ und $\sqrt{2}$ sind inkommensurabel}\uncover<4->{
+$\Rightarrow$
+$R$ dicht in $\mathbb{R}$}
+\end{block}
+\uncover<5->{%
+\begin{block}{Algebraische Konstruktion}
+\uncover<8->{%
+Das Polynom $X^2-2$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$}
+\[
+\uncover<8->{\mathbb{Z}[X]/(X^2-2)
+=}
+{\color{red}\{a+bX\;|\;a,b\in\mathbb{Z}\}}
+\]\uncover<7->{%
+mit Rechenregel: $X^2=2$}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Körper}
+$\mathbb{Q}(\sqrt{2}) = \mathbb{Q}[X]/(X^2-2)$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+\begin{scope}
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\foreach \x in {-10,...,10}{
+	\pgfmathparse{int(\x/sqrt(2))-5}
+	\xdef\s{\pgfmathresult}
+	\pgfmathparse{int(\x/sqrt(2))+5}
+	\xdef\t{\pgfmathresult}
+	\foreach \y in {\s,...,\t}{
+		\uncover<4->{
+			\fill[color=blue] ({\x-\y*sqrt(2)},0)
+				circle[radius=0.05];
+		}
+		\uncover<6->{
+			\draw[color=blue,line width=0.1pt] 
+				({\x-\y*sqrt(2)-3.2},3.2) 
+				--
+				({\x-\y*sqrt(2)+3.2},-3.2);
+		}
+	}
+}
+\end{scope}
+
+\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
+
+\uncover<5->{
+	\draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}];
+
+	\foreach \x in {-3,...,3}{
+		\foreach \y in {-2,...,2}{
+			\fill[color=red]
+				({\x},{\y*sqrt(2)}) circle[radius=0.08];
+		}
+	}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
new file mode 100644
index 0000000..ad1081e
--- /dev/null
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -0,0 +1,22 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter4 =								\
+	../slides/4/ggt.tex						\
+	../slides/4/euklidmatrix.tex					\
+	../slides/4/euklidbeispiel.tex					\
+	../slides/4/euklidtabelle.tex					\
+	../slides/4/fp.tex						\
+	../slides/4/division.tex					\
+	../slides/4/gauss.tex						\
+	../slides/4/dh.tex						\
+	../slides/4/divisionpoly.tex					\
+	../slides/4/euklidpoly.tex					\
+	../slides/4/polynomefp.tex					\
+	../slides/4/schieberegister.tex					\
+	../slides/4/alpha.tex						\
+	../slides/4/chapter.tex
+
diff --git a/vorlesungen/slides/4/alpha.tex b/vorlesungen/slides/4/alpha.tex
new file mode 100644
index 0000000..3cd54c0
--- /dev/null
+++ b/vorlesungen/slides/4/alpha.tex
@@ -0,0 +1,54 @@
+%
+% alpha.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\frametitle{Was ist $\alpha$?}
+$m(X)$ ein irreduzibles Polynome in $\mathbb{F}_2[X]$
+
+Beispiel: $m(X) = X^8{\color{red}\mathstrut+X^4+X^3+X^2+1}\in\mathbb{F}_2[X]$
+\begin{columns}[t]
+\begin{column}{0.40\textwidth}
+\uncover<2->{%
+\begin{block}{Abstrakt}
+$\alpha$ ist ein ``imaginäres''
+Objekt mit der Rechenregel $m(\alpha)=0$
+\begin{align*}
+\alpha^8 &= {\color{red}\alpha^4+\alpha^3+\alpha^2+1}\\
+\uncover<3->{
+\alpha^9 &= \alpha^5+\alpha^4+\alpha^3+\alpha}\\
+\uncover<4->{
+\alpha^{10}&= \alpha^6+\alpha^5+\alpha^4+\alpha^2}\\
+\uncover<5->{
+\alpha^{11}&= \alpha^7+\alpha^6+\alpha^5+\alpha^3}\\
+\uncover<6->{
+\alpha &= \alpha^7+\alpha^3+\alpha^2+\alpha}
+\\
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.54\textwidth}
+\uncover<7->{%
+\begin{block}{Matrix}
+Eine konkretes Element in $M_n(\mathbb{F}_2)$
+\[
+\alpha
+=
+\begin{pmatrix}
+0& 0& 0& 0& 0& 0& 0& {\color{red}1}\\
+1& 0& 0& 0& 0& 0& 0& {\color{red}0}\\
+0& 1& 0& 0& 0& 0& 0& {\color{red}1}\\
+0& 0& 1& 0& 0& 0& 0& {\color{red}1}\\
+0& 0& 0& 1& 0& 0& 0& {\color{red}1}\\
+0& 0& 0& 0& 1& 0& 0& {\color{red}0}\\
+0& 0& 0& 0& 0& 1& 0& {\color{red}0}\\
+0& 0& 0& 0& 0& 0& 1& {\color{red}0}
+\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
new file mode 100644
index 0000000..a10712a
--- /dev/null
+++ b/vorlesungen/slides/4/chapter.tex
@@ -0,0 +1,18 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{4/ggt.tex}
+\folie{4/euklidmatrix.tex}
+\folie{4/euklidbeispiel.tex}
+\folie{4/euklidtabelle.tex}
+\folie{4/fp.tex}
+\folie{4/division.tex}
+\folie{4/gauss.tex}
+\folie{4/dh.tex}
+\folie{4/divisionpoly.tex}
+\folie{4/euklidpoly.tex}
+\folie{4/polynomefp.tex}
+\folie{4/alpha.tex}
+\folie{4/schieberegister.tex}
diff --git a/vorlesungen/slides/4/dh.tex b/vorlesungen/slides/4/dh.tex
new file mode 100644
index 0000000..b0a88e5
--- /dev/null
+++ b/vorlesungen/slides/4/dh.tex
@@ -0,0 +1,62 @@
+%
+% dh.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Diffie-Hellmann Schlüsselaushandlung}
+
+\begin{center}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\skala{0.95}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\def\l{2.5}
+\fill[color=blue!20] (-7,-6.5) rectangle (7,0.5);
+\fill[color=red!20] (-\l,-6.5) rectangle (\l,0.501);
+\node[color=red] at (0,-1.5) {öffentliches Netzwerk};
+\node[color=blue] at (-7,0.2) [right] {privat};
+\node[color=blue] at (7,0.2) [left] {privat};
+\coordinate (A) at (-\l,-2.5);
+\coordinate (C) at (\l,-5.0);
+\coordinate (B) at (\l,-2.5);
+\coordinate (D) at (-\l,-5.0);
+\node at (0,0) {$p\in\mathbb{N},g\in\mathbb{F}_p$ aushandeln};
+\fill[color=white] (-\l,-0.7) circle[radius=0.3];
+\draw (-\l,-0.7) circle[radius=0.3];
+\fill[color=white] (\l,-0.7) circle[radius=0.3];
+\draw (\l,-0.7) circle[radius=0.3];
+\node at (-\l,-0.7) {$A$};
+\node at (\l,-0.7) {$B$};
+\uncover<2->{
+	\node at (-\l,-1.5) [left] {$a$ auswählen\strut};
+	\node at (-\l,-2.0) [left] {$x=g^a\in\mathbb{F}_p$\strut};
+	\node at (\l,-1.5) [right] {$b$ auswählen\strut};
+	\node at (\l,-2.0) [right] {$y=g^b\in\mathbb{F}_p$\strut};
+}
+\draw[->] (-\l,-1) -- (-\l,-6);
+\draw[->] (\l,-1) -- (\l,-6);
+\uncover<3->{
+	\draw[->] (A) -- (C);
+	\draw[->] (B) -- (D);
+	\fill (A) circle[radius=0.08];
+	\fill (B) circle[radius=0.08];
+	\node at ($0.8*(A)+0.2*(C)+(-0.4,0)$) [above right] {$x=g^a$};
+	\node at ($0.8*(B)+0.2*(D)+(0.4,0)$) [above left] {$y=g^b$};
+}
+\uncover<4->{
+	\node at (-\l,-5.0) [left] {$s=g^{ab}=y^a\in\mathbb{F}_p$};
+	\node at (-\l,-5.5) [left] {ausrechnen};
+	\node at (\l,-5.0) [right] {$s=g^{ab}=x^b\in\mathbb{F}_p$};
+	\node at (\l,-5.5) [right] {ausrechnen};
+}
+\uncover<5->{
+	\fill[rounded corners=0.3cm,color=darkgreen!20]
+		({-\l-1.7},-7) rectangle ({\l+1.7},-6);
+	\draw[rounded corners=0.3cm] ({-\l-1.7},-7) rectangle ({\l+1.7},-6);
+	\node at (0,-6.5) {$A$ und $B$ haben den gemeinsamen Schlüssel $s$};
+}
+\end{tikzpicture}
+
+\end{center}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/division.tex b/vorlesungen/slides/4/division.tex
new file mode 100644
index 0000000..846738f
--- /dev/null
+++ b/vorlesungen/slides/4/division.tex
@@ -0,0 +1,65 @@
+%
+% division.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Division in $\mathbb{F}_p$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Inverse {\bf berechnen}}
+Gegeben $a\in\mathbb{F}_p$, finde $b=a^{-1}\in\mathbb{F}_p$
+\begin{align*}
+\uncover<2->{&& a{\color{blue}b} &\equiv 1 \mod p}
+\\
+\uncover<3->{&\Leftrightarrow& a{\color{blue}b}&=1 + {\color{blue}n}p}
+\\
+\uncover<4->{&&a{\color{blue}b}-{\color{blue}n}p&=1}
+\end{align*}
+\uncover<5->{Wegen
+$\operatorname{ggT}(a,p)=1$ gibt es
+$s$ und $t$ mit
+\[
+{\color{red}s}a+{\color{red}t}p=1
+\Rightarrow
+{\color{blue}b}={\color{red}s},\;
+{\color{blue}n}=-{\color{red}t}
+\]}
+\uncover<6->{%
+$\Rightarrow$ Die Inverse kann mit dem euklidischen Algorithmus
+berechnet werden}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Beispiel in $\mathbb{F}_{1291}$}
+Finde $47^{-1}\in\mathbb{F}_{1291}$
+%\vspace{-10pt}
+\begin{center}
+\begin{tabular}{|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|}
+\hline
+k& a_k& b_k&q_k&  c_k& d_k\\
+\hline
+ &    &    &   &    1&   0\\
+0&  47&1291&\uncover<8->{  0}&    0&   1\\
+1&\uncover<9->{ 1291&  47}&\uncover<11->{ 27}&\uncover<10->{    1&   0}\\
+2&\uncover<12->{  47&  22}&\uncover<14->{  2}&\uncover<13->{  -27&   1}\\
+3&\uncover<15->{  22&   3}&\uncover<17->{  7}&\uncover<16->{   55&  -2}\\
+4&\uncover<18->{   3&   1}&\uncover<20->{  3}&\uncover<19->{{\color{red}-412}&{\color{red}15}}\\
+5&\uncover<21->{   1&   0}&   &\uncover<22->{ 1291& -47}\\
+\hline
+\end{tabular}
+\end{center}
+\uncover<23->{%
+\[
+{\color{red}-412}\cdot 47 +{\color{red}15}\cdot 1291 = 1
+\uncover<24->{\;\Rightarrow\;
+47^{-1}={\color{red}879}}
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/divisionpoly.tex b/vorlesungen/slides/4/divisionpoly.tex
new file mode 100644
index 0000000..5e71c95
--- /dev/null
+++ b/vorlesungen/slides/4/divisionpoly.tex
@@ -0,0 +1,37 @@
+%
+% divisionpoly.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynomdivision in $\mathbb{F}_3[X]$}
+Rechenregeln in $\mathbb{F}_3$: $1+2=0$, $2\cdot 2 = 1$
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcrcrcrc}
+\llap{$ ($}X^4&+&X^3&+& X^2&+& X&+&1\rlap{$)$}&\;\;:&(X^2&+&X&+&2)&=&\uncover<2->{X^2}&\uncover<5->{+&2=q}\\
+\uncover<3->{\llap{$-($}X^4&+&X^3&+&2X^2\rlap{$)$}}& & & &           &     &    & & & &  & &   & &  &   \\
+\uncover<4->{              & &   & &2X^2&+& X&+& 1}        &     &    & & & &  & &   & &   \\
+\uncover<6->{              & &   & &\llap{$-($}2X^2&+&2X&+&          2\rlap{$)$}}&     &    & & & &  & &   & &   \\
+\uncover<7->{              & &   & &    & &2X&+&2\rlap{$\mathstrut=r$}&     &    & & & &  & &  &  &}
+\end{array}
+\]
+\uncover<8->{%
+Kontrolle:
+\[
+\arraycolsep=1.4pt
+\begin{array}{rclcrcr}
+(\underbrace{X^2+2}_{\displaystyle=q})
+(X^2+X+2)
+              &=&\rlap{$\uncover<9->{X^4+X^3+2X^2}\uncover<10->{ + 2X^2+2X+2}$}
+\\
+\uncover<11->{&=& X^4+X^3+X^2&+&2X&+&2}
+\\
+\uncover<12->{& &            &&\llap{$r=\mathstrut$}2X&+&2}
+\\
+\uncover<13->{&=& X^4+X^3+X^2&+&1X&+&1}
+\end{array}
+\]
+}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidbeispiel.tex b/vorlesungen/slides/4/euklidbeispiel.tex
new file mode 100644
index 0000000..366a7a6
--- /dev/null
+++ b/vorlesungen/slides/4/euklidbeispiel.tex
@@ -0,0 +1,78 @@
+%
+% euklidmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Euklidischer Algorithmus: Beispiel}
+\setlength{\abovedisplayskip}{0pt}
+\setlength{\belowdisplayskip}{0pt}
+\vspace{-0pt}
+\begin{block}{Finde $\operatorname{ggT}(25,15)$}
+\vspace{-12pt}
+\begin{align*}
+a_0&=25 & b_0 &= 15 &\uncover<2->{25&=15 \cdot {\color{orange}      1} + 10 &q_0 &= {\color{orange}1} & r_0 &= 10}\\
+\uncover<3->{a_1&=15 & b_1 &= 10}&\uncover<4->{15&=10 \cdot {\color{darkgreen}1} + \phantom{0}5  &q_1 &= {\color{darkgreen}1} & r_1 &= \phantom{0}5}\\
+\uncover<5->{a_2&=10 & b_2 &= \phantom{0}5}&\uncover<6->{10&=\phantom{0}5  \cdot {\color{blue}     2} + \phantom{0}0  &q_2 &= {\color{blue}2} & r_2 &= \phantom{0}0 }
+\end{align*}
+\end{block}
+\vspace{-5pt}
+\uncover<7->{%
+\begin{block}{Matrix-Operationen}
+\begin{align*}
+Q
+&=
+\uncover<9->{Q({\color{blue}2})}
+\uncover<8->{Q({\color{darkgreen}1})}
+Q({\color{orange}1})
+=
+\uncover<9->{
+\begin{pmatrix*}[r]0&1\\1&-{\color{blue}2}\end{pmatrix*}
+}
+\uncover<8->{
+\begin{pmatrix*}[r]0&1\\1&-{\color{darkgreen}1}\end{pmatrix*}
+}
+\begin{pmatrix*}[r]0&1\\1&-{\color{orange}1}\end{pmatrix*}
+=
+\ifthenelse{\boolean{presentation}}{
+\only<7>{
+\begin{pmatrix*}[r]\phantom{-}0&1\\1&-1\end{pmatrix*}
+}
+\only<8>{
+\begin{pmatrix*}[r]
+1&-1\\-1&2
+\end{pmatrix*}
+}
+}{}
+\only<9->{
+\begin{pmatrix*}[r]
+{\color{red}-1}&{\color{red}2}\\3&-5
+\end{pmatrix*}}
+\end{align*}
+\end{block}}
+\vspace{-5pt}
+\uncover<10->{%
+\begin{block}{Relationen ablesen}
+\[
+\begin{pmatrix}
+\operatorname{ggT}(a,b)\\0
+\end{pmatrix}
+=
+Q
+\begin{pmatrix}a\\b\end{pmatrix}
+\uncover<11->{%
+\quad
+\Rightarrow\quad
+\left\{
+\begin{aligned}
+\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 =
+{\color{red}-1}\cdot {\usebeamercolor[fg]{title}25} + {\color{red}2}\cdot {\usebeamercolor[fg]{title}15} \\
+ 0                        &= \phantom{5=-}3\cdot {\usebeamercolor[fg]{title}25} -5\cdot {\usebeamercolor[fg]{title}15}
+\end{aligned}
+\right.}
+\]
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex
new file mode 100644
index 0000000..be5b3ca
--- /dev/null
+++ b/vorlesungen/slides/4/euklidmatrix.tex
@@ -0,0 +1,108 @@
+%
+% euklidmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Matrixform des euklidischen Algorithmus}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.52\textwidth}
+\begin{block}{Einzelschritt}
+\vspace{-10pt}
+\[
+a_k = b_kq_k + r_k
+\uncover<2->{
+\;\Rightarrow\;
+\left\{
+\begin{aligned}
+a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\
+b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k 
+\end{aligned}
+\right.}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.44\textwidth}
+\uncover<3->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-10pt}
+\begin{align*}
+\begin{pmatrix}
+a_{k+1}\\
+b_{k+1}
+\end{pmatrix}
+&=
+\begin{pmatrix}
+b_k\\r_k
+\end{pmatrix}
+=
+\uncover<4->{
+\underbrace{\begin{pmatrix}
+\uncover<5->{0&1}\\
+\uncover<6->{1&-q_k}
+\end{pmatrix}}_{\uncover<7->{\displaystyle =Q(q_k)}}
+}
+\begin{pmatrix}
+a_k\\b_k
+\end{pmatrix}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<8->{%
+\begin{block}{Ende des Algorithmus}
+\vspace{-10pt}
+\begin{align*}
+\uncover<9->{
+\begin{pmatrix}
+a_{n+1}\\
+b_{n+1}\\
+\end{pmatrix}
+&=}
+\begin{pmatrix}
+r_{n-1}\\
+r_{n}
+\end{pmatrix}
+=
+\begin{pmatrix}
+\operatorname{ggT}(a,b) \\
+0
+\end{pmatrix}
+\uncover<11->{
+=
+\underbrace{\uncover<15->{Q(q_n)}
+\uncover<14->{\dots}
+\uncover<13->{Q(q_1)}
+\uncover<12->{Q(q_0)}}_{\displaystyle =Q}}
+\uncover<10->{
+\begin{pmatrix} a_0\\ b_0\end{pmatrix}
+\uncover<6->{
+=
+Q\begin{pmatrix}a\\b\end{pmatrix}
+}
+}
+\end{align*}
+\end{block}}
+\uncover<16->{%
+\begin{block}{Konsequenzen}
+\[
+Q=\begin{pmatrix}
+q_{11}&q_{12}\\
+q_{21}&q_{22}
+\end{pmatrix}
+\quad\Rightarrow\quad
+\left\{
+\quad
+\begin{aligned}
+\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b = {\color{red}s}a+{\color{red}t}b\\
+                      0 &= q_{21}a + q_{22}b
+\end{aligned}
+\right.
+\]
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidpoly.tex b/vorlesungen/slides/4/euklidpoly.tex
new file mode 100644
index 0000000..432b6b4
--- /dev/null
+++ b/vorlesungen/slides/4/euklidpoly.tex
@@ -0,0 +1,47 @@
+%
+% euklidpoly.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Euklidischer Algorithmus in $\mathbb{F}_2[X]$}
+Gegeben: $m(X)=X^4+X+1$, $b(X) = {\color{blue}X^2+1}$
+\\
+\uncover<2->{Berechne $s,t\in\mathbb{F}_2[X]$ derart, dass $sm+tb=1$}
+\uncover<3->{%
+\begin{center}
+\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|}
+\hline
+k&    a_k&  b_k&  q_k&r_k&  c_k&    d_k\\
+\hline
+ &       &     &     &   &    1&      0\\
+0&X^4+X+1&{\color{blue}X^2+1}&\uncover<4->{X^2+1}&\uncover<4->{X}&    0&      1\\
+1&\uncover<5->{X^2+1 }&\uncover<5->{X}&\uncover<5->{X}&\uncover<5->{1}&\uncover<5->{1}&\uncover<5->{X^2+1}\\
+2&\uncover<6->{X     }&\uncover<6->{1}&\uncover<6->{X}&\uncover<6->{0}&\uncover<6->{{\color{red}X}}&\uncover<6->{{\color{red}X^3+X+1}}\\
+3&\uncover<7->{1     }&\uncover<7->{0}&&&\uncover<7->{X^2+1}&\uncover<7->{X^4+X+1} \\
+\hline
+\end{tabular}
+\end{center}}
+\ifthenelse{\boolean{presentation}}{
+\only<8->{%
+\begin{block}{Kontrolle}
+\vspace{-10pt}
+\begin{align*}
+{\color{red}X}\cdot (X^4+X+1) + ({\color{red}X^3+X+1})({\color{blue}X^2+1})
+&\uncover<9->{=
+(X^5+X^2+X)}\\
+&\qquad \uncover<10->{+ (X^5+X^3+X^2+X^3+X+1)}
+\\
+&\uncover<11->{=(X^5+X^2+X) + (X^5+X^2+X+1)}
+\\
+&\uncover<12->{=1}
+\end{align*}
+\end{block}}}{}
+\begin{block}{Rechenregeln in $\mathbb{F}_2$}
+$1+1=0$,
+$2=0$, $+1=-1$.
+\end{block}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidtabelle.tex b/vorlesungen/slides/4/euklidtabelle.tex
new file mode 100644
index 0000000..3f1b8d7
--- /dev/null
+++ b/vorlesungen/slides/4/euklidtabelle.tex
@@ -0,0 +1,73 @@
+%
+% euklidtabelle.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Durchführung des euklidischen Algorithmus}
+Problem: Berechnung der Produkte $Q(q_k)\cdots Q(q_1)Q(q_0)$ für $k=0,1,\dots,n$
+\uncover<2->{%
+\begin{block}{Multiplikation mit $Q(q_k)$}
+\vspace{-12pt}
+\begin{align*}
+Q(q_k)
+\ifthenelse{\boolean{presentation}}{
+\only<-3>{
+\begin{pmatrix}
+u&v\\c&d
+\end{pmatrix}
+=\begin{pmatrix}
+0&1\\1&-q_k
+\end{pmatrix}
+}}{}
+\begin{pmatrix}
+u&v\\c&d
+\end{pmatrix}
+&\uncover<3->{=
+\begin{pmatrix}
+c&d\\
+u-q_kc&v-q_kd
+\end{pmatrix}}
+&&\uncover<5->{\Rightarrow&
+\begin{pmatrix}
+c_k&d_k\\c_{k+1}&d_{k+1}
+\end{pmatrix}
+&=
+Q(q_k)
+%\begin{pmatrix}
+%0&1\\1&-q_k
+%\end{pmatrix}
+\begin{pmatrix}
+c_{k-1}&d_{k-1}\\c_{k}&d_{k}
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<6->{%
+\begin{equation*}
+\begin{tabular}{|>{\tiny$}r<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|}
+\hline
+k  &q_k    &                     c_k    &                     d_k    \\
+\hline
+-1 &       &                     1      &                     0      \\
+ 0 &\uncover<7->{q_0    }&                     0      &                     1      \\
+ 1 &\uncover<9->{q_1    }&\uncover<8->{c_{-1} -q_0    \cdot c_0    &d_{-1} -q_0    \cdot d_0    }\\
+ 2 &\uncover<11->{q_2    }&\uncover<10->{c_0    -q_1    \cdot c_1    &d_0    -q_1    \cdot d_1    }\\
+\vdots&\uncover<12->{\vdots}&\uncover<12->{\vdots                    &\vdots                      }\\
+ n &\uncover<14->{q_n    }&\uncover<13->{{\color{red}c_{n-2}-q_{n-1}\cdot c_{n-1}}&{\color{red}d_{n-2}-q_{n-1}\cdot d_{n-1}}}\\
+n+1&       &\uncover<15->{c_{n-1}-q_{n}  \cdot c_{n}  &d_{n-1}-q_{n}  \cdot d_{n}  }\\
+\hline
+\end{tabular}
+\uncover<16->{
+\Rightarrow
+\left\{
+\begin{aligned}
+\rlap{${\color{red}c_{n}}$}\phantom{c_{n+1}} a + \rlap{${\color{red}d_n}$}\phantom{d_{n+1}}b &= \operatorname{ggT}(a,b)
+\\
+c_{n+1} a + d_{n+1} b &= 0
+\end{aligned}
+\right.}
+\end{equation*}}
+\end{frame}
diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex
new file mode 100644
index 0000000..968b777
--- /dev/null
+++ b/vorlesungen/slides/4/fp.tex
@@ -0,0 +1,178 @@
+%
+% fp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\feld#1#2#3{
+	\node at ({#1},{5-#2}) {$#3$};
+}
+\def\geld#1#2#3{
+	\node at ({#1},{6-#2}) {$#3$};
+}
+\def\rot#1#2{
+	\fill[color=red!20] ({#1-0.5},{5-#2-0.5}) rectangle ({#1+0.5},{5-#2+0.5});
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\gruen#1#2{
+	\fill[color=darkgreen!20] ({#1-0.5},{6-#2-0.5}) rectangle ({#1+0.5},{6-#2+0.5});
+}
+\def\inverse#1#2{
+	\node at (9,{6-#1}) {$#1^{-1}=#2\mathstrut$};
+}
+\begin{frame}[t]
+\frametitle{Galois-Körper}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Restklassenring$\mathstrut$}
+$\mathbb{Z}/n\mathbb{Z}
+=\{ \llbracket r\rrbracket\;|\; 0\le r < n \} \mathstrut$
+ist ein Ring
+\end{block}
+\uncover<2->{%
+\begin{block}{Nullteiler}
+Falls $n=n_1n_2$, dann sind $\llbracket n_1\rrbracket$ und
+$\llbracket n_2\rrbracket$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$:
+\[
+\llbracket n_1\rrbracket
+\llbracket n_2\rrbracket
+=
+\llbracket n_1n_2 \rrbracket
+=
+\llbracket n\rrbracket
+=
+\llbracket 0 \rrbracket
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Galois-Körper $\mathbb{F}_p\mathstrut$}
+$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\mathstrut$
+\end{block}}
+\uncover<4->{%
+\begin{block}{$n$ prim}
+Für $n=p$ prim ist $\mathbb{Z}/n\mathbb{Z}$ nullteilerfrei
+\medskip
+
+\uncover<5->{
+$\Rightarrow \quad \mathbb{F}_p$ ist ein Körper
+}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.45]
+\fill[color=white] (-12,0) circle[radius=0.1];
+\fill[color=white] (12,0) circle[radius=0.1];
+\uncover<3->{
+\begin{scope}[xshift=-8cm]
+\rot{2}{3}
+\rot{4}{3}
+\rot{3}{2}
+\rot{3}{4}
+\fill[color=gray!40] (-0.5,5.5) rectangle (5.5,6.5);
+\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,5.5);
+\foreach \x in {-0.5,5.5}{
+	\draw (\x,-0.5) -- (\x,6.5);
+}
+\foreach \x in {0.5,...,4.5}{
+	\draw[line width=0.3pt] (\x,-0.5) -- (\x,6.5);
+}
+\foreach \y in {0.5,...,5.5}{
+	\draw[line width=0.3pt] (-1.5,\y) -- (5.5,\y);
+}
+\foreach \y in {-0.5,5.5}{
+	\draw (-1.5,\y) -- (5.5,\y);
+}
+\draw (-1.5,-0.5) -- (-1.5,5.5);
+\draw (-0.5,6.5) -- (5.5,6.5);
+\foreach \x in {0,...,5}{
+	\node at (\x,6) {$\x$};
+	\node at (-1,{5-\x}) {$\x$};
+}
+\foreach \x in {0,...,5}{
+	\feld{\x}{0}{0}
+	\feld{0}{\x}{0}
+}
+\foreach \x in {2,...,5}{
+	\feld{\x}{1}{\x}
+	\feld{1}{\x}{\x}
+}
+\feld{1}{1}{1}
+\feld{2}{2}{4}
+\feld{2}{3}{0} \feld{3}{2}{0}
+\feld{2}{4}{2} \feld{4}{2}{2}
+\feld{2}{5}{4} \feld{5}{2}{4}
+\feld{3}{3}{3}
+\feld{4}{3}{0} \feld{3}{4}{0}
+\feld{5}{3}{3} \feld{3}{5}{3}
+\feld{4}{4}{4}
+\feld{4}{5}{2} \feld{5}{4}{2}
+\feld{5}{5}{1}
+\end{scope}}
+\uncover<6->{
+\begin{scope}[xshift=6cm]
+\uncover<7->{ \gruen{1}{1} }
+\uncover<8->{ \gruen{4}{2} }
+\uncover<9->{ \gruen{5}{3} }
+\uncover<10->{ \gruen{2}{4} }
+\uncover<11->{ \gruen{3}{5} }
+\uncover<12->{ \gruen{6}{6} }
+\fill[color=gray!40] (-0.5,6.5) rectangle (6.5,7.5);
+\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,6.5);
+\foreach \x in {-0.5,6.5}{
+	\draw (\x,-0.5) -- (\x,7.5);
+}
+\foreach \x in {0.5,...,5.5}{
+	\draw[line width=0.3pt] (\x,-0.5) -- (\x,7.5);
+}
+\foreach \y in {0.5,...,6.5}{
+	\draw[line width=0.3pt] (-1.5,\y) -- (6.5,\y);
+}
+\foreach \y in {-0.5,6.5}{
+	\draw (-1.5,\y) -- (6.5,\y);
+}
+\draw (-1.5,-0.5) -- (-1.5,6.5);
+\draw (-0.5,7.5) -- (6.5,7.5);
+\foreach \x in {0,...,6}{
+	\node at (\x,7) {$\x$};
+	\node at (-1,{6-\x}) {$\x$};
+}
+\foreach \x in {0,...,6}{
+	\geld{\x}{0}{0}
+	\geld{0}{\x}{0}
+}
+\foreach \x in {2,...,6}{
+	\geld{\x}{1}{\x}
+	\geld{1}{\x}{\x}
+}
+\geld{1}{1}{1}
+\geld{2}{2}{4}
+\geld{2}{3}{6} \geld{3}{2}{6}
+\geld{2}{4}{1} \geld{4}{2}{1}
+\geld{2}{5}{3} \geld{5}{2}{3}
+\geld{2}{6}{5} \geld{6}{2}{5}
+\geld{3}{3}{2}
+\geld{4}{3}{5} \geld{3}{4}{5}
+\geld{5}{3}{1} \geld{3}{5}{1}
+\geld{6}{3}{4} \geld{3}{6}{4}
+\geld{4}{4}{2}
+\geld{5}{4}{6} \geld{4}{5}{6}
+\geld{6}{4}{3} \geld{4}{6}{3}
+\geld{5}{5}{4}
+\geld{6}{5}{2} \geld{5}{6}{2}
+\geld{6}{6}{1}
+\uncover<7->{ \inverse{1}{1} }
+\uncover<8->{ \inverse{2}{4} }
+\uncover<9->{ \inverse{3}{5} }
+\uncover<10->{ \inverse{4}{2} }
+\uncover<11->{ \inverse{5}{3} }
+\uncover<12->{ \inverse{6}{6} }
+\end{scope}}
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex
new file mode 100644
index 0000000..23cdfee
--- /dev/null
+++ b/vorlesungen/slides/4/gauss.tex
@@ -0,0 +1,143 @@
+%
+% gauss.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\def\ds{0.5}
+\def\punkt#1#2{({(#1)*\ds},{-(#2)*\ds})}
+\def\tabelle{
+	\foreach \x in {-0.5,0.5,3.5}{
+		\draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5};
+		\draw \punkt{-0.5}{\x} -- \punkt{3.5}{\x};
+	}
+	\node at \punkt{0}{1} {$0$};
+	\node at \punkt{0}{2} {$1$};
+	\node at \punkt{0}{3} {$2$};
+	\node at \punkt{1}{0} {$0$};
+	\node at \punkt{2}{0} {$1$};
+	\node at \punkt{3}{0} {$2$};
+}
+\begin{frame}[t]
+\frametitle{Gauss-Algorithmus in $\mathbb{F}_3$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Additions-/Multiplikationstabelle}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}[xshift=-1.6cm]
+\tabelle
+\node at \punkt{0}{0} {$+$};
+\node at \punkt{1}{1} {$0$};
+\node at \punkt{1}{2} {$1$};
+\node at \punkt{1}{3} {$2$};
+\node at \punkt{2}{1} {$1$};
+\node at \punkt{2}{2} {$2$};
+\node at \punkt{2}{3} {$0$};
+\node at \punkt{3}{1} {$2$};
+\node at \punkt{3}{2} {$0$};
+\node at \punkt{3}{3} {$1$};
+\end{scope}
+\begin{scope}[xshift=1.6cm]
+\tabelle
+\node at \punkt{0}{0} {$\cdot$};
+\node at \punkt{1}{1} {$0$};
+\node at \punkt{1}{2} {$0$};
+\node at \punkt{1}{3} {$0$};
+\node at \punkt{2}{1} {$0$};
+\node at \punkt{2}{2} {$1$};
+\node at \punkt{2}{3} {$2$};
+\node at \punkt{3}{1} {$0$};
+\node at \punkt{3}{2} {$2$};
+\node at \punkt{3}{3} {$1$};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<2->{%
+\begin{block}{Gleichungssystem\uncover<9->{/Lösung}}
+\[
+\left.
+\begin{array}{rcrcrcrcr}
+ x&+&y&+2z&=&1\\
+2x& & &+ z&=&2\\
+ x&+&y&   &=&2
+\end{array}
+\uncover<9->{
+\right\}
+\Rightarrow
+\left\{
+\begin{aligned}
+x&=2\\
+y&=0\\
+z&=1
+\end{aligned}
+\right.}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Gauss-Algorithmus}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {\begin{minipage}{13cm}%
+\[
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 2 & 1 \\
+ 2 & 0 & 1 & 2 \\
+ 1 & 1 & 0 & 2 \\
+\hline
+\end{tabular}
+\uncover<5->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 2 & 1 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\uncover<7->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 0 & 2 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\uncover<9->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 0 & 0 & 2 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\]
+\end{minipage}};
+\begin{scope}[yshift=0.2cm]
+\uncover<4->{
+\draw[color=red] (-5.6,0.3) circle[radius=0.2];
+\draw[color=blue] (-5.4,-0.8) -- (-5.4,-0.2) arc (0:180:0.2) -- (-5.8,-0.8);
+}
+\uncover<6->{
+\draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5);
+}
+\uncover<8->{
+\draw[color=blue] (1.05,0.5) -- (1.05,0.2) arc (180:360:0.2) -- (1.45,0.5);
+}
+\end{scope}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex
new file mode 100644
index 0000000..ef97182
--- /dev/null
+++ b/vorlesungen/slides/4/ggt.tex
@@ -0,0 +1,75 @@
+%
+% ggt.tex -- GGT, Definition und Algorithmus
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschuöe
+%
+\begin{frame}[t]
+\frametitle{Grösster gemeinsamer Teiler}
+\vspace{-15pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gegeben: $a,b\in\mathbb Z$
+\\
+Gesucht: grösster gemeinsamer Teiler $\operatorname{ggT}(a,b)$
+\end{block}
+\uncover<4->{%
+\begin{block}{Euklidischer Algorithmus}
+$a_0 = a$, $b_0=b$
+\begin{align*}
+\uncover<5->{
+a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0}\\
+\uncover<6->{
+a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1}\\
+\uncover<7->{
+a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2}\\
+\uncover<8->{
+   &\;\vdots      &     &     &    &    }\\
+\uncover<9->{
+a_n&=b_nq_n + r_n & r_n &= 0  & r_{n-1}&=\operatorname{ggT}(a,b)}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{ggT}(15,25) = 5$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.09]
+\draw[->] (-1,0) -- (65,0) coordinate[label={$a$}];
+\draw[->] (0,-1) -- (0,65) coordinate[label={right:$b$}];
+\begin{scope}
+\clip (-1,-1) rectangle (65,65);
+\foreach \x in {0,...,4}{
+	\draw[line width=0.2pt] ({\x*15},-2) -- ({\x*15},65);
+}
+\foreach \y in {0,...,2}{
+	\draw[line width=0.2pt] (-2,{\y*25}) -- (65,{\y*25});
+}
+\uncover<3->{
+	\foreach \x in {0,5,...,120}{
+		\draw[color=blue] ({\x+2},-2) -- ({\x+2-70},{-2+70});
+		\node[color=blue] at ({0.5*\x-0.5},{0.5*\x-0.5})
+			[rotate=-45,above] {\tiny $\x$};
+	}
+}
+\uncover<2->{
+	\foreach \x in {0,...,4}{
+		\foreach \y in {0,...,2}{
+			\fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8];
+		}
+	}
+}
+\uncover<3->{
+	\foreach \x in {0,5,...,60}{
+		\fill[color=blue] (\x,0) circle[radius=0.5];
+		\node at (\x,0) [below] {\tiny $\x$};
+	}
+}
+\end{scope}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/polynomefp.tex b/vorlesungen/slides/4/polynomefp.tex
new file mode 100644
index 0000000..1db50e1
--- /dev/null
+++ b/vorlesungen/slides/4/polynomefp.tex
@@ -0,0 +1,62 @@
+%
+% polynomefp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynome über $\mathbb{F}_p[X]$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Polynomring}
+$\mathbb{F}_p[X]$ sind Polynome
+\[
+p(X)
+=
+a_0+a_1X+\dots+a_nX^n
+\]
+mit $a_i\in\mathbb{F}_p$.
+\uncover<2->{ObdA: $a_n=1$}%
+
+\end{block}
+\uncover<3->{%
+\begin{block}{Irreduzible Polynome}
+$m(X)$ ist irreduzibel, wenn es keine Faktorisierung
+$m(X)=p(X)q(X)$ mit $p,q\in\mathbb{F}_p[X]$ gibt
+\end{block}}
+\uncover<4->{%
+\begin{block}{Rest modulo $m(X)$}
+$X^{n+k}$ kann immer reduziert werden:
+\[
+X^{n+k} = -(a_0+a_1X+\dots+a_{n-1}X^{n-1})X^k
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Körper $\mathbb{F}_p/(m(X))$}
+Wenn $m(X)$ irreduzibel ist, dann ist
+$\mathbb{F}_p[X]$ nullteilerfrei.
+\medskip
+
+\uncover<6->{$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist}
+\begin{enumerate}
+\item<7->
+$\operatorname{ggT}(a,m) = 1$
+\item<8->
+Es gibt $s,t\in\mathbb{F}_p[X]$ mit
+\[
+s(X)m(X)+t(X)a(X) = 1
+\]
+(aus dem euklidischen Algorithmus)
+\item<9->
+$a^{-1} = t(X)$
+\end{enumerate}
+\uncover<9->{$\Rightarrow$ $\mathbb{F}_p[X]/(m(X))$ ist ein Körper
+mit genau $p^n$ Elementen}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/schieberegister.tex b/vorlesungen/slides/4/schieberegister.tex
new file mode 100644
index 0000000..f349337
--- /dev/null
+++ b/vorlesungen/slides/4/schieberegister.tex
@@ -0,0 +1,120 @@
+%
+% schieberegister.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\ds{0.7}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\punkt#1#2{({(#1)*\ds},{(#2)*\ds})}
+\def\rahmen{
+	\draw ({-0.5*\ds},{-0.5*\ds}) rectangle ({7.5*\ds},{0.5*\ds});
+	\foreach \x in {0.5,1.5,...,6.5}{
+		\draw ({\x*\ds},{-0.5*\ds}) rectangle ({\x*\ds},{0.5*\ds});
+	}
+}
+\def\polynom#1#2#3#4#5#6#7#8{
+	\node at \punkt{0}{0} {$#1$};
+	\node at \punkt{1}{0} {$#2$};
+	\node at \punkt{2}{0} {$#3$};
+	\node at \punkt{3}{0} {$#4$};
+	\node at \punkt{4}{0} {$#5$};
+	\node at \punkt{5}{0} {$#6$};
+	\node at \punkt{6}{0} {$#7$};
+	\node at \punkt{7}{0} {$#8$};
+}
+\begin{frame}[t]
+\frametitle{Implementation der Multiplikation in $\mathbb{F}_2(\alpha)$\uncover<10->{: Schieberegister}}
+Rechnen in $\mathbb{F}_2[X]$\only<5->{ und $\mathbb{F}_2(\alpha)$}
+ist speziell einfach
+\\
+Minimalpolynom von $\alpha$: ${\color{darkgreen}m(X) = X^8 + X^4+X^3+X+1}$
+(aus dem AES Standard)
+
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\uncover<4->{
+	\fill[color=blue!20]
+		\punkt{-0.5}{-0.5} rectangle \punkt{7.5}{0.5};
+}
+
+\uncover<2->{
+\begin{scope}
+	\rahmen
+	\node at \punkt{-0.5}{1} [left] {$p(X)=\mathstrut$};
+	\node at \punkt{0}{1} {$X^7$\strut};
+	\node at \punkt{2.5}{1}{$+$\strut};
+	\node at \punkt{3}{1} {$X^4$\strut};
+	\node at \punkt{4.5}{1}{$+$\strut};
+	\node at \punkt{5}{1} {$X^2$\strut};
+	\node at \punkt{6.5}{1}{$+$\strut};
+	\node at \punkt{7}{1} {$1$\strut};
+	\polynom10010101
+\end{scope}}
+
+\uncover<3->{
+	\draw[->] ({7.7*\ds},-0.2) to[out=-45,in=45] ({7.7*\ds},-1.8);
+	\node at ({8*\ds},-1) [right] {$\mathstrut\cdot X = \text{Shift}$};
+}
+\uncover<4->{
+	\foreach \x in {0,...,7}{
+		\draw[->,color=blue!40]
+			({\x*\ds},{-0.6*\ds}) -- ({(\x-1)*\ds},{-2+0.6*\ds});
+	}
+}
+
+\fill[color=white] (-4.65,0) circle[radius=0.01];
+
+\uncover<3->{
+	\begin{scope}[yshift=-2cm]
+		\uncover<4->{
+			\fill[color=blue!20]
+				\punkt{-1.5}{-0.5} rectangle \punkt{6.5}{0.5};
+			\rahmen
+			\polynom00101010
+		}
+		\node at \punkt{2}{1} {$X^5$\strut};
+		\node at \punkt{3.5}{1}{$+$\strut};
+		\node at \punkt{4}{1} {$X^3$\strut};
+		\node at \punkt{5.5}{1}{$+$\strut};
+		\node at \punkt{6}{1} {$X$\strut};
+		\begin{scope}[xshift=0.4cm]
+			\node at \punkt{-1}{1} [left]
+				{$\uncover<5->{{\color{darkgreen}\alpha^4+\alpha^3+\alpha+1=\alpha^8}}\only<-4>{X^8}$\strut};
+		\end{scope}
+		\node at \punkt{-1}{0} {$1$\strut};
+	\end{scope}
+}
+
+\uncover<6->{
+	{\color<8->{red}
+		\draw[->] (-2.5,-1.5) to[out=-90,in=180] (-0.5,-2.7);
+	}
+	\begin{scope}[yshift=-2.7cm]
+		\rahmen
+		{\color{darkgreen}
+			\polynom00011011
+		}
+	\end{scope}
+}
+
+\uncover<7->{
+	\node at ({3.5*\ds},-3.45) {$\|$};
+
+	\begin{scope}[yshift=-4.2cm]
+	\rahmen
+	\polynom00110001
+	\node at \punkt{7.6}{0} [right] {$\mathstrut=\alpha\cdot p(\alpha)$};
+	\end{scope}
+}
+
+\uncover<8->{
+	\node[color=red] at (-3.0,-2.5) {Feedback};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/Aiteration.tex b/vorlesungen/slides/5/Aiteration.tex
new file mode 100644
index 0000000..3078c55
--- /dev/null
+++ b/vorlesungen/slides/5/Aiteration.tex
@@ -0,0 +1,59 @@
+%
+% Aiteration.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Iteration von $A$}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.34\textwidth}
+\begin{block}{$\varrho(A) > 1\uncover<4->{\Rightarrow \|A^k\|\to\infty}$}
+\uncover<2->{%
+Eigenvektor $v$, $\|v\|=1$, zum Eigenwert $\lambda$ mit $|\lambda| > 1$}
+\uncover<3->{%
+\[
+\|A^kv\| = |\lambda|^k\to \infty
+\]}
+\uncover<4->{$\Rightarrow \|A\|^k\to\infty$}
+
+\end{block}
+\end{column}
+\begin{column}{0.63\textwidth}
+\begin{block}{$\varrho(A) < 1\uncover<12->{\Rightarrow \|A\|^k\to 0}$}
+\uncover<5->{%
+$A$ setzt sich zusammen aus Jordanblöcken:
+\[
+J(\lambda)^k
+=
+\renewcommand{\arraystretch}{1.2}
+\begin{pmatrix}
+\lambda^k&\binom{k}{1}\lambda^{k-1}&\binom{k}{2}\lambda^{k-2}
+	&\dots&\binom{k}{n-1}\lambda^{k-n+1}\\
+    0    &\lambda^k&\binom{k}{1}\lambda^{k-1}
+	&\dots&\binom{k}{n-2}\lambda^{k-n+2}\\
+    0    &    0    &\lambda^k&\dots  &\binom{k}{n-3}\lambda^{k-n+3}\\
+ \vdots  & \vdots  & \vdots  &\ddots &\vdots\\
+    0    &    0    &    0    &\dots  &\lambda^k
+\end{pmatrix}
+\]}
+\uncover<6->{Alle Matrixelemente konvergieren gegen $0$:}
+\[
+\uncover<7->{\binom{k}{s} \le k^s}
+\uncover<8->{\Rightarrow
+\underbrace{\binom{k}{s}}_{\text{\uncover<9->{polynomiell $\to \infty$}}}
+\underbrace{\lambda^{k-s}}_{\text{\uncover<10->{exponentiell $\to 0$}}}
+}
+\uncover<11->{\to 0}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\uncover<13->{%
+{\usebeamercolor[fg]{title}Folgerung:}
+Es gibt $m,M$ derart, dass
+$m\varrho(A)^k \le \|A^k\|  \le M \varrho(A)^k$
+}
+\end{frame}
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
new file mode 100644
index 0000000..4ca3de4
--- /dev/null
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -0,0 +1,44 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter5 =								\
+	../slides/5/verzerrung.tex					\
+	../slides/5/motivation.tex					\
+	../slides/5/charpoly.tex					\
+	../slides/5/kernbildintro.tex					\
+	../slides/5/kernbilder.tex					\
+	../slides/5/kernbild.tex					\
+	../slides/5/ketten.tex						\
+	../slides/5/dimension.tex					\
+	../slides/5/folgerungen.tex					\
+	../slides/5/injektiv.tex					\
+	../slides/5/nilpotent.tex					\
+	../slides/5/eigenraeume.tex					\
+	../slides/5/zerlegung.tex					\
+	../slides/5/normalnilp.tex					\
+	../slides/5/bloecke.tex						\
+	../slides/5/jordanblock.tex					\
+	../slides/5/jordan.tex						\
+	../slides/5/reellenormalform.tex				\
+	../slides/5/cayleyhamilton.tex					\
+									\
+	../slides/5/spektrum.tex					\
+	../slides/5/normal.tex						\
+	../slides/5/unitaer.tex						\
+									\
+	../slides/5/konvergenzradius.tex				\
+	../slides/5/krbeispiele.tex					\
+	../slides/5/spektralgelfand.tex					\
+	../slides/5/Aiteration.tex					\
+	../slides/5/satzvongelfand.tex					\
+									\
+	../slides/5/stoneweierstrass.tex				\
+	../slides/5/potenzreihenmethode.tex				\
+	../slides/5/logarithmusreihe.tex				\
+	../slides/5/exponentialfunktion.tex				\
+	../slides/5/hyperbolisch.tex					\
+	../slides/5/chapter.tex
+
diff --git a/vorlesungen/slides/5/beispiele/Makefile b/vorlesungen/slides/5/beispiele/Makefile
new file mode 100644
index 0000000..05bd5b5
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/Makefile
@@ -0,0 +1,32 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	kern bild kb kombiniert.jpg leer.jpg drei.jpg
+
+kern:	kern1.jpg kern2.jpg
+bild:	bild1.jpg bild2.jpg
+kb:	kernbild1.jpg kernbild2.jpg
+
+JK1.inc:	kernbild.m
+	octave kernbild.m
+
+kernbild1.png:	JK1.inc common.inc kernbild1.pov
+kernbild2.png:	JK1.inc common.inc kernbild2.pov
+bild1.png:	JK1.inc common.inc bild1.pov
+bild2.png:	JK1.inc common.inc bild2.pov
+kern1.png:	JK1.inc common.inc kern1.pov
+kern2.png:	JK1.inc common.inc kern2.pov
+kombiniert.png:	JK1.inc common.inc kombiniert.pov
+leer.png:	JK1.inc common.inc leer.pov
+drei.png:	JK1.inc common.inc drei.pov
+
+%.png:	%.pov 
+	povray +A0.1 -W1920 -H1080 -O$@ $<
+
+%.jpg:	%.png
+	convert -extract 1080x1080+420+0 $< $@ 
+
+clean:
+	rm -f *.png *.jpg
diff --git a/vorlesungen/slides/5/beispiele/bild1.jpg b/vorlesungen/slides/5/beispiele/bild1.jpg
new file mode 100644
index 0000000..879fae8
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild1.jpg
diff --git a/vorlesungen/slides/5/beispiele/bild1.pov b/vorlesungen/slides/5/beispiele/bild1.pov
new file mode 100644
index 0000000..fd814f1
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild1.pov
@@ -0,0 +1,13 @@
+//
+// bild1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/bild2.jpg b/vorlesungen/slides/5/beispiele/bild2.jpg
new file mode 100644
index 0000000..2597c95
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild2.jpg
diff --git a/vorlesungen/slides/5/beispiele/bild2.pov b/vorlesungen/slides/5/beispiele/bild2.pov
new file mode 100644
index 0000000..6e3c6dd
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild2.pov
@@ -0,0 +1,17 @@
+//
+// bild2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j11, 0.7 * at, orange1)
+arrow(O, j12, 0.7 * at, orange1)
+ebene(j11, j12, orange1)
+
+arrow(O, j21, at, orange2)
+gerade(j21, orange2)
+
+
diff --git a/vorlesungen/slides/5/beispiele/common.inc b/vorlesungen/slides/5/beispiele/common.inc
new file mode 100644
index 0000000..ffcff60
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/common.inc
@@ -0,0 +1,134 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.25;
+#declare O = <0, 0, 0>;
+#declare at = 0.02;
+
+camera {
+	location <3, 2, -10>
+	look_at <0, 0, 0>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+//light_source {
+//	<-14, 20, -50> color White
+//	area_light <1,0,0> <0,0,1>, 10, 10
+//	adaptive 1
+//	jitter
+//}
+
+light_source {
+	<41, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.0 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+#declare r = 1.1;
+
+arrow(< -r-0.2,    0.0,    0  >, <  r+0.2,    0.0,    0.0 >, at, Gray)
+arrow(<    0.0,    0.0, -r-0.2>, <    0.0,    0.0,  r+0.2 >, at, Gray)
+arrow(<    0.0, -r-0.2,    0  >, <    0.0,  r+0.2,    0.0 >, at, Gray)
+
+#declare gruen1  = rgb<0.0,0.4,0.0>;
+#declare gruen2  = rgb<0.0,0.4,0.8>;
+#declare orange1 = rgb<1.0,0.6,0.0>;
+#declare orange2 = rgb<0.8,0.0,0.4>;
+
+#macro ebene(v1, v2, farbe)
+	intersection {
+		box { <-r,-r,-r>, <r,r,r> }
+		plane { vnormalize(vcross(v1, v2)), 0.004 }
+		plane { vnormalize(-vcross(v1, v2)), 0.004 }
+		pigment {
+			color rgbt<farbe.x, farbe.y, farbe.z, 0.5>
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro gerade(v1, farbe)
+	intersection {
+		box { <-r,-r,-r>, <r,r,r> }
+		cylinder { -2 * r * vnormalize(v1),
+				2 * r * vnormalize(v1), 0.80*at }
+		pigment {
+			color farbe
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro kasten() 
+	difference {
+		box { <-r-0.01,-r-0.01,-r-0.01>, <r+0.01,r+0.01,r+0.01> }
+		union {
+			box {	<  -r,        -r,        -r     >,
+				<   r,         r,         r     >	}
+			box {	<-2*r,        -r+0.03,   -r+0.03>,
+				< 2*r,         r-0.03,    r-0.03>	}
+			box {	<  -r+0.03, -2*r,        -r+0.03>,
+				<   r-0.03,  2*r,         r-0.03>	}
+			box {	<  -r+0.03,   -r+0.03, -2*r     >,
+				<   r-0.03,    r-0.03,  2*r     >	}
+		}
+		pigment {
+			color rgb<0.8,0.8,0.8>
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
diff --git a/vorlesungen/slides/5/beispiele/drei.jpg b/vorlesungen/slides/5/beispiele/drei.jpg
new file mode 100644
index 0000000..35f9034
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/drei.jpg
diff --git a/vorlesungen/slides/5/beispiele/drei.pov b/vorlesungen/slides/5/beispiele/drei.pov
new file mode 100644
index 0000000..bdc9630
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/drei.pov
@@ -0,0 +1,22 @@
+//
+// drei.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j21, at, orange2)
+//arrow(O, k21, at, gruen2)
+//arrow(O, k22, at, gruen2)
+gerade(j21, orange2)
+//ebene(k21, k22, gruen2)
+
+#declare at = 0.7 * at;
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+arrow(O, k11, at, gruen1)
+ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/kern1.jpg b/vorlesungen/slides/5/beispiele/kern1.jpg
new file mode 100644
index 0000000..5c99664
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern1.jpg
diff --git a/vorlesungen/slides/5/beispiele/kern1.pov b/vorlesungen/slides/5/beispiele/kern1.pov
new file mode 100644
index 0000000..8e61d8d
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern1.pov
@@ -0,0 +1,12 @@
+//
+// kern1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, k11, at, gruen1)
+gerade(k11, gruen1)
+
diff --git a/vorlesungen/slides/5/beispiele/kern2.jpg b/vorlesungen/slides/5/beispiele/kern2.jpg
new file mode 100644
index 0000000..87d18ac
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern2.jpg
diff --git a/vorlesungen/slides/5/beispiele/kern2.pov b/vorlesungen/slides/5/beispiele/kern2.pov
new file mode 100644
index 0000000..70127a2
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern2.pov
@@ -0,0 +1,17 @@
+//
+// kern2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, k21, at, gruen2)
+arrow(O, k22, at, gruen2)
+ebene(k21, k22, gruen2)
+
+#declare at = 0.7 * at;
+arrow(O, k11, at, gruen1)
+gerade(k11, gruen1)
+
diff --git a/vorlesungen/slides/5/beispiele/kernbild.m b/vorlesungen/slides/5/beispiele/kernbild.m
new file mode 100644
index 0000000..28cd552
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild.m
@@ -0,0 +1,79 @@
+#
+# kernbild.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+rand("seed", 1291)
+rand("seed", 4711)
+
+lambda1 = 1;
+lambda2 = 1.8;
+
+A = [
+	lambda1,  0,  0;
+	0,  lambda2,  1;
+	0,  0, lambda2
+];
+
+B = eye(3) + rand(3,3);
+det(B)
+
+
+C = B*A*inverse(B)
+rank(C)
+
+# Eigenwert lambda1
+E2 = C - lambda1 * eye(3)
+rref(E2)
+
+# Eigenwert lambda2, k = 1
+E1 = C - lambda2 * eye(3)
+D = rref(E1);
+K1 = [
+	-D(1,3);
+	-D(2,3);
+	1
+];
+K1(:,1) = K1(:,1) / norm(K1(:,1));
+K1
+
+f = fopen("JK.inc", "w");
+fprintf(f, "//\n// JK.inc\n//\n// (c) 2021 Prof Dr Andreas Müller\n//\n\n");
+fprintf(f, "// Kern und Bild von C - %.3f I\n", lambda2);
+fprintf(f, "#declare k11 = < %.5f, %.5f, %.5f>;\n", K1(1,1), K1(2,1), K1(3,1));
+fprintf(f, "#declare j11 = < %.5f, %.5f, %.5f>;\n", E1(1,1), E1(2,1), E1(3,1));
+fprintf(f, "#declare j12 = < %.5f, %.5f, %.5f>;\n", E1(1,2), E1(2,2), E1(3,2));
+fprintf(f, "\n");
+
+# k = 2
+E12 = E1 * E1
+D = rref(E12);
+K2 = [
+	-D(1,2), -D(1,3);
+	      1,       0;
+	      0,       1
+]
+K2(:,1) = K2(:,1) / norm(K2(:,1));
+K2(:,2) = K2(:,2) / norm(K2(:,2));
+K2
+
+fprintf(f, "// Kern und Bild von (C - %.3f I)^2\n", lambda2);
+fprintf(f, "#declare k21 = < %.5f, %.5f, %.5f>;\n", K2(1,1), K2(2,1), K2(3,1));
+fprintf(f, "#declare k22 = < %.5f, %.5f, %.5f>;\n", K2(1,2), K2(2,2), K2(3,2));
+fprintf(f, "#declare j21 = < %.5f, %.5f, %.5f>;\n", E12(1,1), E12(2,1), E12(3,1));
+fprintf(f, "\n");
+
+fclose(f);
+
+# Verifikation
+x = K2 \ K1
+K2 * x
+
+eig(C)
+
+[U, S, V] = svd(C)
+
+
+s = rand("seed")
+
diff --git a/vorlesungen/slides/5/beispiele/kernbild1.jpg b/vorlesungen/slides/5/beispiele/kernbild1.jpg
new file mode 100644
index 0000000..87e874e
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild1.jpg
diff --git a/vorlesungen/slides/5/beispiele/kernbild1.pov b/vorlesungen/slides/5/beispiele/kernbild1.pov
new file mode 100644
index 0000000..425f299
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild1.pov
@@ -0,0 +1,15 @@
+//
+// kernbild1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+arrow(O, k11, at, gruen1)
+ebene(j11, j12, orange1)
+
+//kasten()
diff --git a/vorlesungen/slides/5/beispiele/kernbild2.jpg b/vorlesungen/slides/5/beispiele/kernbild2.jpg
new file mode 100644
index 0000000..1160b31
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild2.jpg
diff --git a/vorlesungen/slides/5/beispiele/kernbild2.pov b/vorlesungen/slides/5/beispiele/kernbild2.pov
new file mode 100644
index 0000000..ae67ea1
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild2.pov
@@ -0,0 +1,21 @@
+//
+// kernbild2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j21, at, orange2)
+arrow(O, k21, at, gruen2)
+arrow(O, k22, at, gruen2)
+gerade(j21, orange2)
+ebene(k21, k22, gruen2)
+
+//arrow(O, j11, at, orange1)
+//arrow(O, j12, at, orange1)
+//arrow(O, k11, at, gruen1)
+//gerade(k11, gruen1)
+//ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg
new file mode 100644
index 0000000..9cb789c
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kombiniert.jpg
diff --git a/vorlesungen/slides/5/beispiele/kombiniert.pov b/vorlesungen/slides/5/beispiele/kombiniert.pov
new file mode 100644
index 0000000..c187d08
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kombiniert.pov
@@ -0,0 +1,22 @@
+//
+// kombiniert.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j21, at, orange2)
+arrow(O, k21, at, gruen2)
+arrow(O, k22, at, gruen2)
+gerade(j21, orange2)
+ebene(k21, k22, gruen2)
+
+#declare at = 0.7 * at;
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+arrow(O, k11, at, gruen1)
+ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/leer.jpg b/vorlesungen/slides/5/beispiele/leer.jpg
new file mode 100644
index 0000000..9789887
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/leer.jpg
diff --git a/vorlesungen/slides/5/beispiele/leer.pov b/vorlesungen/slides/5/beispiele/leer.pov
new file mode 100644
index 0000000..f4653d9
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/leer.pov
@@ -0,0 +1,9 @@
+//
+// leer.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
diff --git a/vorlesungen/slides/5/bloecke.tex b/vorlesungen/slides/5/bloecke.tex
new file mode 100644
index 0000000..974f238
--- /dev/null
+++ b/vorlesungen/slides/5/bloecke.tex
@@ -0,0 +1,141 @@
+%
+% bloecke.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1}
+\def\sy{0.1}
+\def\block#1#2{
+	\fill[color=red] ({#1},{-#1}) rectangle ({#1+#2},{-#1-#2});
+}
+\def\kreuz#1{
+	\draw[color=white,line width=0.1pt] (0,{-#1})--(60,{-#1});
+	\draw[color=white,line width=0.1pt] (#1,0)--(#1,-60);
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Blockgrössen aus $\dim\mathcal{K}^k(A)$ ablesen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.56\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\coordinate (A) at ({1*\sx},{20*\sy});
+\coordinate (B) at ({2*\sx},{(20+15)*\sy});
+\coordinate (C) at ({3*\sx},{(20+15+10)*\sy});
+\coordinate (D) at ({4*\sx},{(20+15+10+8)*\sy});
+\coordinate (E) at ({5*\sx},{(20+15+10+8+5)*\sy});
+\coordinate (F) at ({6*\sx},{(20+15+10+8+5+2)*\sy});
+\fill[color=darkgreen!20] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F)
+	-- ({6*\sx},0) -- cycle;
+
+\fill[color=darkgreen!40] (0,0) -- ({1*\sx},0) -- (A) -- cycle;
+\fill[color=darkgreen!40] (A) -- ({2*\sx},{20*\sy}) -- (B) -- cycle;
+\fill[color=darkgreen!40] (B) -- ({3*\sx},{(20+15)*\sy}) -- (C) -- cycle;
+\fill[color=darkgreen!40] (C) -- ({4*\sx},{(20+15+10)*\sy}) -- (D) -- cycle;
+\fill[color=darkgreen!40] (D) -- ({5*\sx},{(20+15+10+8)*\sy}) -- (E) -- cycle;
+\fill[color=darkgreen!40] (E) -- ({6*\sx},{(20+15+10+8+5)*\sy}) -- (F) -- cycle;
+
+\draw[color=darkgreen,line width=1.4pt] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F);
+
+\draw[color=gray] (A) -- (0,{20*\sy});
+\draw[color=gray] (B) -- (0,{(20+15)*\sy});
+\draw[color=gray] (C) -- (0,{(20+15+10)*\sy});
+\draw[color=gray] (D) -- (0,{(20+15+10+8)*\sy});
+\draw[color=gray] (E) -- (0,{(20+15+10+8+5)*\sy});
+\draw[color=gray] (F) -- (0,{(20+15+10+8+5+2)*\sy});
+
+\node at ({0.5*\sx},{0.5*20*\sy})
+	[right] {$d_1 = \dim\mathcal{K}^1(A)-\dim\mathcal{K}^0(A)$};
+\node at ({1.5*\sx},{0.5*(20+20+15)*\sy})
+	[right] {$d_2 = \dim\mathcal{K}^2(A)-\dim\mathcal{K}^1(A)$};
+\node at ({2.5*\sx},{0.5*(2*20+2*15+1*10)*\sy}) [right] {$d_3$};
+\node at ({3.5*\sx},{0.5*(2*20+2*15+2*10+8)*\sy}) [right] {$d_4$};
+\node at ({4.5*\sx-0.1},{0.5*(2*20+2*15+2*10+2*8+5)*\sy+0.2}) [below right] {$d_5$};
+\node at ({5.5*\sx},{0.5*(2*20+2*15+2*10+2*8+2*5+2)*\sy+0.1}) [below] {$d_6$};
+
+\fill (A) circle[radius=0.08];
+\fill (B) circle[radius=0.08];
+\fill (C) circle[radius=0.08];
+\fill (D) circle[radius=0.08];
+\fill (E) circle[radius=0.08];
+\fill (F) circle[radius=0.08];
+
+\draw[->] (-0.1,0) -- ({6*\sx+1},0) coordinate[label={$k$}];
+\draw[->] (0,-0.1) -- (0,6.5) coordinate[label={right:$\dim\mathcal{K}^k(A)$}];
+
+\foreach \x in {0,1,...,6}{
+	\draw ({\sx*\x},{-0.05}) -- ({\sx*\x},0.05);
+	\node at ({\sx*\x},{-0.1}) [below] {$\x$};
+}
+
+\node at (0,{60*\sy}) [left] {\llap{$n$}};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.43\textwidth}
+\vspace{-10pt}
+\begin{center}
+\begin{tabular}{>{$}c<{$}|>{$}r<{$}|>{$}c<{$}|>{$}c<{$}}
+k&d_k&\# M_k(\Bbbk)\text{-Blöcke}&\text{Beispiel}\\
+\hline
+0&  0&        &\\
+1& 20& d_1-d_2&5\\
+2& 15& d_2-d_3&5\\
+3& 10& d_3-d_4&2\\
+4&  8& d_4-d_5&3\\
+5&  5& d_5-d_6&3\\
+6&  2& d_6    &2\\
+\end{tabular}
+\end{center}
+\vspace{-13pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.05]
+\fill[color=gray!40] (0,0) rectangle (60,-60);
+\node[color=white] at (30,-30) [scale=6] {$A$};
+\kreuz{5}
+\kreuz{15}
+\kreuz{21}
+\kreuz{33}
+\kreuz{48}
+\node at (0,-2.5) [left] {$k=1$};
+\node at (60,-2.5) [right] {$5$ Blöcke};
+\node at (0,-10) [left] {$k=2$};
+\node at (60,-10) [right] {$5$ Blöcke};
+\node at (0,-18) [left] {$k=3$};
+\node at (60,-18) [right] {$2$ Blöcke};
+\node at (0,-27) [left] {$k=4$};
+\node at (60,-27) [right] {$3$ Blöcke};
+\node at (0,-40.5) [left] {$k=5$};
+\node at (60,-40.5) [right] {$3$ Blöcke};
+\node at (0,-54) [left] {$k=6$};
+\node at (60,-54) [right] {$2$ Blöcke};
+\block{0}{1}
+\block{1}{1}
+\block{2}{1}
+\block{3}{1}
+\block{4}{1}
+\block{5}{2}
+\block{7}{2}
+\block{9}{2}
+\block{11}{2}
+\block{13}{2}
+\block{15}{3}
+\block{18}{3}
+\block{21}{4}
+\block{25}{4}
+\block{29}{4}
+\block{33}{5}
+\block{38}{5}
+\block{43}{5}
+\block{48}{6}
+\block{54}{6}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/cayleyhamilton.tex b/vorlesungen/slides/5/cayleyhamilton.tex
new file mode 100644
index 0000000..c0813be
--- /dev/null
+++ b/vorlesungen/slides/5/cayleyhamilton.tex
@@ -0,0 +1,91 @@
+%
+% cayleyhamilton.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Satz von Cayley-Hamilton}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ein Eigenwert $\lambda$\strut}
+$A$ besteht aus
+$b$ Blöcken $J_\lambda$ mit maximaler Dimension $l$:
+\phantom{blubb\strut}
+\begin{align*}
+\uncover<2->{
+\chi_{A}(X)
+&=
+\det (A-XI) = (\lambda-X)^n
+}
+\\
+\uncover<3->{
+m_{A}(X)
+&=
+(\lambda-X)^l
+}
+\\
+\uncover<4->{
+b&= \ker A
+}
+\end{align*}
+\uncover<5->{%
+Wegen $l \le n$ folgt
+\[
+m_A(X) | \chi_A(X)
+\uncover<6->{\quad\Rightarrow\quad
+\chi_A(A) = 0}
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{$A=A_1\oplus\dots\oplus A_k$}
+\uncover<8->{%
+$A_i\in M_{n_i}(\Bbbk)$ mit EW $\lambda_i$,
+$A_i$ besteht aus
+$b_i$ Blöcken $J_{\lambda_i}$ mit max.~Dimension $l_i$\strut:}
+\begin{align*}
+\uncover<9->{
+\chi_A(X)
+&=
+(\lambda_1-X)^{n_1}
+\dots
+(\lambda_k-X)^{n_k}
+}
+\\
+\uncover<10->{
+m_A(X)
+&=
+(\lambda_1-X)^{l_1}
+\dots
+(\lambda_k-X)^{l_k}
+}
+\\
+\uncover<11->{
+b_i &= \ker (A-\lambda_iI)
+}
+\end{align*}
+\uncover<12->{%
+$A=A_1\oplus\dots\oplus A_k$}
+\begin{align*}
+\uncover<13->{
+\chi_{A_i}(A_i)&=0\;\forall i
+}
+\\
+\uncover<14->{%
+\chi_A(A) &=
+\chi_{A_1}(A)\dots\chi_{A_k}(A)
+ = 0}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<15->{%
+\begin{block}{Satz}
+Für jede Matrix $A\in M_n(\Bbbk)$ gilt
+$m_A(X) | \chi_A(X)$ oder $\chi_A(A)=0$
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
new file mode 100644
index 0000000..96eea29
--- /dev/null
+++ b/vorlesungen/slides/5/chapter.tex
@@ -0,0 +1,36 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{5/verzerrung.tex}
+\folie{5/motivation.tex}
+\folie{5/charpoly.tex}
+\folie{5/kernbildintro.tex}
+\folie{5/kernbilder.tex}
+\folie{5/kernbild.tex}
+\folie{5/ketten.tex}
+\folie{5/dimension.tex}
+\folie{5/folgerungen.tex}
+\folie{5/injektiv.tex}
+\folie{5/nilpotent.tex}
+\folie{5/eigenraeume.tex}
+\folie{5/zerlegung.tex}
+\folie{5/normalnilp.tex}
+\folie{5/bloecke.tex}
+\folie{5/jordanblock.tex}
+\folie{5/jordan.tex}
+\folie{5/reellenormalform.tex}
+\folie{5/cayleyhamilton.tex}
+\folie{5/konvergenzradius.tex}
+\folie{5/krbeispiele.tex}
+\folie{5/spektralgelfand.tex}
+\folie{5/Aiteration.tex}
+\folie{5/satzvongelfand.tex}
+\folie{5/stoneweierstrass.tex}
+\folie{5/potenzreihenmethode.tex}
+\folie{5/logarithmusreihe.tex}
+\folie{5/exponentialfunktion.tex}
+\folie{5/hyperbolisch.tex}
+\folie{5/spektrum.tex}
+\folie{5/normal.tex}
diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex
new file mode 100644
index 0000000..63bfee5
--- /dev/null
+++ b/vorlesungen/slides/5/charpoly.tex
@@ -0,0 +1,78 @@
+%
+% charpoly.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Charakteristisches Polynom über $\mathbb{C}$}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenwerte}
+Nur diejenigen $\mu$ kommen in Frage, für die
+$A-\mu I$ singulär ist:
+\[
+\chi_{A}(\mu)
+=
+\det (A-\mu I) = 0
+\]
+$\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$
+\end{block}
+\uncover<2->{%
+\begin{block}{Zerlegung in Linearfaktoren}
+$\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$:
+\[
+\chi_A(X)
+=
+(X-\mu_1)\dots (X-\mu_n)
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Fundamentalsatz der Algebra}
+Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in
+Linearfaktoren
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Minimalpolynom}
+Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen
+\end{block}}
+\uncover<5->{%
+\begin{proof}[Beweis]
+\begin{enumerate}
+\item<6->
+$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$
+\item<7->
+$A-\lambda I$ ist regulär
+\end{enumerate}
+\uncover<8->{%
+\begin{align*}
+&\Rightarrow&
+m_A(A)&=0
+\\
+&&
+\uncover<9->{
+(A-\lambda)^{-1}m_A(A) &=0
+}
+\\
+&&
+\uncover<10->{
+\prod_{i\in I}(A-\mu_i)&=0,
+}
+\end{align*}}
+\uncover<11->{%
+d.~h.~\(
+\displaystyle
+\overline{m}_A(X)
+=
+\prod_i{i\in I}(X-\mu_i)
+\in
+\mathbb{C}[X]
+\)}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/dimension.tex b/vorlesungen/slides/5/dimension.tex
new file mode 100644
index 0000000..ff687b3
--- /dev/null
+++ b/vorlesungen/slides/5/dimension.tex
@@ -0,0 +1,68 @@
+%
+% dimension.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Dimension von $\mathcal{K}^k(f)$ und $\mathcal{J}^k(f)$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\pfad{
+	(0,0) -- (1,0.3) -- (2,0.9) 
+	--
+	(4,2.4) -- (5,2.7) -- (6,3.3)
+	--
+	(8,3.7) -- (9,4) -- (10,4) -- (11,4) -- (12,4)
+}
+
+\fill[color=darkgreen!20] \pfad -- (12,0) -- cycle;
+\fill[color=orange!20] \pfad -- (12,6) -- (0,6) -- cycle;
+
+\fill[color=darkgreen!40] (9,0) -- (12,0) -- (12,4) -- (9,4) -- cycle;
+\fill[color=orange!40] (9,4) -- (12,4) -- (12,6) -- (9,6) -- cycle;
+
+\node[color=orange] at (10.5,5) {$\mathcal{J}(f)$};
+\node[color=darkgreen] at (10.5,2) {$\mathcal{K}(f)$};
+
+\node[color=orange] at (5.5,4.5) {$\mathcal{J}^k(f)\supset\mathcal{J}^{k+1}(f)$};
+\node[color=darkgreen] at (5.5,1.5) {$\mathcal{K}^k(f)\subset\mathcal{K}^{k+1}(f)$};
+
+\draw[line width=1.4pt] \pfad;
+
+\draw[->] (-0.1,6) -- (12.5,6) coordinate[label={$k$}];
+\draw[->] (-0.1,0) -- (12.5,0) coordinate[label={$k$}];
+\node at (-0.1,6) [left] {$n$};
+\node at (-0.1,0) [left] {$0$};
+\foreach \x in {0,1,2,4,5,6,8,9,10,11,12}{
+	\fill (\x,0) circle[radius=0.05];
+	\fill (\x,6) circle[radius=0.05];
+}
+\node at (0,0) [below] {$0$};
+\node at (1,0) [below] {$1$};
+\node at (2,0) [below] {$2$};
+
+\node at (4,0) [below] {$k-1$};
+\node at (5,0) [below] {$k$};
+\node at (6,0) [below] {$k+1$};
+
+\node at (8,0) [below] {$l-1$};
+\node at (9,0) [below] {$l$};
+\node at (10,0) [below] {$l+1$};
+\node at (11,0) [below] {$l+2$};
+\node at (12,0) [below] {$l+3$};
+
+\fill (9,4) circle[radius=0.05];
+
+\node[color=orange] at (-0.2,3) [rotate=90] {$\dim\mathcal{J}^k(f)$};
+\node[color=darkgreen] at (12.2,2) [rotate=-90] {$\dim\mathcal{K}^k(f)$};
+
+\node[color=orange] at (9,5) [rotate=-90] {$\dim\mathcal{J}(f)$};
+\node[color=darkgreen] at (9,2) [rotate=-90] {$\dim\mathcal{K}(f)$};
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex
new file mode 100644
index 0000000..fd4803c
--- /dev/null
+++ b/vorlesungen/slides/5/eigenraeume.tex
@@ -0,0 +1,48 @@
+%
+% eigenraeume.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Eigenräume}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenraum}
+Für $\lambda\in\Bbbk$ heisst
+\begin{align*}
+E_\lambda(f)
+&=
+\ker (f-\lambda)
+\\
+\uncover<2->{
+&=
+\{v\in V\;|\; f(v) = \lambda v\}
+}
+\end{align*}
+\uncover<3->{%
+{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.}
+\end{block}
+\uncover<4->{%
+$E_\lambda(f)\subset V$ ist ein Unterraum}
+
+\uncover<5->{%
+\begin{block}{Eigenwert}
+Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$.
+\end{block}}
+
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{verallgemeinerter Eigenraum}
+Für $\lambda\in \Bbbk$ heisst
+\[
+\mathcal{E}_\lambda(f)
+=
+\mathcal{K}(f-\lambda)
+\]
+verallgemeinerter Eigenraum
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/exponentialfunktion.tex b/vorlesungen/slides/5/exponentialfunktion.tex
new file mode 100644
index 0000000..caae16b
--- /dev/null
+++ b/vorlesungen/slides/5/exponentialfunktion.tex
@@ -0,0 +1,131 @@
+%
+% exponentialfunktion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Exponentialfunktion}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\only<1-6>{%
+\ifthenelse{\boolean{presentation}}{
+\begin{column}{0.48\textwidth}
+\begin{block}{$x(t) \in\mathbb{R}$}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}x(t) &= ax(t) &a&\in\mathbb{R}
+\\
+x(0) &= c&&\in\mathbb{R}
+\intertext{\uncover<2->{Lösung:}}
+\uncover<2->{x(t) &= ce^{at}}
+\end{align*}
+\end{block}
+\end{column}}{}}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{$X(t) \in M_n(\mathbb{R})$}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}X(t)
+&=
+A
+X(t)&A&\in M_n(\mathbb{R})
+\\
+X(0)&=C&&\in M_n(\mathbb{R})
+\intertext{\uncover<4->{gekoppelte Differentialgleichung für
+vier Funktionen $x_{ij}(t)$}}
+\uncover<5->{\dot{x}_{11} &= \rlap{$a_{11} x_{11}(t) + a_{12} x_{21}(t)$}}\\
+\uncover<5->{\dot{x}_{12} &= \rlap{$a_{11} x_{12}(t) + a_{12} x_{22}(t)$}}\\
+\uncover<5->{\dot{x}_{21} &= \rlap{$a_{21} x_{11}(t) + a_{22} x_{21}(t)$}}\\
+\uncover<5->{\dot{x}_{22} &= \rlap{$a_{21} x_{12}(t) + a_{22} x_{22}(t)$}}\\
+\intertext{\uncover<6->{Lösung:}}
+\uncover<6->{X(t) &= \exp(At) C}
+\end{align*}
+\end{block}}
+\end{column}
+\only<7-9>{%
+\ifthenelse{\boolean{presentation}}{
+\begin{column}{0.48\textwidth}
+\begin{block}{Beispiel: Diagonalmatrix}
+%$D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$
+\begin{align*}
+\frac{d}{dt}X&=DX &&\uncover<8->{\Rightarrow &\dot{x}_{ij}(t) &= \lambda_i x_{ij}(t)}
+\\
+X(0)&=C
+&&\uncover<8->{\Rightarrow&x_{ij}(t)&=c_{ij}}
+\end{align*}
+\uncover<9->{%
+Lösung:
+\[
+x_{ij}(t) =c_{ij}e^{\lambda_i t}
+\]}
+\end{block}
+\end{column}}{}}
+\uncover<10->{%
+\begin{column}{0.48\textwidth}
+\begin{block}{Beispiel: Jordan-Block}
+\vspace{-10pt}
+\begin{align*}
+A&=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}
+\rlap{$\displaystyle,\;
+X(t)
+=
+\ifthenelse{\boolean{presentation}}{
+\only<22>{
+	e^{\lambda t}
+	\begin{pmatrix} 1&t/\lambda\\ 0&1 \end{pmatrix}
+}}{}
+\only<23>{
+	\frac{e^{\lambda t}}{\lambda}
+	\begin{pmatrix} \lambda&t\\ 0&\lambda \end{pmatrix}
+}
+C
+$}
+\\
+\uncover<11->{
+\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + \phantom{\lambda}x_{2i}(t),&&x_{1i}(0)&=c_{1i}
+}
+\\
+\uncover<12->{
+\dot{x}_{2i}(t)&=\phantom{\lambda x_{1i}(t)+\mathstrut}\lambda x_{2i}(t),&&x_{2i}(0)&=c_{2i}
+}
+\end{align*}
+\uncover<13->{%
+Lösung:}
+\begin{align*}
+\uncover<14->{
+x_{2i}(t)&=c_{2i}e^{\lambda t}
+}
+\\
+\uncover<15->{
+\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + c_{2i}e^{\lambda t}
+}
+\\
+\ifthenelse{\boolean{presentation}}{
+\only<16-17>{x_{1i\only<16>{,h}}(t)}}{}
+\only<18->{\dot{x}_{1i}(t)}
+&
+\ifthenelse{\boolean{presentation}}{
+\only<16-17>{=c\only<17>{(t)}\lambda e^{\lambda t}}
+\only<18>{=\dot{c}(t)\lambda e^{\lambda t}
++
+c(t)\lambda^2 e^{\lambda t}}
+}{}
+\only<19->{=\lambda x_{1i}(t) + \dot{c}(t)\lambda e^{\lambda t}}
+\\
+\uncover<20->{\Rightarrow
+\dot{c}(t)&= c_{2i}/\lambda
+\Rightarrow
+c(t) = c_{2i}(0) +tc_{2i}/\lambda
+}
+\\
+\uncover<21->{
+x_{1i}(t) & =c_{1i}e^{\lambda t} + t(c_{2i}/\lambda)e^{\lambda t}
+}
+\end{align*}
+\end{block}
+\end{column}}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex
new file mode 100644
index 0000000..4a8dbe6
--- /dev/null
+++ b/vorlesungen/slides/5/folgerungen.tex
@@ -0,0 +1,84 @@
+%
+% folgerungen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Folgerungen}
+\vspace{-10pt}
+\begin{columns}[t]
+\begin{column}{0.30\textwidth}
+\begin{block}{Zunahme}
+Für alle $k<l$ gilt
+\begin{align*}
+\mathcal{J}^k(f) &\supsetneq \mathcal{J}^{k+1}(f)
+\\
+\mathcal{K}^k(f) &\subsetneq \mathcal{K}^{k+1}(f)
+\end{align*}
+Für $k\ge l$ gilt
+\begin{align*}
+\mathcal{J}^k(f) &= \mathcal{J}^{k+1}(f)
+\\
+\mathcal{K}^k(f) &= \mathcal{K}^{k+1}(f)
+\end{align*}
+Ausserdem ist $l\le n$
+\end{block}
+\end{column}
+\begin{column}{0.66\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\pfad{
+	({0*\sx},6) --
+	({1*\sx},4.5) --
+	({2*\sx},3.5) --
+	({3*\sx},2.9) --
+	({4*\sx},2.6) --
+	({5*\sx},2.4) --
+	({6*\sx},2.4)
+}
+
+\fill[color=orange!20] \pfad -- ({6*\sx},0) -- (0,0) -- cycle;
+\fill[color=darkgreen!20] \pfad -- ({6*\sx},6) -- cycle;
+\fill[color=orange!40] ({5*\sx},0) rectangle ({6*\sx},2.4);
+\fill[color=darkgreen!40] ({5*\sx},6) rectangle ({6*\sx},2.4);
+
+\draw[color=darkgreen,line width=2pt] ({3*\sx},6) -- ({3*\sx},2.9);
+\node[color=darkgreen] at ({3*\sx},4.45) [rotate=90,above] {$\dim\mathcal{K}^k(A)$};
+\draw[color=orange,line width=2pt] ({3*\sx},0) -- ({3*\sx},2.9);
+\node[color=orange] at ({3*\sx},1.45) [rotate=90,above] {$\dim\mathcal{J}^k(A)$};
+
+\node[color=orange] at ({5.5*\sx},1.2) [rotate=90] {bijektiv};
+\node[color=darkgreen] at ({5.5*\sx},4.2) [rotate=90] {konstant};
+
+\fill ({0*\sx},6) circle[radius=0.08];
+\fill ({1*\sx},4.5) circle[radius=0.08];
+\fill ({2*\sx},3.5) circle[radius=0.08];
+\fill ({3*\sx},2.9) circle[radius=0.08];
+\fill ({4*\sx},2.6) circle[radius=0.08];
+\fill ({5*\sx},2.4) circle[radius=0.08];
+\fill ({6*\sx},2.4) circle[radius=0.08];
+
+\draw \pfad;
+
+\draw[->] (-0.1,0) -- ({6*\sx+0.5},0) coordinate[label={$k$}];
+\draw[->] (-0.1,6) -- ({6*\sx+0.5},6);
+
+\foreach \x in {0,...,6}{
+	\draw (\x,-0.05) -- (\x,0.05);
+}
+\foreach \x in {0,...,3}{
+	\node at ({\x*\sx},-0.05) [below] {$\x$};
+}
+\node at ({4*\sx},-0.05) [below] {$\dots\mathstrut$};
+\node at ({5*\sx},-0.05) [below] {$l$};
+\node at ({6*\sx},-0.05) [below] {$l+1$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/hyperbolisch.tex b/vorlesungen/slides/5/hyperbolisch.tex
new file mode 100644
index 0000000..905082a
--- /dev/null
+++ b/vorlesungen/slides/5/hyperbolisch.tex
@@ -0,0 +1,105 @@
+%
+% hyperbolisch.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Hyperbolische Funktionen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Differentialgleichung}
+\vspace{-10pt}
+\begin{align*}
+\ddot{y} &= y
+\;\Rightarrow\;
+\frac{d}{dt}
+\begin{pmatrix}y\\y_1\end{pmatrix}
+=
+\begin{pmatrix}0&1\\1&0\end{pmatrix}
+\begin{pmatrix}y\\y_1\end{pmatrix}
+\\
+y(0)&=a,\qquad y'(0)=b
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Lösung}
+\vspace{-13pt}
+\begin{align*}
+\lambda^2-1&=0
+\uncover<3->{
+\qquad\Rightarrow\qquad \lambda=\pm 1
+}
+\\
+\uncover<4->{
+y(t)&=Ae^t+Be^{-t}}
+\uncover<5->{
+\Rightarrow
+\left\{
+\arraycolsep=1.4pt
+\begin{array}{rcrcr}
+A&+&B&=&a\\
+A&-&B&=&b
+\end{array}
+\right.}
+\\
+&\uncover<6->{
+=\frac{a+b}2e^t + \frac{a-b}2e^{-t}}
+\\
+&\uncover<7->{=
+a{\color{darkgreen}\frac{e^t+e^{-t}}2} + b{\color{red}\frac{e^t-e^{-t}}2}}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.49\textwidth}
+\uncover<8->{%
+\begin{block}{Potenzreihe}
+\vspace{-12pt}
+\begin{align*}
+K&=\begin{pmatrix}0&1\\1&0\end{pmatrix}
+\uncover<10->{\quad\Rightarrow\quad K^2=I}
+\\
+\uncover<9->{
+e^{Kt}
+&=
+I+K+\frac1{2!}K^2 + \frac{1}{3!}K^3 + \frac{1}{4!}K^4+\dots
+}
+\\
+\uncover<11->{
+&=
+\biggl( 1+\frac{t^2}{2!} + \frac{t^4}{4!}+\dots \biggr)I
+}
+\\
+\uncover<11->{
+&\qquad
++\biggl(t+\frac{t^3}{3!}+\frac{t^5}{5!}+\dots\biggr)K
+}
+\\
+\uncover<12->{
+&=
+I{\,\color{darkgreen}\cosh t}  + K{\,\color{red}\sinh t}
+}
+\\
+\uncover<13->{
+\begin{pmatrix}y(t)\\y_1(t)\end{pmatrix}
+&=
+e^{Kt}\begin{pmatrix}a\\b\end{pmatrix}
+}
+\uncover<14->{
+=
+\begin{pmatrix}
+a{\,\color{darkgreen}\cosh t} + b{\,\color{red}\sinh t}\\
+a{\,\color{red}\sinh t} + b{\,\color{darkgreen}\cosh t}
+\end{pmatrix}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex
new file mode 100644
index 0000000..90cbcd6
--- /dev/null
+++ b/vorlesungen/slides/5/injektiv.tex
@@ -0,0 +1,81 @@
+%
+% injektiv.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1.05}
+\begin{frame}[t]
+\frametitle{$f$ injektiv auf $\mathcal{J}(f)$}
+\setlength{\abovedisplayskip}{8pt}
+\setlength{\belowdisplayskip}{8pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=orange!20]
+	({0*\sx},-3.0) -- ({1*\sx},-2.0) -- ({2*\sx},-1.5) --
+	({3*\sx},-1.1) -- ({4*\sx},-0.9) -- ({5*\sx},-0.8) --
+	({6*\sx},-0.8) --
+	({6*\sx},0.8) -- ({5*\sx},0.8) -- ({4*\sx},0.9) --
+	({3*\sx},1.1) -- ({2*\sx},1.5) -- ({1*\sx},2.0) --
+	({0*\sx},3.0) -- cycle;
+\fill[color=orange!40] (0,-0.8) rectangle ({6*\sx},0.8);
+
+\foreach \x in {0,...,6}{
+	\draw[color=gray,line width=3pt] ({\x*\sx},-3)--({\sx*\x},3);
+}
+\foreach \x in {0,1,2,3}{
+	\node at ({\sx*\x},-3) [below] {$\x$};
+}
+\node at ({\sx*5},-3) [below] {$l$};
+\node at ({\sx*6},-3) [below] {$l+1$};
+\draw[->] (-0.1,-3.5) -- ({6*\sx+0.4},-3.5) coordinate[label={below:$k$}];
+
+\draw[line width=3pt,color=orange] ({0*\sx},-3.0) -- ({0*\sx},3.0);
+\draw[line width=3pt,color=orange] ({1*\sx},-2.0) -- ({1*\sx},2.0);
+\draw[line width=3pt,color=orange] ({2*\sx},-1.5) -- ({2*\sx},1.5);
+\draw[line width=3pt,color=orange] ({3*\sx},-1.1) -- ({3*\sx},1.1);
+\draw[line width=3pt,color=orange] ({4*\sx},-0.9) -- ({4*\sx},0.9);
+\draw[line width=3pt,color=orange] ({5*\sx},-0.8) -- ({5*\sx},0.8);
+\draw[line width=3pt,color=orange] ({6*\sx},-0.8) -- ({6*\sx},0.8);
+
+\foreach \x in {0,1,2,3}{
+	\node at ({\x*\sx},0) [rotate=90] {$\mathcal{J}^{\x}(A)$};
+}
+\node at ({4*\sx},0) {$\cdots$};
+\node at ({5*\sx},0) [rotate=90] {$\mathcal{J}^{l}(A)$};
+\node at ({6*\sx},0) [rotate=90] {$\mathcal{J}^{l+1}(A)$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.38\textwidth}
+\begin{block}{stationär}
+$l$ der $k$-Wert, ab dem gilt
+\begin{align*}
+\mathcal{J}^l(A) &= \mathcal{J}^{l+1}(A) = A\mathcal{J}^l(A)
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Dimension}
+\vspace{-10pt}
+\[
+\dim \mathcal{J}^l(A) = \dim\mathcal{J}^{l+1}(A)
+\]
+\uncover<3->{%
+d.~h.~$A$ ist bijektiv als Selbstabbildung von
+$\mathcal{J}(A)$}
+\uncover<4->{%
+\[
+\Downarrow
+\]
+$A|\mathcal{J}(A)$ ist injektiv}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/jordan.tex b/vorlesungen/slides/5/jordan.tex
new file mode 100644
index 0000000..e6ece47
--- /dev/null
+++ b/vorlesungen/slides/5/jordan.tex
@@ -0,0 +1,138 @@
+%
+% jordan.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\L#1{
+	\node at ({#1-0.5},{0.5-#1}) {$\lambda$};
+}
+\def\E#1{
+	\node at ({#1-0.5},{1.5-#1}) {$1$};
+}
+
+\begin{frame}[t]
+\frametitle{Jordan Normalform}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Wahl der Basis}
+\begin{enumerate}
+\item<2-> Zerlegung in verallgemeinerte Eigenräume
+\begin{align*}
+V
+&=
+\mathcal{E}_{{\color{blue}\lambda}}(A)
+\oplus
+\mathcal{E}_{{\color{darkgreen}\lambda}}(A)
+\oplus
+\mathcal{E}_{{\color{red}\lambda}}(A)
+%\oplus
+%\dots
+\\
+\llap{$A\mathcal{E}_{{\color{blue}\lambda}}$}(A)
+&\subset
+\mathcal{E}_{{\color{blue}\lambda}}(A)
+\\
+\llap{$A\mathcal{E}_{{\color{darkgreen}\lambda}}$}(A)
+&\subset
+\mathcal{E}_{{\color{darkgreen}\lambda}}(A)
+\\
+\llap{$A\mathcal{E}_{{\color{red}\lambda}}$}(A)
+&\subset
+\mathcal{E}_{{\color{red}\lambda}}(A),
+\dots
+\end{align*}
+\item<3-> In jedem Eigenraum: Zerlegung in Jordan-Blöcke
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.33]
+\fill[color=gray!20] (0,-20) rectangle (20,0);
+\node[color=white] at (10,-10) [scale=12] {$A$};
+
+\uncover<2->{
+	\fill[color=blue!20,opacity=0.5] (0,0) rectangle (8,-8);
+	\fill[color=darkgreen!20,opacity=0.5] (8,-8) rectangle (15,-15);
+	\fill[color=red!20,opacity=0.5] (15,-15) rectangle (20,-20);
+	\fill[color=blue!20] (0,0) rectangle (8,2);
+	\fill[color=blue!20] (-2,-8) rectangle (0,0);
+	\fill[color=darkgreen!20] (8,0) rectangle (15,2);
+	\fill[color=darkgreen!20] (-2,-15) rectangle (0,-8);
+	\fill[color=red!20] (15,0) rectangle (20,2);
+	\fill[color=red!20] (-2,-20) rectangle (0,-15);
+}
+
+\uncover<3->{
+	\draw[color=gray] (0,0) rectangle (5,-5);
+	\draw[color=gray] (5,-5) rectangle (8,-8);
+	\draw[color=gray] (8,-8) rectangle (15,-15);
+	\draw[color=gray] (15,-15) rectangle (16,-16);
+	\draw[color=gray] (16,-16) rectangle (17,-17);
+	\draw[color=gray] (17,-17) rectangle (20,-20);
+}
+
+\uncover<2->{
+	\draw[color=gray] (8,0) -- (8,-20);
+	\draw[color=gray] (15,0) -- (15,-20);
+	\draw[color=gray] (0,-8) -- (20,-8);
+	\draw[color=gray] (0,-15) -- (20,-15);
+
+	\node at (0,-4) [above,rotate=90]
+		{$\mathcal{E}_{{\color{blue}\lambda}}(A)$};
+	\node at (4,0) [above]
+		{$\mathcal{E}_{{\color{blue}\lambda}}(A)$};
+	\node at (0,-11.5) [above,rotate=90]
+		{$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$};
+	\node at (11.5,0) [above]
+		{$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$};
+	\node at (0,-18.5) [above,rotate=90]
+		{$\mathcal{E}_{{\color{red}\lambda}}(A)$};
+	\node at (18.5,0) [above]
+		{$\mathcal{E}_{{\color{red}\lambda}}(A)$};
+}
+
+\uncover<2->{
+	{\color{blue}
+	\foreach \x in {1,...,8}{ \L{\x} }
+	}
+	{\color{darkgreen}
+	\foreach \x in {9,...,15}{ \L{\x} }
+	}
+	{\color{red}
+	\foreach \x in {16,...,20}{ \L{\x} }
+	}
+}
+
+\uncover<3->{
+\E{2}
+\E{3}
+\E{4}
+\E{5}
+
+\E{7}
+\E{8}
+
+\E{10}
+\E{11}
+\E{12}
+\E{13}
+\E{14}
+\E{15}
+
+\E{19}
+\E{20}
+}
+
+\draw (0,-20) rectangle (20,0);
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/5/jordanblock.tex b/vorlesungen/slides/5/jordanblock.tex
new file mode 100644
index 0000000..1c3bce9
--- /dev/null
+++ b/vorlesungen/slides/5/jordanblock.tex
@@ -0,0 +1,68 @@
+%
+% jordanblock.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\def\NL{
+\ifthenelse{\boolean{presentation}}{
+\only<-8>{\phantom{\lambda}\llap{$0$}}\only<9->{\lambda}
+}{
+\lambda
+}
+}
+
+\begin{frame}[t]
+\frametitle{Jordan-Block}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Gegeben}
+Matrix $A\in M_n(\Bbbk)$ derart, dass
+\begin{itemize}
+\item<2->
+$A-\lambda I$ nilpotent
+\item<5->
+$A^{n-1}\ne 0$
+\end{itemize}
+\end{block}
+\vspace{-5pt}
+\uncover<3->{
+\begin{block}{Folgerungen}
+Es gibt eine Basis derart, dass
+\begin{enumerate}
+\item<4->
+$A-\lambda I$ hat Normalform einer nilpotenten Matrix
+\item<6->
+Es gibt nur einen Block, da $\dim\ker(A-\lambda I)=1$
+\end{enumerate}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{\ifthenelse{\boolean{presentation}}{\only<-8>{Normalform einer nilpotenten Matrix\strut}}{}\only<9->{Normalform: genau ein Eigenwert\strut}}
+\[
+A\uncover<-8>{-\lambda I}=\begin{pmatrix}
+\NL &1& & & & & & & \\
+ &\NL &1& & & & & & \\
+ & &\NL &\uncover<7->{{\color<7>{red}1}}& & & & & \\
+ & & &\NL &1& & & & \\
+ & & & &\NL &1& & & \\
+ & & & & &\NL &1& & \\
+ & & & & & &\NL &\uncover<7->{{\color<7>{red}1}}& \\
+ & & & & & & &\NL &\uncover<7->{{\color<7>{red}1}}\\
+ & & & & & & & &\NL
+\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-5pt}
+\uncover<8->{%
+\begin{block}{Jordan-Normalform}
+In dieser Basis hat $A$ Jordan-Normalform
+\end{block}}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex
new file mode 100644
index 0000000..3890717
--- /dev/null
+++ b/vorlesungen/slides/5/kernbild.tex
@@ -0,0 +1,86 @@
+%
+% kernbild.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kern und Bild}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<1->{%
+\begin{block}{Kern}
+Lineare Abbildung $f\colon V\to V$
+\[
+\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\}
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Kern von $A^k$}
+\[
+\mathcal{K}^k(f) = \operatorname{ker} f^k
+\]
+\begin{align*}
+\uncover<5->{
+\mathcal{K}^k(f)
+&=
+\{v\in V\;|\; f^{k}(v)=0\}
+}
+\\
+\uncover<6->{
+&\subset
+\{v\in V\;|\; f^{k+1}(v)=0\}
+}
+\\
+\uncover<7->{
+&=\mathcal{K}^{k+1}(f)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Bild}
+Lineare Abbildung $f\colon V\to V$
+\[
+\operatorname{im}f
+=
+\mathcal{J}(f)
+=
+\{f(v)\;|\; v\in V\}
+\]
+\end{block}}
+\uncover<4->{%
+\begin{block}{Bild von $A^k$}
+\[
+\mathcal{J}^k(f) = \operatorname{im}f^k
+\]
+\begin{align*}
+\uncover<8->{
+\mathcal{J}^k(f)
+&=
+\operatorname{im}f^k
+=
+\operatorname{im}(f^{k}\circ f)
+}
+\\
+\uncover<9->{
+&=
+\{f^{k-1} w\;|\; w = f(v)\}
+}
+\\
+\uncover<10->{
+&\subset
+\{f^{k-1} w\;|\; w \in V\}
+}
+\\
+\uncover<11->{
+&=\mathcal{J}^{k-1}(f)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/kernbilder.tex b/vorlesungen/slides/5/kernbilder.tex
new file mode 100644
index 0000000..08581ff
--- /dev/null
+++ b/vorlesungen/slides/5/kernbilder.tex
@@ -0,0 +1,68 @@
+%
+% kernbilder.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{grueneins}{rgb}{0.0,0.4,0.0}
+\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8}
+\definecolor{orangeeins}{rgb}{1.0,0.6,0.0}
+\definecolor{orangezwei}{rgb}{0.8,0.0,0.4}
+
+\begin{frame}[t]
+\frametitle{Kerne und Bilder}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\uncover<2->{
+\begin{scope}[xshift=-4cm,yshift=1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}};
+\node[color=orangeeins] at (1.6,1.3) [right] {$\mathcal{J}^1(A)$};
+\end{scope}
+}
+
+\uncover<3->{
+\begin{scope}[xshift=-4cm,yshift=-1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}};
+\node[color=orangezwei] at (0.9,0.5) {$\mathcal{J}^2(A)$};
+\end{scope}
+}
+
+\begin{scope}[xshift=0cm,yshift=0cm]
+\uncover<1>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/leer.jpg}};
+}
+\uncover<2>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}};
+}
+\uncover<3>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}};
+}
+\uncover<4>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/drei.jpg}};
+}
+\uncover<5->{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kombiniert.jpg}};
+}
+\end{scope}
+
+\uncover<4->{
+\begin{scope}[xshift=4cm,yshift=1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern1.jpg}};
+\node[color=grueneins] at (1.0,1.3) [right] {$\mathcal{K}^1(A)$};
+\end{scope}
+}
+
+\uncover<5->{
+\begin{scope}[xshift=4cm,yshift=-1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern2.jpg}};
+\node[color=gruenzwei] at (0.7,-0.6) {$\mathcal{K}^2(A)$};
+\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/kernbildintro.tex b/vorlesungen/slides/5/kernbildintro.tex
new file mode 100644
index 0000000..9fd7849
--- /dev/null
+++ b/vorlesungen/slides/5/kernbildintro.tex
@@ -0,0 +1,89 @@
+%
+% kernbildintro.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{grueneins}{rgb}{0.0,0.4,0.0}
+\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8}
+\definecolor{orangeeins}{rgb}{1.0,0.6,0.0}
+\definecolor{orangezwei}{rgb}{0.8,0.0,0.4}
+
+\begin{frame}[t]
+\frametitle{Bilder und Kerne}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\begin{scope}[xshift=-3.4cm]
+
+\only<1>{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/leer.jpg}};
+}
+\only<2-3>{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild1.jpg}};
+}
+\uncover<4->{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild2.jpg}};
+}
+\uncover<2->{
+	\fill[color=white,opacity=0.7]  (0.1,2.18) rectangle (4,2.64);
+	\node[color=orangeeins] at (0,2.4) [right]
+		{$\operatorname{im} A = \{Av\;|v\in\mathbb{R}^n\}$};
+}
+\uncover<4->{
+	\node[color=orangezwei] at (4,0.7) [left]
+		{$\operatorname{im} A^2 = \{A^2v\;|v\in\mathbb{R}^n\}$};
+}
+\end{scope}
+
+\begin{scope}[xshift=3.4cm]
+
+\uncover<2->{
+\fill[color=orangeeins!40] (-1,0.5) rectangle (1.8,2);
+}
+\uncover<4->{
+\fill[color=orangezwei!40] (-1.1,-1.7) rectangle (-0.,-0.3);
+}
+
+\node at (0,0) {\begin{minipage}{6cm}
+\begin{align*}
+A&={\scriptstyle\begin{pmatrix*}[r]
+  -0.979& -0.142&  0.917\\
+  -0.260& -0.643&  1.069\\
+  -0.285& -0.449&  0.823
+\end{pmatrix*}}
+\\
+\operatorname{Rang}A&=2
+\\
+\uncover<3->{
+A^2&={\scriptstyle\begin{pmatrix*}[r]
+   0.734& -0.181& -0.295\\
+   0.118& -0.029& -0.047\\
+   0.161& -0.039& -0.065
+\end{pmatrix*}}}\\
+\uncover<3->{
+\operatorname{Rang}A^2&=1}
+\end{align*}
+\end{minipage}};
+
+\only<5>{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern1.jpg}};
+}
+
+\uncover<6->{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern2.jpg}};
+\node[color=gruenzwei] at (-1.35,-3.0) [right] {$\ker A^2 = \{v\;|\; A^2v=0\}$};
+}
+
+\uncover<5->{
+\node[color=grueneins] at (-0.9,3.1) [right] {$\ker A = \{v\;|\; Av=0\}$};
+}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex
new file mode 100644
index 0000000..1116a83
--- /dev/null
+++ b/vorlesungen/slides/5/ketten.tex
@@ -0,0 +1,79 @@
+%
+% ketten.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ketten von Unterräumen}
+\begin{block}{Schachtelung}
+Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt:
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcccc}
+0       &=&\mathcal{K}^0(f)
+	&\subset&\mathcal{K}^1(f)
+	&\subset&\dots
+	&\subset&\mathcal{K}^k(f)
+	&\subset&\mathcal{K}^{k+1}(f)
+	&\subset&\dots
+	&\subset&\displaystyle\bigcup_{k=0}^\infty \mathcal{K}^k(f)
+	&=:&\mathcal{K}(f)
+\\[14pt]
+\Bbbk^n &=&\mathcal{J}^0(f)
+	&\supset&\mathcal{J}^1(f)
+	&\supset&\dots
+	&\supset&\mathcal{J}^{k}(f)
+	&\supset&\mathcal{J}^{k+1}(f)
+	&\supset&\dots
+	&\supset&\displaystyle\bigcap_{k=0}^\infty \mathcal{J}^k(f)
+	&=:&\mathcal{J}(f)
+\end{array}
+\]
+\end{block}
+\vspace{-20pt}
+\uncover<2->{%
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Abildung der Kerne}
+\vspace{-10pt}
+\begin{align*}
+f \mathcal{K}^k(f)
+&=
+\{f(v)\;|\; f^k(v) = 0\}
+\\
+&\subset
+\{ v\;|\; f^{k+1}(v)=0\}
+\\
+&=
+\mathcal{K}^{k+1}(f)
+\\
+\Rightarrow
+f\mathcal{K}(f)&= f\mathcal{K}(f)
+\quad\text{invariant}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Abbildung der Bild}
+\vspace{-10pt}
+\begin{align*}
+f\mathcal{J}^k(f)
+&=
+\{f(f^{k}(v))\;|\; v\in V\}
+\\
+&=
+\{f^{k+1}(v)\;|\; v\in V\}
+\\
+&=
+\mathcal{J}^{k+1}(f)
+\\
+\Rightarrow
+f\mathcal{J}(f)&= \mathcal{J}(f)
+\quad\text{invariant}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}}
+\end{frame}
diff --git a/vorlesungen/slides/5/konvergenzradius.tex b/vorlesungen/slides/5/konvergenzradius.tex
new file mode 100644
index 0000000..a0b4b3a
--- /dev/null
+++ b/vorlesungen/slides/5/konvergenzradius.tex
@@ -0,0 +1,109 @@
+%
+% konvergenzradius.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\setbeamercolor{column}{bg=blue!20}
+\def\punkt#1{
+	\fill[color=blue!30] #1 circle[radius=0.05];
+	\draw[color=blue] #1 circle[radius=0.05];
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Konvergenzradius}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Potenzreihen}
+$f\colon\mathbb{C}\to\mathbb{C}$ (komplex differenzierbar)
+\begin{equation}
+f(z) = \sum_{k=0}^\infty a_kz^k
+\label{reihe}
+\end{equation}
+\end{block}
+\vspace{-8pt}
+\uncover<2->{%
+\begin{block}{Konvergenz}
+\eqref{reihe} konvergiert für $|z| < {\color{darkgreen}R}$,
+\[
+\frac{1}{{\color{darkgreen}R}}
+=
+\limsup_{k\to\infty} |a_k|^{\frac1k}
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Polstellen}
+{\color{darkgreen}$R$} ist der Radius des grössten Kreises um $O$,
+auf dessen Rand eine
+{\color{blue}Polstelle} der Funktion $f(z)$ liegt
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2.5}
+\uncover<2->{
+	\fill[color=red!20] (0,0) circle[radius=\r];
+	\draw[color=red] (0,0) circle[radius=\r];
+}
+\draw[->] (-2.6,0) -- (2.9,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-2.6) -- (0,2.9) coordinate[label={$\operatorname{Im}z$}];
+
+\uncover<2->{
+	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (100:\r);
+	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (220:\r);
+	\node[color=darkgreen] at ($0.5*(100:\r)$) [left] {$R$};
+	\node[color=darkgreen] at ($0.5*(220:\r)+(-0.1,0.1)$)
+		[below right] {$R$};
+
+	\fill[color=white] (0,0) circle[radius=0.05];
+	\draw (0,0) circle[radius=0.05];
+}
+
+\node at (2.8,2.8) {$\mathbb{C}$};
+
+\uncover<3->{
+	\punkt{(100:\r)}
+	\punkt{(220:\r)}
+
+	\begin{scope}
+		\clip (-2.6,-2.6) rectangle (2.9,2.9);
+
+		\punkt{(144.2527:2.7232)}
+		%\punkt{(226.1822:2.5164)}
+		\punkt{(173.7501:3.4140)}
+		\punkt{(267.4103,2.7668)}
+		\punkt{(137.7328:3.1683)}
+		%\punkt{(30.1155:3.3629)}
+		%\punkt{(139.1036:2.5366)}
+		\punkt{(167.4964:3.0503)}
+		\punkt{(289.2650:3.4324)}
+		\punkt{(120.1911:3.2966)}
+		%\punkt{(292.3422:2.7550)}
+		\punkt{(141.4877:2.6494)}
+		\punkt{(70.8326:2.9005)}
+		\punkt{(56.0758:3.2098)}
+		\punkt{(99.0585:3.2340)}
+		\punkt{(299.7242:2.5990)}
+		\punkt{(158.8802:2.6539)}
+		\punkt{(235.2721:2.9476)}
+		\punkt{(108.0584:2.8344)}
+		\punkt{(220.0117:2.7679)}
+
+	\end{scope}
+
+	\begin{scope}[yshift=-3.2cm,xshift=-1.0cm]
+		\punkt{(0,-0.05)}
+		\node at (0,0) [right] {$=$ Polstelle};
+	\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/krbeispiele.tex b/vorlesungen/slides/5/krbeispiele.tex
new file mode 100644
index 0000000..b51df78
--- /dev/null
+++ b/vorlesungen/slides/5/krbeispiele.tex
@@ -0,0 +1,99 @@
+%
+% krbeispiele.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Konvergenzradius --- Beispiele}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Exponentialreihe}
+\vspace{-20pt}
+\begin{align*}
+e^z &= \sum_{k=0}^\infty \frac{z^k}{k!}
+\\
+\uncover<2->{
+\frac1k\log k!
+}
+&\uncover<3->{=\frac1k\sum_{x=1}^k {\color{blue}\log x}}
+\uncover<6->{>\frac1k\int_1^k{\color{red}\log x}\,dx}
+\\
+&
+\ifthenelse{\boolean{presentation}}{
+\only<7>{=\frac1k[x\log x -x]_1^k}
+}{}
+\only<8->{=
+\log k -1 +\frac1k}
+\uncover<9->{\to \infty\phantom{\frac1k}}
+\\
+\uncover<10->{(k!)^{\frac1k}
+&\to\infty}\uncover<11->{ \quad\Rightarrow\quad R = \infty}
+\end{align*}
+\vspace{-40pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.7]
+\uncover<4->{
+\foreach \x in {2,...,9}{
+	\fill[color=blue!20] ({\x-1},0) rectangle ({\x},{ln(\x)});
+	\draw[color=blue] ({\x-1},0) rectangle ({\x},{ln(\x)});
+	\node at ({\x-0.5},{ln(\x)}) [above] {\tiny $\log\x$};
+	\draw (\x,-0.1) -- (\x,0.1);
+	\node at (\x,0) [below] {\tiny$\x$};
+}
+\draw (1,-0.1) -- (1,0.1);
+\uncover<5->{
+\begin{scope}
+	\clip (0,-1) rectangle (9.5,2.5);
+	\fill[color=red!40,opacity=0.5] (0,0) -- (0,-1)
+		-- plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)})
+		-- (9.1,0) -- cycle;
+	\draw[color=red] plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)});
+\end{scope}
+}
+\draw[->] (-0.2,0) -- (9.4,0) coordinate[label={$x$}];
+\draw[->] (0,-1) -- (0,2.5) coordinate[label={right:$y$}];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Geometrische Reihe}
+\vspace{-15pt}
+\begin{align*}
+\uncover<13->{
+\frac{1}{{\color{blue}1}-z}
+&=
+\sum_{k=0}^\infty
+z^k}
+\\
+\uncover<14->{
+a_k&=1}
+\uncover<15->{\quad\Rightarrow\quad
+|a_k|^{\frac1k}=1}
+\\
+\uncover<16->{
+\limsup_{k\to\infty} &= |a_k|^{\frac1k}=1}\uncover<17->{ = \frac1R}
+\uncover<18->{\quad\Rightarrow\quad R=1}
+\end{align*}
+%\uncover<19->{Polstelle bei $z=1$ limitiert Konvergenzradius}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+\clip (-2.2,-1.5) rectangle (2.2,1.5);
+\fill[color=red!20] (0,0) circle[radius=2];
+\draw[color=red] (0,0) circle[radius=2];
+\end{scope}
+\draw[->] (-2.2,0) -- (2.5,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-1.6) -- (0,1.8) coordinate[label={right:$\operatorname{Im}z$}];
+\fill[color=blue!20] (2,0) circle[radius=0.08];
+\draw[color=blue] (2,0) circle[radius=0.08];
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/logarithmusreihe.tex b/vorlesungen/slides/5/logarithmusreihe.tex
new file mode 100644
index 0000000..85ba0ef
--- /dev/null
+++ b/vorlesungen/slides/5/logarithmusreihe.tex
@@ -0,0 +1,53 @@
+%
+% logarithmus.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Logarithmusreihe}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Integralgleichung}
+\vspace{-5pt}
+\begin{align*}
+\log(1+x)&=\int_0^x \frac{1}{1+t}\,dt
+\\
+&\uncover<5->{=
+\int_0^x
+1-t+t^2-t^3+\dots\,dt
+}
+\\
+\uncover<6->{
+&=
+x-\frac{x^2}2+\frac{x^3}{3}-\frac{x^4}{4}+\dots
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Geometrische Reihe}
+\vspace{-5pt}
+\begin{align*}
+\frac{1}{1-q}&=1+q+q^2+q^3+\dots
+\\
+\uncover<3->{
+\frac{1}{1+q}&=1-q+q^2-q^3+\dots
+}
+\end{align*}
+\uncover<4->{Konvergenzradius $1$}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<7->{%
+\begin{block}{Matrix-Logarithmus}
+Für $\operatorname{Sp}(A)\subset \{z\in\mathbb{C}\;|\;|z-1|<1\}$ konvergiert
+\[
+\log A
+=
+(A-I) - \frac12(A-I)^2 + \frac13(A-I)^3 - \frac14(A-I)^4 + \dots
+\]
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex
new file mode 100644
index 0000000..b0a1d82
--- /dev/null
+++ b/vorlesungen/slides/5/motivation.tex
@@ -0,0 +1,67 @@
+%
+% movitation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Motivation}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Matrix $A$ analysieren}
+Matrix $A$ mit Minimalpolynom $m_A(X)$ vom
+Grad $s$
+\end{block}
+\uncover<2->{%
+\begin{block}{Faktorisieren}
+Minimalpolynom faktorisieren
+\[
+m_A(X)
+=
+(X-\mu_1)(X-\mu_2)\dots(X-\mu_s)
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Vertauschen}
+$\sigma\in S_s$ eine Permutation von $1,\dots,s$
+ist
+\begin{align*}
+m_A(X)
+&=
+(X-\mu_{\sigma(1)})
+%(X-\mu_{\sigma(2)})
+\dots
+(X-\mu_{\sigma(s)})
+\\
+0
+&=
+(A-\mu_{\sigma(1)})
+%(A-\mu_{\sigma(2)})
+\dots
+(A-\mu_{\sigma(s)})
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Bedingung für $\mu_k$}
+Permutation wählen so dass $\mu_k$ an erster Stelle steht:
+\[
+0=(A-\mu_k) \prod_{i\ne k}(A-\mu_i) v
+\]
+für alle $v\in\Bbbk^n$.
+\end{block}}
+\uncover<5->{%
+\begin{block}{Eigenwerte}
+Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$
+gibt mit
+\[
+(A-\mu)v = 0
+\Rightarrow Av = {\color{red}\mu} v
+\]
+Eigenwertbedingung
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/nilpotent.tex b/vorlesungen/slides/5/nilpotent.tex
new file mode 100644
index 0000000..ca38c40
--- /dev/null
+++ b/vorlesungen/slides/5/nilpotent.tex
@@ -0,0 +1,190 @@
+%
+% nilpotent.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\feld#1{
+	\fill[color=red!20] (#1,0) rectangle ({#1+1},12);
+}
+\begin{frame}[t]
+\frametitle{$\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ für nilpotente Matrizen}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.42\textwidth}
+Matrix mit dem dargestellten Verlauf von
+${\color{red}\dim\mathcal{K}^k(A)}$
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.42]
+
+\only<2->{
+	\feld{0}
+	\feld{1}
+	\feld{2}
+	\feld{3}
+}
+\only<2->{ \feld{4} }
+\only<2->{ \feld{6} }
+\ifthenelse{\boolean{presentation}}{
+\only<3->{ \feld{5} }
+\only<3->{ \feld{7} }
+\only<4->{ \feld{8} }
+\only<5->{ \feld{9} }
+\only<6->{ \feld{10} }
+\only<7->{ \feld{11} }
+
+\only<1>{ \node at (6,0) [below] {$k=0$}; }
+}{}
+\only<2>{ \node at (6,0) [below] {$k=1$}; }
+\ifthenelse{\boolean{presentation}}{
+\only<3>{ \node at (6,0) [below] {$k=2$}; }
+\only<4>{ \node at (6,0) [below] {$k=3$}; }
+\only<5>{ \node at (6,0) [below] {$k=4$}; }
+\only<6>{ \node at (6,0) [below] {$k=5$}; }
+\only<7>{ \node at (6,0) [below] {$k=6$}; }
+}{}
+
+\draw (0,0) rectangle (12,12);
+\ifthenelse{\boolean{presentation}}{
+\only<1>{
+	\foreach \x in {1,...,12}{
+		\node at ({\x-0.5},{12-\x+0.5}) {$1$};
+	}
+}
+}{}
+\only<2->{
+	\foreach \x in {1,...,12}{
+		\node at ({\x-0.5},{12-\x+0.5}) {$0$};
+	}
+}
+\only<2>{
+	\foreach \x in {7,...,11}{
+		\node at ({\x+0.5},{12-\x+0.5}) {$1$};
+	}
+}
+\ifthenelse{\boolean{presentation}}{
+\only<3->{
+	\foreach \x in {7,...,11}{
+		\node at ({\x+0.5},{12-\x+0.5}) {$0$};
+	}
+}
+\only<3>{
+	\foreach \x in {8,...,11}{
+		\node at ({\x+0.5},{13-\x+0.5}) {$1$};
+	}
+}
+\only<4->{
+	\foreach \x in {8,...,11}{
+		\node at ({\x+0.5},{13-\x+0.5}) {$0$};
+	}
+}
+\only<4>{
+	\foreach \x in {9,...,11}{
+		\node at ({\x+0.5},{14-\x+0.5}) {$1$};
+	}
+}
+\only<5->{
+	\foreach \x in {9,...,11}{
+		\node at ({\x+0.5},{14-\x+0.5}) {$0$};
+	}
+}
+\only<5>{
+	\foreach \x in {10,...,11}{
+		\node at ({\x+0.5},{15-\x+0.5}) {$1$};
+	}
+}
+\only<6->{
+	\foreach \x in {10,...,11}{
+		\node at ({\x+0.5},{15-\x+0.5}) {$0$};
+	}
+}
+\only<6>{
+	\foreach \x in {11,...,11}{
+		\node at ({\x+0.5},{16-\x+0.5}) {$1$};
+	}
+}
+\only<7->{
+	\foreach \x in {11,...,11}{
+		\node at ({\x+0.5},{16-\x+0.5}) {$0$};
+	}
+}
+}{}
+\draw[line width=0.1pt]
+	(0,11) -- (2,11) -- (2,9) -- (4,9) -- (4,6) -- (12,6);
+\draw[line width=0.1pt]
+	(1,12) -- (1,10) -- (3,10) -- (3,8) -- (6,8) -- (6,0);
+\only<2>{
+	\node at (5.5,7.5) {$1$};
+}
+\ifthenelse{\boolean{presentation}}{
+\only<3->{
+	\node at (5.5,7.5) {$0$};
+}
+}{}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.56\textwidth}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\pfad{
+	(0,0) -- (1,3) -- (2,4) -- (3,4.5) -- (4,5) -- (5,5.5) -- (6,6)
+}
+\fill[color=orange!20] \pfad -- (0,6) -- cycle;
+\fill[color=darkgreen!20] \pfad -- (6,0) -- cycle;
+\foreach \y in {0.5,1,...,5.75}{
+	\draw[line width=0.1pt] (0,\y) -- (6,\y);
+}
+\draw[line width=1.4pt] \pfad;
+\draw[->] (-0.1,6) -- (6.5,6); \node at (-0.1,6) [left] {$n$};
+\draw[->] (-0.1,0) -- (6.5,0); \node at (-0.1,0) [left] {$0$};
+\fill (0,0) circle[radius=0.05];
+\fill (1,3) circle[radius=0.05];
+\fill (2,4) circle[radius=0.05];
+\fill (3,4.5) circle[radius=0.05];
+\fill (4,5) circle[radius=0.05];
+\fill (5,5.5) circle[radius=0.05];
+\fill (6,6) circle[radius=0.05];
+\ifthenelse{\boolean{presentation}}{
+\only<1>{
+	\fill[color=red] (0,0) circle[radius=0.08];
+}
+}{}
+\only<2>{
+	\fill[color=red] (1,3) circle[radius=0.08];
+	\draw[color=red] (0,3) -- (1,3);
+	\node[color=red] at (0,3) [left] {$6$};
+}
+\ifthenelse{\boolean{presentation}}{
+\only<3>{
+	\fill[color=red] (2,4) circle[radius=0.08];
+	\draw[color=red] (0,4) -- (2,4);
+	\node[color=red] at (0,4) [left] {$8$};
+}
+\only<4>{
+	\fill[color=red] (3,4.5) circle[radius=0.08];
+	\draw[color=red] (0,4.5) -- (3,4.5);
+	\node[color=red] at (0,4.5) [left] {$9$};
+}
+\only<5>{
+	\fill[color=red] (4,5.0) circle[radius=0.08];
+	\draw[color=red] (0,5.0) -- (4,5.0);
+	\node[color=red] at (0,5.0) [left] {$10$};
+}
+\only<6>{
+	\fill[color=red] (5,5.5) circle[radius=0.08];
+	\draw[color=red] (0,5.5) -- (5,5.5);
+	\node[color=red] at (0,5.5) [left] {$11$};
+}
+\only<7>{
+	\fill[color=red] (6,6.0) circle[radius=0.08];
+}
+}{}
+\draw[color=white] (-0.7,0) -- (-0.7,6);
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/normal.tex b/vorlesungen/slides/5/normal.tex
new file mode 100644
index 0000000..7245608
--- /dev/null
+++ b/vorlesungen/slides/5/normal.tex
@@ -0,0 +1,69 @@
+%
+% normal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Normale Operatoren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Frage}
+$f,g\colon \mathbb{C}\to\mathbb{C}$.
+\\
+In welchen Punkten müssen $f$ und $g$ übereinstimmen, damit
+$f(A)=g(A)$?
+\end{block}
+\uncover<2->{%
+\begin{block}{Definition $f(A)$}
+$f$ durch eine Folge von Polynomen
+appoximieren: $p_n\to f$
+\[
+f(A) = \lim_{n\to\infty}p_n(A)
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<3->{%
+\begin{block}{Vermutung}
+Falls $f(z)=g(z)$ für $z\in\operatorname{Sp}(A)$,
+dann ist $f(A)=g(A)$
+
+\smallskip
+\uncover<4->{%
+{\usebeamercolor[fg]{title}Stimmt für: } $A$ diagonalisierbar
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Beispiel}
+\[
+A=\begin{pmatrix}2&1\\0&2\end{pmatrix}, \quad
+\operatorname{Sp}(A)=\{2\}
+\]
+\uncover<6->{%
+\begin{align*}
+f(z)&=(z-2)^2 &g(z)&=z-2
+\\
+\uncover<7->{
+f(A)&=0&g(A)&=\begin{pmatrix}0&1\\0&0\end{pmatrix}
+}
+\end{align*}}
+\end{block}}
+\vspace{-18pt}
+\uncover<8->{%
+\begin{block}{Normal}
+$A$ heisst {\em normal}, wenn $AA^*=A^*A$
+\begin{itemize}
+\item<9->
+symmetrische Matrizen: $A=A^*$
+\item<10->
+unitäre Matrizen: $A^*=A^{-1}\Rightarrow
+AA^*=AA^{-1}=A^{-1}A=A^*A$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex
new file mode 100644
index 0000000..9457136
--- /dev/null
+++ b/vorlesungen/slides/5/normalnilp.tex
@@ -0,0 +1,237 @@
+%
+% normalnilp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\sx{1.9}
+\def\sy{0.6}
+\def\punkt#1#2#3{
+	\foreach \y in {0,...,#2}{
+	}
+}
+\def\block#1#2{
+	\fill[rounded corners=2pt,color=white]
+		({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05});
+	\draw[rounded corners=2pt]
+		({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05});
+}
+\def\teilmenge#1#2#3{
+	\fill[rounded corners=2pt,color=white]
+		({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=gray]
+		({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\rot#1#2#3{
+	\fill[rounded corners=2pt,color=red!20]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=red]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\hellblau#1#2#3{
+	\fill[rounded corners=2pt,color=blue!20]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=blue!40]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\punkt#1#2{
+	\fill[color=blue] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08];
+}
+\def\bildpunkt#1#2{
+	\fill[color=blue!40] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08];
+}
+\def\pfeil#1#2#3{
+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.1cm] 
+	({-#1*\sx},{(#2-0.5)*\sy}) 
+	--
+	({-(#1-1)*\sx},{(#3-0.5)*\sy}) ;
+}
+\begin{frame}[t]
+\frametitle{Normalform einer nilpotenten Matrix}
+{\usebeamercolor[fg]{title}$A^l=0$ $\Rightarrow$ finde eine ``gute'' Basis}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\vspace{-25pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=darkgreen!20,rounded corners=2pt]
+	({-3*\sx+0.35},0) -- (-0.35,0)
+	--
+	({-1*\sx+0.35},{4*\sy}) -- ({-1*\sx-0.35},{4*\sy})
+	--
+	({-2*\sx+0.35},{7*\sy}) -- ({-2*\sx-0.35},{7*\sy})
+	--
+	({-3*\sx+0.35},{8*\sy}) -- cycle;
+
+\block{0}{0}
+
+\block{1}{4}
+\uncover<10->{
+	\rot{1}{0}{1}
+	\node[color=red] at ({-1*\sx-0.28},{0.5*\sy}) [left] {$\mathcal{C}_{l-2}$};
+}
+\uncover<8->{
+	\hellblau{1}{1}{3}
+}
+\uncover<4->{
+	\hellblau{1}{3}{4}
+}
+
+\block{2}{7}
+\uncover<4->{
+	\hellblau{2}{6}{7}
+}
+\uncover<6->{
+	\rot{2}{4}{6}
+	\node[color=red] at ({-2*\sx-0.28},{5*\sy}) [left] {$\mathcal{C}_{l-1}$};
+}
+\teilmenge{2}{0}{4}
+
+\block{3}{8}
+\uncover<2->{
+	\rot{3}{7}{8}
+	\node[color=red] at ({-3*\sx-0.28},{7.5*\sy}) [left] {$\mathcal{C}_l$};
+}
+\teilmenge{3}{0}{7}
+
+\uncover<3->{
+	\punkt{3}{8}
+}
+\uncover<4->{
+	\pfeil{3}{8}{7}
+	\bildpunkt{2}{7}
+	\pfeil{2}{7}{4}
+	\bildpunkt{1}{4}
+}
+
+\uncover<7->{
+	\punkt{2}{5}
+	\punkt{2}{6}
+}
+\uncover<8->{
+	\pfeil{2}{5}{2}
+	\bildpunkt{1}{3}
+	\pfeil{2}{6}{3}
+	\bildpunkt{1}{2}
+}
+
+\uncover<11->{
+\punkt{1}{1}
+}
+
+\node at ({-3*\sx},0) [below] {$\mathcal{K}^l(A)\mathstrut$};
+\node at ({-2*\sx},0) [below] {$\mathcal{K}^{l-1}(A)\mathstrut$};
+\node at ({-1.45*\sx},0) [below] {$\dots\mathstrut$};
+\node at ({-1*\sx},0) [below] {$\mathcal{K}^1(A)\mathstrut$};
+\node at ({-0*\sx},0) [below] {$0=\mathcal{K}^0(A)\mathstrut$};
+\node[color=gray] at ({-2*\sx},{2*\sy}) [rotate=90] {$\mathcal{K}^1(A)$};
+\node[color=gray] at ({-3*\sx},{3.5*\sy}) [rotate=90] {$\mathcal{K}^{l-1}(A)$};
+\foreach \x in {0,1,2}{
+	\draw[->,shorten >= 0.1cm, shorten <= 0.1cm]
+		({-(\x+1)*\sx},{8.7*\sy}) -- ({-(\x)*\sx},{8.7*\sy});
+	\node at ({-(\x+0.5)*\sx},{8.7*\sy}) [above] {$A$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\vspace{-30pt}
+\begin{enumerate}
+\item<2->	\(
+	\mathcal{K}^l(A)=\mathcal{K}^{l-1}\oplus {\color{red}\mathcal{C}_l}
+	\)
+\item<3->	\(
+	{\color{blue}b_1}\in{\color{red}\mathcal{C}_l}
+	\)
+\item<4->	\(
+	\mathcal{B}_l
+	=
+	\{{\color{blue}b_1},{\color{blue!40}Ab_1},{\color{blue!40}A^2b_1},\dots,
+	{\color{blue!40}A^{l-1}b_1}\}
+	\)
+\item<5->	\(
+	\mathcal{K}^{l-1}(A)
+	=
+	\mathcal{K}^{l-2}(A)
+	\oplus
+	{\color{red}\mathcal{C}_{l-1}}
+	\oplus
+	{\color{blue}A\mathcal{C}_l}
+	\)
+\item<6->	\(
+	{\color{blue}b_2},{\color{blue}b_3}\in{\color{red}\mathcal{C}_{l-1}}
+	\)
+\item<7->	\(
+	\mathcal{B}_{l-1}
+	=
+	\{
+	{\color{blue}b_2},{\color{blue}b_3},
+	{\color{blue!40}Ab_2}, {\color{blue!40}Ab_3},\dots
+	\}
+	\)
+\item<8-> \dots
+\end{enumerate}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+
+\uncover<2-4>{
+	\fill[color=red!20] (2,0) rectangle (3,8);
+}
+\uncover<4->{
+	\fill[color=blue!20] (0,6) rectangle (2,8);
+}
+\uncover<5->{
+	\fill[color=red!20] (2,5) rectangle (3,8);
+	\node[color=blue] at (2.5,6.5) {$1$};
+	\node[color=blue] at (1.5,7.5) {$1$};
+	\node[color=gray] at (0.5,7.5) {$0$};
+	\node[color=gray] at (1.5,6.5) {$0$};
+	\node[color=gray] at (2.5,5.5) {$0$};
+	\draw[color=gray] (0.05,5.05) rectangle (2.95,7.95);
+}
+
+\uncover<6-8>{
+	\fill[color=red!20] (4,0) rectangle (5,8);
+	\fill[color=red!20] (6,0) rectangle (7,8);
+}
+\uncover<8->{
+	\fill[color=blue!20] (3,4) rectangle (4,5);
+	\fill[color=blue!20] (5,2) rectangle (6,3);
+}
+\uncover<9->{
+	\fill[color=red!20] (4,3) rectangle (5,5);
+	\node[color=blue] at (4.5,4.5) {$1$};
+	\node[color=gray] at (3.5,4.5) {$0$};
+	\node[color=gray] at (4.5,3.5) {$0$};
+	\draw[color=gray] (3.05,3.05) rectangle (4.95,4.95);
+	\fill[color=red!20] (6,1) rectangle (7,3);
+	\node[color=blue] at (6.5,2.5) {$1$};
+	\node[color=gray] at (5.5,2.5) {$0$};
+	\node[color=gray] at (6.5,1.5) {$0$};
+	\draw[color=gray] (5.05,1.05) rectangle (6.95,2.95);
+}
+
+\uncover<10>{
+	\fill[color=red!20] (7,0) rectangle (8,8);
+}
+\uncover<11->{
+	\fill[color=red!20] (7,0) rectangle (8,1);
+	\node[color=gray] at (7.5,0.5) {$0$};
+	\draw[color=gray] (7.05,0.05) rectangle (7.95,0.95);
+}
+
+\draw (0,0) rectangle (8,8);
+\node at (-0.1,4) [left] {$A=$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/potenzreihenmethode.tex b/vorlesungen/slides/5/potenzreihenmethode.tex
new file mode 100644
index 0000000..0c3503d
--- /dev/null
+++ b/vorlesungen/slides/5/potenzreihenmethode.tex
@@ -0,0 +1,93 @@
+%
+% potenzreihenmethode.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzreihenmethode}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Lineare Differentialgleichung}
+\vspace{-12pt}
+\begin{align*}
+y'&=ay&&\Rightarrow&y'-ay&=0
+\\
+y(0)&=C
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Potenzreihenansatz}
+\vspace{-12pt}
+\begin{align*}
+y(x)
+&=
+a_0+ a_1x + a_2x^2 + \dots
+\\
+y(0)&=a_0=C
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Lösung}
+\vspace{-12pt}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcr}
+\uncover<3->{ y'(x)}
+	\uncover<5->{
+	&=&\phantom{(} a_1\phantom{\mathstrut-aa_0)}
+	&+& 2a_2\phantom{\mathstrut-aa_1)}x
+		&+& 3a_3\phantom{\mathstrut-aa_2)}x^2
+			&+& 4a_4\phantom{\mathstrut-aa_3)}x^3
+				&+& \dots}\\
+\uncover<3->{-ay(x)}
+	\uncover<6->{
+	&=&\mathstrut-aa_0 \phantom{)}
+	&-&  aa_1\phantom{)}x
+		&-&  aa_2\phantom{)}x^2
+			&-&  aa_3\phantom{)}x^3
+				&-& \dots}\\[2pt]
+\hline
+\\[-10pt]
+\uncover<3->{0}
+	\uncover<7->{
+	&=&(a_1-aa_0)
+	&+& (2a_2-aa_1)x
+		&+& (3a_3-aa_2)x^2
+			&+& (4a_4-aa_3)x^3
+				&+& \dots}\\
+\end{array}
+\]
+\begin{align*}
+\uncover<4->{
+a_0&=C}\uncover<8->{,
+\quad
+a_1=aa_0=aC}\uncover<9->{,
+\quad
+a_2=\frac12a^2C}\uncover<10->{,
+\quad
+a_3=\frac16a^3C}\uncover<11->{,
+\dots
+a_k=\frac1{k!}a^kC}
+\hspace{3cm}
+\\
+\uncover<4->{
+\Rightarrow y(x) &= C}\uncover<8->{+Cax}\uncover<9->{ + C\frac12(ax)^2}
+\uncover<10->{ + C \frac16(ac)^3}
+\uncover<11->{ + \dots+C\frac{1}{k!}(ax)^k+\dots}
+\ifthenelse{\boolean{presentation}}{
+\only<12>{
+=
+C\sum_{k=0}^\infty \frac{(ax)^k}{k!}}
+}{}
+\uncover<13->{=
+Ce^{ax}}
+\end{align*}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/reellenormalform.tex b/vorlesungen/slides/5/reellenormalform.tex
new file mode 100644
index 0000000..4ceabe9
--- /dev/null
+++ b/vorlesungen/slides/5/reellenormalform.tex
@@ -0,0 +1,115 @@
+%
+% reellenormalform.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Reelle Normalform}
+$A\in M_n(\mathbb{R})\subset M_n(\mathbb{C})$ hat reelle und Paare von
+konjugiert komplexen Eigenwerten
+\medskip
+
+$\Rightarrow$ Konjugiert komplexe Eigenvektoren $v$ und $\overline{v}$,
+$x=\operatorname{Re}v$ und $y=\operatorname{Im}v$
+\begin{align*}
+\only<-2>{
+\begin{pmatrix}
+Av\\
+A\overline v
+\end{pmatrix}
+=
+\begin{pmatrix}
+Ax+Ay J \\
+Ax-Ay J
+\end{pmatrix}
+&=
+\begin{pmatrix}
+\lambda v\\
+\overline{\lambda}\overline{v}
+\end{pmatrix}
+=
+\begin{pmatrix}
+a+bJ & 0 \\
+  0  & a-bJ
+\end{pmatrix}
+\begin{pmatrix}
+x+ yJ\\
+x- yJ
+\end{pmatrix}
+\\
+}
+\only<2-3>{
+\begin{pmatrix}
+Ax&-Ay\\
+Ay& Ax\\
+Ax& Ay\\
+-Ay&Ax
+\end{pmatrix}
+&=
+\begin{pmatrix}
+a&-b& 0& 0\\
+b& a& 0& 0\\
+0& 0& a& b\\
+0& 0&-b& a
+\end{pmatrix}
+\begin{pmatrix}
+x&-y\\
+y& x\\
+x& y\\
+-y&x
+\end{pmatrix}
+\\
+}
+\only<3-4>{
+\ifthenelse{\boolean{presentation}}{
+\begin{pmatrix}
+Ax&-Ay\\
+Ax& Ay\\
+Ay& Ax\\
+-Ay&Ax
+\end{pmatrix}
+&
+=
+\begin{pmatrix}
+a& 0&-b& 0\\
+0& a& 0& b\\
+b& 0& a& 0\\
+0&-b& 0& a
+\end{pmatrix}
+\begin{pmatrix}
+x&-y\\
+x& y\\
+y& x\\
+-y&x
+\end{pmatrix}
+\Rightarrow
+\\
+}{}
+}
+\only<4->{
+Ax &= ax -by \\
+Ay &= bx +ay
+}
+\end{align*}
+\uncover<5->{%
+D.h. in Basis $x=\operatorname{Re}v,y=\operatorname{Im}v$ hat $A$ die Matrix
+$\begin{pmatrix}a&-b\\b&a\end{pmatrix}$}
+\uncover<6->{%
+\[
+\text{
+Reeller
+Jordan-Block:
+}
+\qquad
+J_{\lambda,\overline{\lambda}}
+=
+\begin{pmatrix}
+a&-b&1& 0&0& 0\\
+b& a&0& 1&0& 0\\
+ &  &a&-b&1& 0\\
+ &  &b& a&0& 1\\
+ &  & &  &a&-b\\
+ &  & &  &b& a
+\end{pmatrix}
+\]}
+\end{frame}
diff --git a/vorlesungen/slides/5/satzvongelfand.tex b/vorlesungen/slides/5/satzvongelfand.tex
new file mode 100644
index 0000000..3cf8710
--- /dev/null
+++ b/vorlesungen/slides/5/satzvongelfand.tex
@@ -0,0 +1,89 @@
+%
+% satzvongelfand.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{0pt}
+\setlength{\belowdisplayskip}{0pt}
+\setbeamercolor{block body}{bg=blue!20}
+\setbeamercolor{block title}{bg=blue!20}
+\frametitle{Satz von Gelfand}
+{\usebeamercolor[fg]{title}Behauptung:} $\varrho(A)=\pi(A)$\uncover<2->{,
+$A(\varepsilon) = \displaystyle\frac{A}{\varrho(A)+\varepsilon}$}\uncover<3->{,
+$\varrho(A(\varepsilon))=\displaystyle\frac{\varrho(A)}{\varrho(A)+\varepsilon}
+\uncover<4->{=\frac{1}{1+\varepsilon/\varrho(A)}}$}
+
+\uncover<5->{%
+%{\usebeamercolor[fg]{title}Beweisidee:}
+%$\displaystyle\pi\biggl(\frac{A}{\varrho(A)+\epsilon}\biggr)
+%=
+%\frac{\pi(A)}{\varrho(A)+\epsilon}$ berechnen
+\vspace{-5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{$\varepsilon < 0$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<6->{
+\varrho(A(\varepsilon))&>1}\uncover<7->{\quad\Rightarrow\quad \|A(\varepsilon)^k\|\to \infty}
+\\
+\uncover<8->{\|A(\varepsilon)^k\| &\ge m\varrho(A(\varepsilon))^k}
+\\
+\uncover<9->{\|A(\varepsilon)^k\|^{\frac1k} &\ge m^{\frac1k} \varrho(A(\varepsilon))}
+\\
+\uncover<10->{\pi(A) &\ge \lim_{k\to\infty}m^{\frac1k}\varrho(A(\varepsilon))}
+\\
+&\uncover<11->{= \varrho(A(\varepsilon))}\uncover<12->{ > 1}
+\\
+\uncover<13->{\frac{ \pi(A(\varepsilon))}{\varrho(A)+\varepsilon} &> 1}
+\\
+\uncover<14->{
+\pi(A) &> \varrho(A)+\varepsilon
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\varepsilon > 0$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<16->{
+\varrho(A(\varepsilon)) &<1}
+\uncover<17->{\quad\Rightarrow\quad \|A(\varepsilon)^k\| \to 0}
+\\
+\uncover<18->{\|A(\varepsilon)^k\|
+&\le M\varrho(A(\varepsilon))^k}
+\\
+\uncover<19->{
+\|A(\varepsilon)^k\|^{\frac1k}
+&\le M^{\frac1k}\varrho(A(\varepsilon))
+}
+\\
+\uncover<20->{
+\pi(A(\varepsilon))
+&\le
+\varrho(A(\varepsilon)) \lim_{k\to\infty} M^{\frac1k}
+}
+\\
+&\uncover<21->{= \varrho(A(\varepsilon))}
+\uncover<22->{ < 1}
+\\
+\uncover<23->{\frac{\pi(A)}{\varrho(A)+\varepsilon}&< 1}
+\\
+\uncover<24->{\pi(A)&< \varrho(A) + \varepsilon}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}}
+\uncover<15->{%
+\vspace{2pt}
+{\usebeamercolor[fg]{title}Folgerung:}
+$\varrho(A)-\varepsilon < \pi(A) \uncover<25->{< \varrho(A)+\varepsilon}\quad\forall\varepsilon>0
+\uncover<26->{
+\qquad\Rightarrow\qquad
+\varrho(A)=\pi(A)}$
+}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/spektralgelfand.tex b/vorlesungen/slides/5/spektralgelfand.tex
new file mode 100644
index 0000000..9342cd6
--- /dev/null
+++ b/vorlesungen/slides/5/spektralgelfand.tex
@@ -0,0 +1,190 @@
+%
+% spektralgelfand.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\eigenwert#1#2{
+	\fill[color=blue!30] (#1:#2) circle[radius=0.05];
+	\draw[color=blue] (#1:#2) circle[radius=0.05];
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spektral- und Gelfand-Radius}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth}
+\begin{block}{Spektralradius}
+\vspace{-10pt}
+\[
+\varrho(A)
+=
+\sup\{|\lambda|\;|\; \text{{\color{blue}$\lambda$} ist EW von $A$}\}
+\]
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<5->{
+	\fill[color=red!30] (0,0) circle[radius=2.2];
+	\draw[color=red] (0,0) circle[radius=2.2];
+}
+
+\uncover<3->{
+	\eigenwert{190.46}{1.3365}
+	%\eigenwert{52.663}{2.1819}
+	\eigenwert{281.94}{1.7305}
+	\eigenwert{21.29}{1.0406}
+	\eigenwert{69.511}{1.56}
+	\eigenwert{63.365}{1.3535}
+	\eigenwert{281.43}{0.31994}
+	\eigenwert{313.1}{1.5419}
+	\eigenwert{118.14}{1.1966}
+	\eigenwert{195.75}{0.41156}
+	\eigenwert{233.42}{1.5613}
+	\eigenwert{25.203}{1.1936}
+	\eigenwert{53.375}{1.4886}
+	\eigenwert{346.13}{2.1073}
+	\eigenwert{246.47}{1.124}
+	\eigenwert{35.451}{1.99}
+	\eigenwert{212.43}{1.9708}
+	\eigenwert{58.479}{0.61602}
+	\eigenwert{344.37}{1.6107}
+	\eigenwert{305.42}{1.7075}
+	\eigenwert{29.693}{0.28791}
+	\eigenwert{195.82}{0.63079}
+	\eigenwert{209.71}{0.25669}
+	\eigenwert{51.355}{0.7247}
+	\eigenwert{356.43}{1.0867}
+	\eigenwert{33.119}{0.7328}
+	\eigenwert{73.131}{1.5021}
+	\eigenwert{345.67}{0.37564}
+	\eigenwert{76.52}{0.71763}
+	%\eigenwert{197.04}{2.1431}
+	\eigenwert{217.87}{1.7704}
+	\eigenwert{172.93}{1.1204}
+	\eigenwert{339.19}{1.5305}
+	\eigenwert{272.86}{2.04}
+	\eigenwert{168.8}{1.6289}
+	\eigenwert{248.68}{0.70879}
+	\eigenwert{237.98}{0.71097}
+	\eigenwert{81.411}{1.8461}
+	\eigenwert{224.65}{1.0827}
+	\eigenwert{357.54}{0.291}
+	\eigenwert{325.26}{1.2778}
+	\eigenwert{150.97}{0.32358}
+	\eigenwert{260.68}{1.4077}
+	\eigenwert{116.29}{1.0715}
+	\eigenwert{358.25}{0.99667}
+	\eigenwert{276.2}{0.077375}
+	\eigenwert{316.16}{0.77763}
+	\eigenwert{69.398}{1.2818}
+	\eigenwert{353.5}{0.74099}
+	\eigenwert{4.7935}{1.391}
+	\eigenwert{136.98}{1.7572}
+	\eigenwert{45.62}{1.9649}
+	\eigenwert{299.96}{0.19199}
+	\eigenwert{187.32}{0.63805}
+	\eigenwert{272.88}{1.1467}
+	\eigenwert{231.85}{1.5763}
+	\eigenwert{124.24}{0.77024}
+	\eigenwert{196.24}{2.0375}
+	\eigenwert{186.33}{1.0656}
+	%\eigenwert{22.812}{2.1616}
+	\eigenwert{37.982}{0.038956}
+	\eigenwert{142.36}{1.7944}
+	\eigenwert{56.863}{1.8952}
+	\eigenwert{4.6281}{1.1857}
+	\eigenwert{71.674}{0.07642}
+	\eigenwert{94.049}{1.8985}
+	\eigenwert{97.294}{0.23412}
+	\eigenwert{84.739}{0.31209}
+	\eigenwert{147.42}{1.8434}
+	\eigenwert{160.67}{0.76956}
+	\eigenwert{292.5}{0.85697}
+	\eigenwert{308.1}{1.7061}
+	\eigenwert{68.669}{2.111}
+	\eigenwert{86.866}{1.1271}
+	\eigenwert{124.72}{1.3019}
+	\eigenwert{267.36}{0.7462}
+	\eigenwert{295.78}{1.0425}
+	\eigenwert{44.972}{0.65363}
+	\eigenwert{34.534}{1.2817}
+	\eigenwert{357.78}{2.0592}
+	\eigenwert{147.52}{0.020535}
+	%\eigenwert{28.502}{2.1964}
+	\eigenwert{343.48}{2.0968}
+	\eigenwert{129.96}{0.80371}
+	\eigenwert{254.75}{1.5775}
+	\eigenwert{89.91}{0.88605}
+	\eigenwert{20.35}{0.66065}
+	\eigenwert{60.382}{1.7585}
+	\eigenwert{158.87}{0.68399}
+	\eigenwert{328.44}{1.504}
+	\eigenwert{189.41}{0.33879}
+	\eigenwert{273.47}{0.11109}
+	\eigenwert{285.99}{0.66704}
+	\eigenwert{311.42}{2.0266}
+	\eigenwert{32.636}{0.5713}
+	\eigenwert{221.35}{2.1329}
+	\eigenwert{50.983}{1.1957}
+	\eigenwert{53.298}{1.2982}
+	\eigenwert{101.4}{1.9051}
+	\eigenwert{71.999}{0.25671}
+}
+
+\uncover<2->{
+	\draw[->] (-2.4,0) -- (2.7,0)
+		coordinate[label={$\operatorname{Re}z$}];
+	\draw[->] (0,-2.4) -- (0,2.5)
+		coordinate[label={right:$\operatorname{Im}z$}];
+}
+\uncover<4->{
+	\fill[color=darkgreen] (0,0) circle[radius=0.05];
+	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (150:2.2);
+	\node[color=darkgreen] at ($(150:1.85)+(0.4,0)$)
+		[below left] {$\varrho(A)$};
+}
+\uncover<3->{
+	\eigenwert{150}{2.2}
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Gelfand-Radius}
+\[
+\pi(A)
+=
+\lim_{k\to\infty} \|A^k\|^{\frac{1}{k}}
+\]
+\end{block}}
+\vspace{-8pt}
+\uncover<7->{%
+\begin{block}{Konvergenz der Neumann-Reihe}
+$
+\uncover<8->{t<1/\pi(A)\;}
+\uncover<10->{\Rightarrow\; \exists q}
+\uncover<11->{,N}$
+\begin{align*}
+\uncover<9->{ t\pi(A) & \only<10->{< q} < 1 }
+\\
+\uncover<11->{ \|(tA)^k\|^{\frac1k} &\le q }
+\\
+\uncover<12->{
+\|(tA)^k\|
+&\le
+(t\pi(A))^k<q^k
+}
+\end{align*}
+\uncover<11->{für $k>N$.}
+\uncover<13->{
+$\Rightarrow$
+$(1-tA)^{-1}=\displaystyle\sum_{k=0}^\infty (tA)^k$ konvergiert für $t<1/\pi(A)$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/spektrum.tex b/vorlesungen/slides/5/spektrum.tex
new file mode 100644
index 0000000..6cbdd7f
--- /dev/null
+++ b/vorlesungen/slides/5/spektrum.tex
@@ -0,0 +1,76 @@
+%
+% spektrum.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spektrum}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+$A\colon V\to V$ beschränkter Operator zwischen Banach-Räumen
+\[
+\operatorname{Sp}A
+=
+\left\{
+\lambda\in\mathbb{C}
+\;\left|\;
+\begin{minipage}{2cm}\raggedright
+$A-\lambda I$ nicht invertierbar
+\end{minipage}
+\right.
+\right\}
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Endlichdimensionale Räume}
+\vspace{-15pt}
+\begin{align*}
+&\lambda\in\operatorname{Sp}A
+\\
+\uncover<3->{
+\Leftrightarrow\quad&\text{$(A-\lambda I)$ nicht invertierbar}
+}
+\\
+\uncover<4->{
+\Leftrightarrow\quad&\text{$(A-\lambda I)$ singulär}
+}
+\\
+\uncover<5->{
+\Leftrightarrow\quad&\ker(A-\lambda I)\ne 0
+}
+\\
+\uncover<6->{
+\Leftrightarrow\quad&\exists v\in V, v\ne 0, Av=\lambda v
+}
+\end{align*}
+\uncover<7->{%
+$\Rightarrow$ $\operatorname{Sp}A$ ist die Menge der Eigenwerte
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Unendlichdimensional}
+Es gibt eine Folge $x_n\in V$ von Einheitsvektoren
+$\|x_n\|=1$
+mit
+\begin{align*}
+\lim_{n\to\infty} (A - \lambda)x_n &= 0
+\end{align*}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Spektrum und Norm}
+\[
+\operatorname{Sp}(A)
+\subset
+\{\lambda\in\mathbb{C}\;|\;
+|\lambda|\le \|A\|\}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/stoneweierstrass.tex b/vorlesungen/slides/5/stoneweierstrass.tex
new file mode 100644
index 0000000..3f9cab5
--- /dev/null
+++ b/vorlesungen/slides/5/stoneweierstrass.tex
@@ -0,0 +1,11 @@
+%
+% stoneweierstrass.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Stone-Weierstrass}
+
+TODO XXX
+
+\end{frame}
diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex
new file mode 100644
index 0000000..f0c4401
--- /dev/null
+++ b/vorlesungen/slides/5/unitaer.tex
@@ -0,0 +1,75 @@
+%
+% unitaer.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Unitäre Matrizen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenwerte}
+$U$ unitär lässt das Skalarprodukt invariant
+\[
+\langle Ux,Uy\rangle
+=
+\langle x,y\rangle
+\]
+\uncover<2->{%
+$\lambda$ ein Eigenwert mit Eigenvektor $v$:
+\begin{align*}
+\langle v,v\rangle
+&=
+\langle Uu,Uv\rangle
+\uncover<3->{= \langle \lambda v,\lambda v\rangle}
+\uncover<4->{= |\lambda|^2 \langle v,v\rangle}
+\\
+\uncover<5->{\Rightarrow\;|\lambda|&=1}
+\end{align*}}
+\end{block}
+\uncover<6->{%
+\begin{block}{Diagonalisierbar}
+Unitäre Matrizen sind über $\mathbb{C}$ diagonalisierbar
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Grosse Jordan-Blöcke?}
+Falls es Vektoren $v,w$ gibt mit
+\begin{align*}
+\uncover<7->{ Uv&=\lambda v}
+\\
+\uncover<8->{ Uw&=\lambda w + v}
+\intertext{\uncover<9->{Skalarprodukt:}}
+\uncover<10->{
+\langle v,w\rangle
+&=
+\langle Uv,Uw\rangle}
+\\
+\uncover<11->{
+&=
+\langle \lambda v,\lambda w\rangle
++
+\langle\lambda v,v\rangle}
+\\
+\uncover<12->{
+&=
+|\lambda|^2 \langle v,w\rangle
++
+\langle\lambda v,v\rangle}
+\\
+\uncover<13->{
+&=
+\langle v,w\rangle
++
+\lambda \| v\|^2}
+\\
+\uncover<14->{
+\Rightarrow\quad
+0&=\|v\|^2\quad\Rightarrow\quad \|v\|=0}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/verzerrung.tex b/vorlesungen/slides/5/verzerrung.tex
new file mode 100644
index 0000000..8d6514c
--- /dev/null
+++ b/vorlesungen/slides/5/verzerrung.tex
@@ -0,0 +1,121 @@
+%
+% verzerrung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\r{1.10}
+\def\s{1.12}
+\def\q{1.23}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Verzerrung}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.49\textwidth}
+\begin{block}{Abbildung $A\colon v\mapsto Av$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=2.5]
+\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1];
+
+\coordinate (a1) at (0.974,0.171);
+\coordinate (a2) at (0.037,1.018);
+
+\coordinate (v1) at (-0.5216,0.8532);
+\coordinate (v2) at (-0.3343,-0.9425);
+
+\foreach \a in {0,5,...,355}{
+	\draw[color=red,line width=1.2pt] 
+		($cos(\a)*(a1)+sin(\a)*(a2)$) --
+		($cos(\a+5)*(a1)+sin(\a+5)*(a2)$);
+}
+\foreach \a in {1,...,144}{
+	\only<\a>{
+		\fill[color=red,line width=1.4pt]
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03];
+		\draw[->,color=red,line width=1.4pt] (0,0) --
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$);
+		\draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1);
+		\fill[color=blue] ({5*\a}:1) circle[radius=0.03];
+		\node[color=blue] at ({5*\a}:\r) {$v$};
+		\node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$)
+			{$Av$};
+	}
+}
+
+\begin{scope}
+\clip (-1.2,-1.1) rectangle (1.2,1.1);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2);
+\end{scope}
+
+\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}];
+\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}];
+
+\node[color=darkgreen] at (v1) [above left] {$v_1$};
+\node[color=darkgreen] at (v2) [below left] {$v_2$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.49\textwidth}
+\uncover<73->{%
+\begin{block}{Abbildung $A\colon v\mapsto (A-\lambda)v$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=2.5]
+\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1];
+
+\coordinate (a1) at (0.121,0.343);
+\coordinate (a2) at (0.074,0.209);
+
+\coordinate (v1) at (-0.5216,0.8532);
+\coordinate (v2) at (-0.3343,-0.9425);
+
+\begin{scope}
+\clip (-1.2,-1.2) rectangle (1.2,1.2);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$);
+\end{scope}
+
+\foreach \a in {0,5,...,355}{
+	\draw[color=red!60,line width=4pt] 
+		($cos(\a)*(a1)+sin(\a)*(a2)$) --
+		($cos(\a+5)*(a1)+sin(\a+5)*(a2)$);
+}
+\foreach \a in {73,...,144}{
+	\only<\a>{
+		\fill[color=red,line width=1.4pt]
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03];
+		\draw[->,color=red,line width=1.4pt] (0,0) --
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$);
+		\draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1);
+		\fill[color=blue] ({5*\a}:1) circle[radius=0.03];
+		\node[color=blue] at ({5*\a}:\r) {$v$};
+		\node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$)
+			{$(A-\lambda)v$};
+	}
+}
+
+\begin{scope}
+\clip (-1.2,-1.1) rectangle (1.2,1.1);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2);
+\end{scope}
+
+\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}];
+\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}];
+
+\node[color=darkgreen] at (v1) [above left] {$v_1$};
+\node[color=darkgreen] at (v2) [below left] {$v_2$};
+
+\end{tikzpicture}
+\end{center}
+
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/verzerrung/verzerrung.m b/vorlesungen/slides/5/verzerrung/verzerrung.m
new file mode 100644
index 0000000..028e7f9
--- /dev/null
+++ b/vorlesungen/slides/5/verzerrung/verzerrung.m
@@ -0,0 +1,13 @@
+#
+# verzerrung.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+rand("seed", 4712);
+
+A = eye(2) + 1.0 * (rand(2,2) - 0.5 * ones(2,2))
+
+[V, lambda] = eig(A)
+
+B = A - lambda(1,1) * eye(2)
diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex
new file mode 100644
index 0000000..a734d69
--- /dev/null
+++ b/vorlesungen/slides/5/zerlegung.tex
@@ -0,0 +1,105 @@
+%
+% zerlegung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Zerlegung in Eigenräume}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.38]
+\uncover<2->{
+\fill[color=blue!20] (0,11) rectangle (4,15);
+\fill[color=red!20] (4,0) rectangle (15,11);
+}
+\uncover<3->{
+\fill[color=red!40] (9,0) rectangle (15,6);
+\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11);
+}
+\uncover<4->{
+\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6);
+\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3);
+}
+
+\uncover<2->{
+\draw[line width=0.1pt] (0,11) -- (15,11);
+\draw[line width=0.1pt] (4,0) -- (4,15);
+}
+
+\uncover<3->{
+\draw[line width=0.1pt] (0,6) -- (15,6);
+\draw[line width=0.1pt] (9,0) -- (9,15);
+}
+
+\uncover<4->{
+\draw[line width=0.1pt] (0,3) -- (15,3);
+\draw[line width=0.1pt] (12,0) -- (12,15);
+}
+\draw (0,0) rectangle (15,15);
+\uncover<2->{
+\node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$};
+\node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$};
+\node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$};
+}
+\uncover<3->{
+\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$};
+\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$};
+}
+\uncover<4->{
+\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$};
+\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$};
+\node at (10.5,4.5) {$(f_2)_{|\mathcal{E}_{\lambda_3}}$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Iteration}
+$\Lambda=\{\lambda_1,\dots,\lambda_s\}$ Eigenwerte
+\begin{align*}
+\uncover<2->{
+V
+&=
+\mathcal{K}(f-\lambda_1)
+\oplus
+\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!20] (0,0) rectangle (1.83,1.1);}}}}
+\underbrace{\mathcal{J}(f-\lambda_1)}_{\displaystyle=V_1}
+}
+\\[-15pt]
+\uncover<2->{
+f_1 &= f_{|V_1}
+}
+\\[10pt]
+\uncover<3->{
+V_1
+&=
+\mathcal{K}(f_1-\lambda_2)
+\oplus
+\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!40] (0,0) rectangle (1.9,1.1);}}}}
+\underbrace{\mathcal{J}(f_1-\lambda_2)}_{\displaystyle=V_2}
+}
+\\[-15pt]
+\uncover<3->{
+f_1 &= f_{|V_1}
+}
+\\
+\uncover<4->{
+&\phantom{0}\vdots
+}
+\end{align*}
+\uncover<5->{%
+$\Rightarrow$ $f$ hat {\color{blue}Blockdiagonalform} für die Zerlegung
+\begin{align*}
+V&=\bigoplus_{\lambda\in\Lambda} \mathcal{E}_{\lambda}
+\end{align*}}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
new file mode 100644
index 0000000..d46dc7f
--- /dev/null
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -0,0 +1,32 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter8 =								\
+	../slides/8/dgraph.tex						\
+	../slides/8/graph.tex						\
+	../slides/8/grad.tex						\
+	../slides/8/inzidenz.tex					\
+	../slides/8/inzidenzd.tex					\
+	../slides/8/diffusion.tex					\
+	../slides/8/laplace.tex						\
+	../slides/8/produkt.tex						\
+	../slides/8/fourier.tex						\
+	../slides/8/spanningtree.tex					\
+	../slides/8/pfade/adjazenz.tex					\
+	../slides/8/pfade/langepfade.tex				\
+	../slides/8/pfade/beispiel.tex					\
+	../slides/8/pfade/gf.tex					\
+	../slides/8/floyd-warshall/problem.tex				\
+	../slides/8/floyd-warshall/rekursion.tex			\
+	../slides/8/floyd-warshall/iteration.tex			\
+	../slides/8/floyd-warshall/wegiteration.tex			\
+	../slides/8/floyd-warshall/wege.tex				\
+	../slides/8/tokyo/google.tex					\
+	../slides/8/tokyo/bahn0.tex					\
+	../slides/8/tokyo/bahn1.tex					\
+	../slides/8/tokyo/bahn2.tex					\
+	../slides/8/chapter.tex
+
diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex
new file mode 100644
index 0000000..6a0b13f
--- /dev/null
+++ b/vorlesungen/slides/8/chapter.tex
@@ -0,0 +1,32 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{8/graph.tex}
+\folie{8/dgraph.tex}
+\folie{8/grad.tex}
+\folie{8/inzidenz.tex}
+\folie{8/inzidenzd.tex}
+\folie{8/diffusion.tex}
+\folie{8/laplace.tex}
+\folie{8/produkt.tex}
+\folie{8/fourier.tex}
+\folie{8/spanningtree.tex}
+
+\folie{8/pfade/adjazenz.tex}
+\folie{8/pfade/langepfade.tex}
+\folie{8/pfade/beispiel.tex}
+\folie{8/pfade/gf.tex}
+
+\folie{8/floyd-warshall/problem.tex}
+\folie{8/floyd-warshall/rekursion.tex}
+\folie{8/floyd-warshall/iteration.tex}
+\folie{8/floyd-warshall/wegiteration.tex}
+\folie{8/floyd-warshall/wege.tex}
+
+\folie{8/tokyo/google.tex}
+\folie{8/tokyo/bahn0.tex}
+\folie{8/tokyo/bahn1.tex}
+\folie{8/tokyo/bahn2.tex}
+
diff --git a/vorlesungen/slides/8/dgraph.tex b/vorlesungen/slides/8/dgraph.tex
new file mode 100644
index 0000000..6b5864a
--- /dev/null
+++ b/vorlesungen/slides/8/dgraph.tex
@@ -0,0 +1,100 @@
+%
+% dgraph.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}
+\frametitle{Gerichteter Graph}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2.4}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\uncover<3->{
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+	\draw[color=white,line width=5pt] (B) -- (D);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+}
+
+\uncover<2->{
+	\draw (A) circle[radius=0.2];
+	\draw (B) circle[radius=0.2];
+	\draw (C) circle[radius=0.2];
+	\draw (D) circle[radius=0.2];
+	\draw (E) circle[radius=0.2];
+
+	\node at (A) {$1$};
+	\node at (B) {$2$};
+	\node at (C) {$3$};
+	\node at (D) {$4$};
+	\node at (E) {$5$};
+}
+\node at (0,0) {$G$};
+
+\uncover<3->{
+	\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+	\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+	\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+	\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+	\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+	\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+	\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+}
+
+\uncover<7->{
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=red]
+                (E) to[out=-18,in=-126,distance=2cm] (E);
+}
+
+\uncover<9->{
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=darkgreen]
+		(D) to[out=120,in=-120] (C);
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Definition}
+Ein gerichteter Graph $G=(V,E)$ ist
+\begin{enumerate}
+\item<2-> Eine Menge $V$ von Knoten (Vertizes)
+$V=\{v_1,v_2,\dots\}$
+\item<3->
+Eine Menge $E$ von gerichteten Kanten
+(Edges)
+\[
+E\subset \{ (v_1,v_2)\;|\; v_i\in V\}
+\]
+\end{enumerate}
+\end{block}
+\vspace{-30pt}
+\uncover<6->{%
+\begin{block}{Achtung}
+\begin{itemize}
+\item<6-> Kanten sind {\em geordnete} Paare
+\uncover<7->{$\Rightarrow$ {\color{red}Schleifen} sind möglich}
+\item<8-> Kanten sind immer ``Einbahnstrassen''
+\item<9-> {\color{darkgreen}Gegenrichtung explizit angeben}
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/diffusion.tex b/vorlesungen/slides/8/diffusion.tex
new file mode 100644
index 0000000..0d07a27
--- /dev/null
+++ b/vorlesungen/slides/8/diffusion.tex
@@ -0,0 +1,89 @@
+%
+% diffusion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Diffusion}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D);
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A);
+
+\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm]
+	(A) -- (E);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(A) -- (C);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(B) -- (D);
+
+\fill[color=red] (A) circle[radius=0.3];
+\fill[color=red!50] (B) circle[radius=0.3];
+\fill[color=white] (C) circle[radius=0.3];
+\fill[color=blue!50] (D) circle[radius=0.3];
+\fill[color=blue] (E) circle[radius=0.3];
+
+\draw (A) circle[radius=0.3];
+\draw (B) circle[radius=0.3];
+\draw (C) circle[radius=0.3];
+\draw (D) circle[radius=0.3];
+\draw (E) circle[radius=0.3];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\begin{block}{Knotenfunktion}
+$f\colon V\to \mathbb{R}$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Fluss}
+Je grösser die Differenz zu den Nachbarn, desto grösser der Fluss in 
+den Knoten:
+\begin{align*}
+\frac{df(v)}{dt}
+&=
+\kappa \sum_{\text{$v'$ Nachbar von $v$}} (f(v')-f(v))
+\end{align*}
+``Wärmeleitungsgleichung''
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/burgerking.png b/vorlesungen/slides/8/floyd-warshall/burgerking.png
new file mode 100644
index 0000000..cf4211d
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/burgerking.png
diff --git a/vorlesungen/slides/8/floyd-warshall/fw.tex b/vorlesungen/slides/8/floyd-warshall/fw.tex
new file mode 100644
index 0000000..99929fb
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/fw.tex
@@ -0,0 +1,680 @@
+%
+% fw.tex -- Durchführung des Floyd-Warshall Algorithmus
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+
+\definecolor{wegelb}{rgb}{1,0.6,0}
+\definecolor{weghell}{rgb}{1,0.9,0.6}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\begin{columns}[t]
+\begin{column}{0.44\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+
+\def\r{2.2}
+\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)});
+\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)});
+\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)});
+\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)});
+\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)});
+
+\def\knoten#1#2#3{
+	\fill[color=#3] #1 circle[radius=0.3];
+	\draw[line width=1pt] #1 circle[radius=0.3];
+	\node at #1 {$#2$};
+}
+
+\def\kante#1#2#3{
+	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] #1 -- #2;
+	\fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22];
+	\node at ($0.5*#1+0.5*#2$) {$#3$};
+}
+
+% Wege über 1
+% 3--1--5
+\only<4>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E);
+}
+
+% Wege über 2
+% 5--2--3
+\only<6>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(C);
+}
+% 5--2--4
+\only<7>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(D);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(D);
+}
+
+% Wege über 3
+% 2--3--1
+\only<9>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A);
+}
+% 2--3--5
+\only<10>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E);
+}
+% 4--3--1
+\only<11>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A);
+}
+% 4--3--5
+\only<12>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E);
+}
+% 5--3--1
+\only<13>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(C)--(A);
+}
+% 5--3--5
+\only<14>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(C)--(A)--(E);
+}
+
+% Wege über 4
+% 2--4--1
+\only<16>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C)--(A);
+}
+% 2--4--3
+\only<17>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C);
+}
+% 2--4--5
+\only<18>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(D)--(C)--(A)--(E);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C)--(A)--(E);
+}
+% 5--4--1
+\only<19>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(B)--(C)--(A);
+}
+% 5--4--3
+\only<20>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(B)--(C);
+}
+% 5--4--5
+\only<21>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(D)--(C)--(A)--(E);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(B)--(C)--(A)--(E);
+}
+% 1--5--1
+\only<23>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(B)--(C)--(A);
+}
+% 1--5--2
+\only<24>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(B);
+}
+% 1--5--3
+\only<25>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(B)--(C);
+}
+% 1--5--4
+\only<26>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(D);
+}
+% 2--5--1
+\only<27>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C)--(A);
+}
+% 2--5--2
+\only<28>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(B);
+}
+% 2--5--3
+\only<29>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C);
+}
+% 2--5--4
+\only<30>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(D);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(D);
+}
+% 3--5--1
+\only<31>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(C)--(A);
+}
+% 3--5--2
+\only<32>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(B);
+}
+% 3--5--3
+\only<33>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(B)--(C);
+}
+% 3--5--4
+\only<34>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(D);
+}
+% 4--5--1
+\only<35>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(D)--(C)--(A);
+}
+% 4--5--2
+\only<36>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(B);
+}
+% 4--5--3
+\only<37>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(D)--(C);
+}
+% 4--5--4
+\only<38>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(D);
+}
+
+
+\uncover<40>{
+	\draw[->,color=red!50,line width=7pt]
+		(B)--(C)--(A)--(E)--(D);
+}
+
+\kante{(A)}{(E)}{1}
+\kante{(B)}{(C)}{2}
+\kante{(B)}{(D)}{13}
+\kante{(C)}{(A)}{3}
+\kante{(D)}{(C)}{6}
+\kante{(E)}{(B)}{5}
+\kante{(E)}{(D)}{6}
+
+\only<1>{
+	\knoten{(A)}{}{white}
+	\knoten{(B)}{}{white}
+	\knoten{(C)}{}{white}
+	\knoten{(D)}{}{white}
+	\knoten{(E)}{}{white}
+}
+
+\only<2->{
+	\knoten{(A)}{1}{white}
+	\knoten{(B)}{2}{white}
+	\knoten{(C)}{3}{white}
+	\knoten{(D)}{4}{white}
+	\knoten{(E)}{5}{white}
+}
+
+\only<4>{
+	\knoten{(A)}{1}{darkgreen!50}
+}
+\only<6-7>{
+	\knoten{(B)}{2}{darkgreen!50}
+}
+\only<9-14>{
+	\knoten{(C)}{3}{darkgreen!50}
+}
+\only<16-21>{
+	\knoten{(D)}{4}{darkgreen!50}
+}
+\only<23-38>{
+	\knoten{(E)}{5}{darkgreen!50}
+}
+
+\end{tikzpicture}
+\end{center}
+\begin{block}{Aufgabe}
+Finde den kürzesten Weg von 2 nach 4
+\end{block}
+\end{column}
+\begin{column}{0.5\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex,scale=0.8]
+
+\def\punkt#1#2{
+	({#2-0.5},{0.5-(#1)})
+}
+\def\punktoff#1#2{
+	({#2-0.7},{0.7-(#1)})
+}
+\def\feld#1#2#3{
+	\ifthenelse{\boolean{wegweiser}}{
+		\fill[color=white]
+			({#2-1},{1-#1}) rectangle ({#2-0.45},{0.45-#1});
+		\node at \punktoff{#1}{#2} {$#3$};
+	}{
+		\fill[color=white]
+			({#2-1},{1-#1}) rectangle ({#2},{-#1});
+		\node at \punkt{#1}{#2} {$#3$};
+	}
+}
+\def\verbindung#1#2#3{
+	\draw[->,line width=5pt,color=red!20,shorten >= 0.2cm,shorten <= 0.2cm]
+		\punkt{#1}{#2}--\punkt{#2}{#3};
+	\node at (5,-5.5) [left] {$#1\rightsquigarrow #2\rightsquigarrow #3$\strut};
+}
+\def\Infty{{}}
+\def\wegweiser#1#2#3{
+	\ifthenelse{\boolean{wegweiser}}{
+		\ifnum #2 = #3
+		\fill[color=weghell]
+			({#2-0.45},{0.45-#1}) rectangle ({#2-0.05},{0.05-#1});
+		\else
+		\fill[color=wegelb]
+			({#2-0.45},{0.45-#1}) rectangle ({#2-0.05},{0.05-#1});
+		\fi
+		\node at ({#2-0.25},{0.25-#1}) {\tiny #3};
+	}{}
+}
+
+% direkte Wege
+\uncover<3->{
+	\feld{1}{1}{\Infty}
+	\feld{1}{2}{\Infty}
+	\feld{1}{3}{\Infty}
+	\feld{1}{4}{\Infty}
+	\feld{1}{5}{1}
+	\wegweiser{1}{5}{5}
+
+	\feld{2}{1}{\Infty}
+	\feld{2}{2}{\Infty}
+	\feld{2}{3}{2}
+	\wegweiser{2}{3}{3}
+	\feld{2}{4}{13}
+	\wegweiser{2}{4}{4}
+	\feld{2}{5}{\Infty}
+
+	\feld{3}{1}{3}
+	\wegweiser{3}{1}{1}
+	\feld{3}{2}{\Infty}
+	\feld{3}{3}{\Infty}
+	\feld{3}{4}{\Infty}
+	\feld{3}{5}{\Infty}
+
+	\feld{4}{1}{\Infty}
+	\feld{4}{2}{\Infty}
+	\feld{4}{3}{6}
+	\wegweiser{4}{3}{3}
+	\feld{4}{4}{\Infty}
+	\feld{4}{5}{\Infty}
+
+	\feld{5}{1}{\Infty}
+	\feld{5}{2}{5}
+	\wegweiser{5}{2}{2}
+	\feld{5}{3}{\Infty}
+	\feld{5}{4}{6}
+	\wegweiser{5}{4}{4}
+	\feld{5}{5}{\Infty}
+}
+
+\uncover<3-3>{
+	\node at (-0.8,-5.5) [right] {direkte Verbindungen};
+}
+
+\uncover<4-4>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $1$:\strut};
+}
+
+% Wege über 1
+% 3-1-5
+\uncover<4>{
+	\verbindung{3}{1}{5}
+	\feld{3}{5}{\color{red}4}
+	\wegweiser{3}{5}{1}
+}
+\uncover<5->{
+	\feld{3}{5}{4}
+	\wegweiser{3}{5}{1}
+}
+
+\uncover<6-7>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $2$:\strut};
+}
+
+% Wege über 2
+% 5-2-3
+\uncover<6>{
+	\verbindung{5}{2}{3}
+	\feld{5}{3}{\color{red}7}
+	\wegweiser{5}{3}{2}
+}
+\uncover<7->{
+	\feld{5}{3}{7}
+	\wegweiser{5}{3}{2}
+}
+% 5-2-4
+\uncover<7>{
+	\verbindung{5}{2}{4}
+	\feld{5}{4}{\color{blue}6}
+}
+
+\uncover<9-14>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $3$:\strut};
+}
+
+% Wege über 3
+% 2-3-1
+\uncover<9>{
+	\verbindung{2}{3}{1}
+	\feld{2}{1}{\color{red}5}
+	\wegweiser{2}{1}{3}
+}
+\uncover<10->{
+	\feld{2}{1}{5}
+	\wegweiser{2}{1}{3}
+}
+% 2-3-5
+\uncover<10>{
+	\verbindung{2}{3}{5}
+	\feld{2}{5}{\color{red}6}
+	\wegweiser{2}{5}{3}
+}
+\uncover<11->{
+	\feld{2}{5}{6}
+	\wegweiser{2}{5}{3}
+}
+% 4-3-1
+\uncover<11>{
+	\verbindung{4}{3}{1}
+	\feld{4}{1}{\color{red}9}
+	\wegweiser{4}{1}{3}
+}
+\uncover<12->{
+	\feld{4}{1}{9}
+	\wegweiser{4}{1}{3}
+}
+% 4-3-5
+\uncover<12>{
+	\verbindung{4}{3}{5}
+	\feld{4}{5}{\color{red}10}
+	\wegweiser{4}{5}{3}
+}
+\uncover<13->{
+	\feld{4}{5}{10}
+	\wegweiser{4}{5}{3}
+}
+% 5-3-1
+\uncover<13>{
+	\verbindung{5}{3}{1}
+	\feld{5}{1}{\color{red}10}
+	\wegweiser{5}{1}{2}
+}
+\uncover<14->{
+	\feld{5}{1}{10}
+	\wegweiser{5}{1}{2}
+}
+% 5-3-5
+\uncover<14>{
+	\verbindung{5}{3}{5}
+	\feld{5}{5}{\color{red}11}
+	\wegweiser{5}{5}{2}
+}
+\uncover<15->{
+	\feld{5}{5}{11}
+	\wegweiser{5}{5}{2}
+}
+
+\uncover<16-21>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $4$:\strut};
+}
+
+% Wege über 4
+% 2-4-1
+\uncover<16>{
+	\verbindung{2}{4}{1}
+	\feld{2}{1}{\color{blue}5}
+}
+% 2-4-3
+\uncover<17>{
+	\verbindung{2}{4}{3}
+	\feld{2}{3}{\color{blue}2}
+}
+% 2-4-5
+\uncover<18>{
+	\verbindung{2}{4}{5}
+	\feld{2}{5}{\color{blue}6}
+}
+% 5-4-1
+\uncover<19>{
+	\verbindung{5}{4}{1}
+	\feld{5}{1}{\color{blue}10}
+}
+% 5-4-3
+\uncover<20>{
+	\verbindung{5}{4}{3}
+	\feld{5}{3}{\color{blue}7}
+}
+% 5-4-5
+\uncover<21>{
+	\verbindung{5}{4}{5}
+	\feld{5}{5}{\color{blue}11}
+}
+
+% Wege über 5
+\uncover<23-38>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $5$:\strut};
+}
+
+% Wege über 5
+% 1-5-1
+\uncover<23>{
+	\verbindung{1}{5}{1}
+	\feld{1}{1}{\color{red}11}
+	\wegweiser{1}{1}{5}
+}
+\uncover<24->{
+	\feld{1}{1}{11}
+	\wegweiser{1}{1}{5}
+}
+% 1-5-2
+\uncover<24>{
+	\verbindung{1}{5}{2}
+	\feld{1}{2}{\color{red}6}
+	\wegweiser{1}{2}{5}
+}
+\uncover<25->{
+	\feld{1}{2}{6}
+	\wegweiser{1}{2}{5}
+}
+% 1-5-3
+\uncover<25>{
+	\verbindung{1}{5}{3}
+	\feld{1}{3}{\color{red}8}
+	\wegweiser{1}{3}{5}
+}
+\uncover<26->{
+	\feld{1}{3}{8}
+	\wegweiser{1}{3}{5}
+}
+% 1-5-4
+\uncover<26>{
+	\verbindung{1}{5}{4}
+	\feld{1}{4}{\color{red}7}
+	\wegweiser{1}{4}{5}
+}
+\uncover<27->{
+	\feld{1}{4}{7}
+	\wegweiser{1}{4}{5}
+}
+% 2-5-1
+\uncover<27>{
+	\verbindung{2}{5}{1}
+	\feld{2}{1}{\color{blue}5}
+}
+% 2-5-2
+\uncover<28>{
+	\verbindung{2}{5}{2}
+	\feld{2}{2}{\color{red}11}
+	\wegweiser{2}{2}{3}
+}
+\uncover<29->{
+	\feld{2}{2}{11}
+	\wegweiser{2}{2}{3}
+}
+% 2-5-3
+\uncover<29>{
+	\verbindung{2}{5}{3}
+	\feld{2}{3}{\color{blue}2}
+}
+% 2-5-4
+\uncover<30>{
+	\verbindung{2}{5}{4}
+	\feld{2}{4}{\color{red}12}
+	\wegweiser{2}{4}{3}
+}
+\uncover<31->{
+	\feld{2}{4}{12}
+	\wegweiser{2}{4}{3}
+}
+% 3-5-1
+\uncover<31>{
+	\verbindung{3}{5}{1}
+	\feld{3}{1}{\color{blue}3}
+}
+% 3-5-2
+\uncover<32>{
+	\verbindung{3}{5}{2}
+	\feld{3}{2}{\color{red}9}
+	\wegweiser{3}{2}{1}
+}
+\uncover<33->{
+	\feld{3}{2}{9}
+	\wegweiser{3}{2}{1}
+}
+% 3-5-3
+\uncover<33>{
+	\verbindung{3}{5}{3}
+	\feld{3}{3}{\color{red}11}
+	\wegweiser{3}{3}{1}
+}
+\uncover<34->{
+	\feld{3}{3}{11}
+	\wegweiser{3}{3}{1}
+}
+% 3-5-4
+\uncover<34>{
+	\verbindung{3}{5}{4}
+	\feld{3}{4}{\color{red}10}
+	\wegweiser{3}{4}{1}
+}
+\uncover<35->{
+	\feld{3}{4}{10}
+	\wegweiser{3}{4}{1}
+}
+% 4-5-1
+\uncover<35>{
+	\verbindung{4}{5}{1}
+	\feld{4}{1}{\color{blue}9}
+}
+% 4-5-2
+\uncover<36>{
+	\verbindung{4}{5}{2}
+	\feld{4}{2}{\color{red}15}
+	\wegweiser{4}{2}{3}
+}
+\uncover<37->{
+	\feld{4}{2}{15}
+	\wegweiser{4}{2}{3}
+}
+% 4-5-3
+\uncover<37>{
+	\verbindung{4}{5}{3}
+	\feld{4}{3}{\color{blue}6}
+}
+% 4-5-4
+\uncover<38>{
+	\verbindung{4}{5}{4}
+	\feld{4}{4}{\color{red}16}
+	\wegweiser{4}{4}{3}
+}
+\uncover<39->{
+	\feld{4}{4}{16}
+	\wegweiser{4}{4}{3}
+}
+
+
+\uncover<3->{
+
+	\foreach \x in {0,...,5}{
+		\draw[line width=0.7pt] (\x,0.8)--(\x,-5);
+	}
+	\foreach \y in {0,...,-5}{
+		\draw[line width=0.7pt] (-0.8,\y)--(5,\y);
+	}
+	\draw[line width=1.4pt] (0,0)--(5,0)--(5,-5)--(0,-5)--cycle;
+
+	\node at (0.5,0.5) {$1$};
+	\node at (1.5,0.5) {$2$};
+	\node at (2.5,0.5) {$3$};
+	\node at (3.5,0.5) {$4$};
+	\node at (4.5,0.5) {$5$};
+
+	\node at (-0.5,-0.5) {$1$};
+	\node at (-0.5,-1.5) {$2$};
+	\node at (-0.5,-2.5) {$3$};
+	\node at (-0.5,-3.5) {$4$};
+	\node at (-0.5,-4.5) {$5$};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\uncover<40>{
+	\vspace{-22pt}
+	\begin{block}{Lösung}
+	Der kürzeste Weg von 2 nach 4 ist 2---3---1---5---4
+	\end{block}
+}
+
+\end{column}
+\end{columns}
+
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/iteration.tex b/vorlesungen/slides/8/floyd-warshall/iteration.tex
new file mode 100644
index 0000000..d7e782d
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/iteration.tex
@@ -0,0 +1,14 @@
+%
+% iteration.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\newboolean{wegweiser}
+\begin{frame}[fragile]
+\frametitle{Floyd-Warshall: Iteration}
+\setboolean{wegweiser}{false}
+\input{../slides/8/floyd-warshall/fw.tex}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/macdonalds.png b/vorlesungen/slides/8/floyd-warshall/macdonalds.png
new file mode 100644
index 0000000..c442dfb
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/macdonalds.png
diff --git a/vorlesungen/slides/8/floyd-warshall/problem.tex b/vorlesungen/slides/8/floyd-warshall/problem.tex
new file mode 100644
index 0000000..93f8229
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/problem.tex
@@ -0,0 +1,146 @@
+%
+% graph.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[fragile]
+\frametitle{Problem: Kürzeste Wege}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\blob#1#2#3{
+	\fill[color=#3] #1 circle[radius=0.2];
+	\draw[line width=0.7pt] #1 circle[radius=0.2];
+	\node at #1 {{\tiny #2}};
+}
+
+\def\kante#1#2{
+	\draw[line width=0.7pt,shorten >= 0.2,shorten >= 0.2] #1 -- #2 ;
+}
+
+\def\a{72}
+\def\r{1.0}
+\def\R{2.0}
+\def\RR{3.5}
+
+\coordinate (A1) at ({\r*cos(0*\a)},{\r*sin(0*\a)});
+\coordinate (B1) at ({\r*cos(1*\a)},{\r*sin(1*\a)});
+\coordinate (C1) at ({\r*cos(2*\a)},{\r*sin(2*\a)});
+\coordinate (D1) at ({\r*cos(3*\a)},{\r*sin(3*\a)});
+\coordinate (E1) at ({\r*cos(4*\a)},{\r*sin(4*\a)});
+
+\coordinate (F1) at ({\R*cos(0*\a)},{\R*sin(0*\a)});
+\coordinate (G1) at ({\R*cos(1*\a)},{\R*sin(1*\a)});
+\coordinate (H1) at ({\R*cos(2*\a)},{\R*sin(2*\a)});
+\coordinate (I1) at ({\R*cos(3*\a)},{\R*sin(3*\a)});
+\coordinate (J1) at ({\R*cos(4*\a)},{\R*sin(4*\a)});
+
+\coordinate (K1) at ({\RR*cos(0.5*\a)},{\RR*sin(0.5*\a)});
+\coordinate (L1) at ({\RR*cos(1.5*\a)},{\RR*sin(1.5*\a)});
+\coordinate (M1) at ({\RR*cos(2.5*\a)},{\RR*sin(2.5*\a)});
+\coordinate (N1) at ({\RR*cos(3.5*\a)},{\RR*sin(3.5*\a)});
+\coordinate (O1) at ({\RR*cos(4.5*\a)},{\RR*sin(4.5*\a)});
+
+\kante{(A1)}{(C1)}
+\kante{(C1)}{(E1)}
+\kante{(E1)}{(B1)}
+\kante{(B1)}{(D1)}
+\kante{(D1)}{(A1)}
+
+\kante{(F1)}{(G1)}
+\kante{(G1)}{(H1)}
+\kante{(H1)}{(I1)}
+\kante{(I1)}{(J1)}
+\kante{(J1)}{(F1)}
+
+\kante{(A1)}{(F1)}
+\kante{(B1)}{(G1)}
+\kante{(C1)}{(H1)}
+\kante{(D1)}{(I1)}
+\kante{(E1)}{(J1)}
+
+\kante{(K1)}{(L1)}
+\kante{(L1)}{(M1)}
+\kante{(M1)}{(N1)}
+\kante{(N1)}{(O1)}
+\kante{(O1)}{(K1)}
+
+\kante{(F1)}{(K1)}
+\kante{(G1)}{(L1)}
+\kante{(H1)}{(M1)}
+\kante{(I1)}{(N1)}
+\kante{(J1)}{(O1)}
+
+\kante{(F1)}{(O1)}
+\kante{(G1)}{(K1)}
+\kante{(H1)}{(L1)}
+\kante{(I1)}{(M1)}
+\kante{(J1)}{(N1)}
+
+\uncover<2>{
+	\draw[line width=2pt,color=red] (M1)--(H1)--(G1)--(B1);
+}
+
+\uncover<3>{
+	\draw[line width=2pt,color=red] (M1)--(L1)--(G1)--(B1);
+}
+
+\uncover<4>{
+	\draw[line width=2pt,color=red] (M1)--(I1)--(D1)--(B1);
+}
+
+\uncover<5>{
+	\draw[line width=2pt,color=red] (M1)--(I1)--(D1)--(A1)--(F1);
+}
+
+\uncover<6->{
+	\draw[line width=2pt,color=red] (M1)--(I1)--(J1)--(F1);
+}
+
+\uncover<2-4>{
+	\blob{(B1)}{1}{red!20}
+	\blob{(M1)}{12}{red!20}
+}
+\uncover<5-8>{
+	\blob{(M1)}{1}{red!20}
+	\blob{(F1)}{12}{red!20}
+}
+
+\blob{(A1)}{0}{white}
+\uncover<1>{
+	\blob{(B1)}{1}{white}
+}
+\uncover<5-8>{
+	\blob{(B1)}{12}{white}
+}
+\blob{(C1)}{2}{white}
+\blob{(D1)}{3}{white}
+\blob{(E1)}{4}{white}
+\uncover<1-4>{
+	\blob{(F1)}{5}{white}
+}
+\blob{(G1)}{6}{white}
+\blob{(H1)}{7}{white}
+\blob{(I1)}{8}{white}
+\blob{(J1)}{9}{white}
+\blob{(K1)}{10}{white}
+\blob{(L1)}{11}{white}
+\uncover<1>{
+	\blob{(M1)}{12}{white}
+}
+\blob{(N1)}{13}{white}
+\blob{(O1)}{14}{white}
+
+\node at (6,0) {\begin{minipage}{5cm}
+\begin{itemize}
+\item<3-> Nicht eindeutig
+\item<5-> geradeste Wege sind nicht unbedingt die kürzesten
+\item<7-> Gewichtung der Kanten
+($\text{Schnellstrassen}\ne\text{Feldwege}$)
+\item<8-> Orientierung der Kanten?
+\end{itemize}
+\end{minipage}};
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
diff --git a/vorlesungen/slides/8/floyd-warshall/rekursion.tex b/vorlesungen/slides/8/floyd-warshall/rekursion.tex
new file mode 100644
index 0000000..c664e41
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/rekursion.tex
@@ -0,0 +1,108 @@
+%
+% rekursion.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[fragile]
+\frametitle{Rekursion}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\blob#1#2#3{
+        \fill[color=#3] #1 circle[radius=0.3];
+        \draw[line width=0.7pt] #1 circle[radius=0.3];
+        \node at #1 {{#2}};
+}
+
+\def\kante#1#2{
+        \draw[line width=0.7pt,shorten >= 0.3,shorten >= 0.3] #1 -- #2 ;
+}
+
+\coordinate (A) at (0,0);
+\coordinate (B1) at (2,2);
+\coordinate (B2) at (2,1);
+\coordinate (B3) at (2,0);
+\coordinate (B4) at (2,-1);
+\coordinate (B5) at (2,-2);
+
+\draw[line width=1.9pt,color=gray] (A)--(B1);
+\draw[line width=1.9pt,color=gray] (A)--(B2);
+\draw[line width=1.9pt,color=gray] (A)--(B3);
+\draw[line width=1.9pt,color=gray] (A)--(B4);
+\draw[line width=1.9pt,color=gray] (A)--(B5);
+
+\coordinate (Z) at (10,0);
+
+\begin{scope}
+\clip (2,-2.3) rectangle (10,2.3);
+\foreach \y in{-10,...,10}{
+	\draw[line width=1.9pt,color=gray]
+		(2,\y)--(10,{\y-8});
+	\draw[line width=1.9pt,color=gray]
+		(2,\y)--(10,{\y+8});
+}
+\end{scope}
+
+\uncover<2>{
+\draw[line width=4pt,color=red] (A)--(B1)--(5,-1)--(8,2)--(Z);
+}
+
+\uncover<3>{
+\draw[line width=4pt,color=red] (A)--(B2)--(3,0)--(4,1)--(5,0)--(6,1)--(8.5,-1.5)--(Z);
+}
+
+\uncover<4>{
+\draw[line width=4pt,color=red] (A)--(B3)--(2.5,0.5)--(3.5,-0.5)--(5,1.0)--(7,-1)--(9,1)--(Z);
+}
+
+\uncover<5>{
+\draw[line width=4pt,color=red] (A)--(B4)--(3,0)--(4,1)--(5,0)--(6,1)--(7,0)
+	--(6.0,-1.0)--(7,-2)--(7.5,-1.5)--(7,-1)--(7.5,-0.5)
+	--(8.5,-1.5)--(Z);
+}
+
+\uncover<6->{
+	\draw[line width=4pt,color=red] (A)--(B5)--(6,2);
+}
+\uncover<7->{
+	\draw[line width=4pt,color=red] (6,2)--(7,1)--(5,-1);
+}
+\uncover<8->{
+	\draw[line width=4pt,color=red] (5,-1)--(6,-2)--(8,0)--(9,-1);
+}
+\uncover<9->{
+	\draw[line width=4pt,color=red] (9,-1)--(Z);
+}
+
+\blob{(A)}{$A$}{red!20}
+\blob{(B1)}{$B_1$}{white}
+\blob{(B2)}{$B_2$}{white}
+\blob{(B3)}{$B_3$}{white}
+\blob{(B4)}{$B_4$}{white}
+\blob{(B5)}{$B_5$}{white}
+
+\blob{(Z)}{$Z$}{red!20}
+
+\uncover<6->{
+	\node at (6,2) {\includegraphics[width=1.5cm]{../slides/8/floyd-warshall/macdonalds.png}};
+}
+
+\uncover<7->{
+	\node at (5,-1) {\includegraphics[width=1.5cm]{../slides/8/floyd-warshall/starbucks.png}};
+}
+
+\uncover<8->{
+	\node at (9,-1) {\includegraphics[width=2cm]{../slides/8/floyd-warshall/burgerking.png}};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\begin{block}{Abstieg}
+Für den kürzesten Weg von $A$ nach $Z$ suche denjenigen Nachbarn $B_i$
+von $A$, der den kürzesten Weg von $B_i$ nach $Z$ hat.
+\uncover<7->{$\Rightarrow$ wir brauchen {\color{red}alle} kürzesten Wege!}
+\end{block}
+
+\end{frame}
diff --git a/vorlesungen/slides/8/floyd-warshall/starbucks.png b/vorlesungen/slides/8/floyd-warshall/starbucks.png
new file mode 100644
index 0000000..a28dbf7
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/starbucks.png
diff --git a/vorlesungen/slides/8/floyd-warshall/wege.tex b/vorlesungen/slides/8/floyd-warshall/wege.tex
new file mode 100644
index 0000000..7ff62a1
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/wege.tex
@@ -0,0 +1,26 @@
+%
+% wege.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{Wege statt Weglänge?}
+\begin{columns}[t]
+\begin{column}{0.48\hsize}
+\begin{block}{Wege speichern?}
+\uncover<3->{Es reicht, einen Wegweiser zum nächsten Knoten zu speichern}
+\end{block}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\hsize}
+\uncover<2->{%
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/floyd-warshall/wegweiser.jpg}
+\end{center}}%
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/floyd-warshall/wegiteration.tex b/vorlesungen/slides/8/floyd-warshall/wegiteration.tex
new file mode 100644
index 0000000..84ec679
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/wegiteration.tex
@@ -0,0 +1,13 @@
+%
+% wegiteration.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\newboolean{wegweiser}
+\begin{frame}[fragile]
+\frametitle{Floyd-Warshall: Wegweiser}
+\setboolean{wegweiser}{true}
+\input{../slides/8/floyd-warshall/fw.tex}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg
new file mode 100644
index 0000000..33aebe3
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg
diff --git a/vorlesungen/slides/8/fourier.tex b/vorlesungen/slides/8/fourier.tex
new file mode 100644
index 0000000..86d8086
--- /dev/null
+++ b/vorlesungen/slides/8/fourier.tex
@@ -0,0 +1,83 @@
+%
+% fourier.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Fourier-Transformation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Algebra}
+Die Laplace-Matrix eines Graphen ist symmetrisch
+\uncover<2->{%
+
+$\Rightarrow$
+Es gibt eine Basis aus Eigenvektoren $g_i\in\mathbb{R}^n$ von $L(G)$:
+\begin{align*}
+L(G)g_i&=\lambda_i g_i
+\end{align*}}
+\end{block}
+\uncover<12->{%
+\vspace{-20pt}
+\begin{block}{Fourier-Transformation}
+Jedes $f\in\mathbb{R}^n$ kann durch die $g_i$ ausgedrückt werden
+\begin{align*}
+\uncover<13->{
+f&= a_1 g_1 + \dots + a_n g_n
+}
+\\
+\uncover<14->{
+&= \hat{f}_1 g_1 + \dots + \hat{f}_ng_n = \sum_{k=1}^n \hat{f}_kg_k
+}
+\end{align*}
+\uncover<15->{%
+Zerlegung nach Zeitkonstante $\lambda_i$
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Anwendung}
+Wärmeleitungsgleichung
+\begin{align*}
+\uncover<4->{
+\frac{d}{dt}f &= L(G) f
+}
+\intertext{\uncover<5->{{\usebeamercolor[fg]{title}Ansatz:}}}
+\uncover<6->{
+f&=a_1g_1T_1(t)+\dots + a_ng_nT_n(t)
+}
+\\
+\uncover<7->{
+\frac{d}{dt}f
+&=
+a_1g_1\dot{T}_1(t) + \dots + a_1g_1 \dot{T}_n(t)
+}
+\\
+\uncover<8->{
+&=
+a_1Lg_1 + \dots + a_nLg_n
+}
+\\
+\uncover<9->{
+&=
+a_1\lambda_1 g_1 + \dots + a_n\lambda_n g_n
+}
+\\
+\uncover<10->{
+\dot{T}_i(t) &= \lambda_i T_i(t)
+}
+\uncover<11->{
+\quad
+\Rightarrow
+\quad
+T_i(t) = e^{\lambda_it} \uncover<-9>{T_i(0)}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/grad.tex b/vorlesungen/slides/8/grad.tex
new file mode 100644
index 0000000..a232828
--- /dev/null
+++ b/vorlesungen/slides/8/grad.tex
@@ -0,0 +1,84 @@
+%
+% grad.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Grad}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+%\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+%\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+%\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+%\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+%\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+%\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+%\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+
+\end{tikzpicture}
+\end{center}
+\begin{block}{Definition}
+Der Grad
+$\deg v$
+eines Knotens $v\in V$ ist die Anzahl der Kanten mit Ende in $v$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Gradmatrix}
+Diagonalmatrix mit $d_{ii}=\deg v_i$
+\[
+D(G)
+=
+\begin{pmatrix}
+3&0&0&0&0\\
+0&3&0&0&0\\
+0&0&3&0&0\\
+0&0&0&2&0\\
+0&0&0&0&1
+\end{pmatrix}
+\]
+\end{block}
+\begin{block}{Satz}
+Die Summe der Grade ist gerade:
+\[
+\sum_{i=1}^n\deg v_i = \operatorname{Spur} D(G) \equiv 0 \mod 2
+\]
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/graph.tex b/vorlesungen/slides/8/graph.tex
new file mode 100644
index 0000000..32150af
--- /dev/null
+++ b/vorlesungen/slides/8/graph.tex
@@ -0,0 +1,117 @@
+%
+% graph.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Graph}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2.4}
+
+\begin{scope}
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\uncover<3->{
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+	\draw[color=white,line width=5pt] (B) -- (D);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+}
+
+\uncover<2->{
+	\draw (A) circle[radius=0.2];
+	\draw (B) circle[radius=0.2];
+	\draw (C) circle[radius=0.2];
+	\draw (D) circle[radius=0.2];
+	\draw (E) circle[radius=0.2];
+
+	\node at (A) {$1$};
+	\node at (B) {$2$};
+	\node at (C) {$3$};
+	\node at (D) {$4$};
+	\node at (E) {$5$};
+}
+\node at (0,0) {$G$};
+
+\uncover<3->{
+	\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$)
+		[above right] {$\scriptstyle 1$};
+	\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$)
+		[above left] {$\scriptstyle 2$};
+	\node at ($0.5*(C)+0.5*(D)+(0.05,0)$)
+		[left] {$\scriptstyle 3$};
+	\node at ($0.5*(D)+0.5*(E)$)
+		[below] {$\scriptstyle 4$};
+	\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$)
+		[below right] {$\scriptstyle 5$};
+	\node at ($0.6*(A)+0.4*(C)$)
+		[above] {$\scriptstyle 6$};
+	\node at ($0.4*(B)+0.6*(D)$)
+		[left] {$\scriptstyle 7$};
+}
+
+\uncover<8->{
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm]
+		(E) to[out=-18,in=-126,distance=2cm] (E);
+
+	\draw[color=red,line width=4pt] ($(E)+(-0.5,-0.5)+(0,-0.5)$)
+		-- ($(E)+(0.5,0.5)+(0,-0.5)$);
+	\draw[color=red,line width=4pt] ($(E)+(-0.5,0.5)+(0,-0.5)$)
+		-- ($(E)+(0.5,-0.5)+(0,-0.5)$);
+}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Ein Graph $G=(V,E)$ ist
+\begin{enumerate}
+\item<2->
+Eine Menge $V$ von Knoten (Vertizes):
+$V=\{v_1,v_2,\dots\}$
+\item<3->
+Eine Menge $E$ von Kanten (Edges):
+\[
+E\subset
+\left\{ e = \{v_1,v_2\}\;\left|\; \begin{minipage}{1.3cm}\raggedright
+$v_i\in V$\\
+$v_1\ne v_2$
+\end{minipage}
+\right.
+\right\}
+\]
+\end{enumerate}
+\end{block}
+\vspace{-20pt}
+\uncover<5->{%
+\begin{block}{Achtung:}
+\begin{itemize}
+\item<6-> Kanten sind Mengen
+\uncover<7->{$\Rightarrow$ zwei verschiedene Knoten}
+\uncover<8->{$\Rightarrow$ Keine Schleifen}
+\item<9-> Kanten sind ungerichtet, keine ``Einbahnstrassen''
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex
new file mode 100644
index 0000000..952c85b
--- /dev/null
+++ b/vorlesungen/slides/8/inzidenz.tex
@@ -0,0 +1,150 @@
+%
+% inzidenz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Adjazenzmatrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+{\color<2->{darkgreen}
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+}
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\only<-2>{
+\fill[color=white] (B) circle[radius=0.2];
+}
+\only<3->{
+\fill[color=red!20] (B) circle[radius=0.2];
+}
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 4$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 5$};
+{\color<2->{darkgreen}
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 6$};
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Definition}
+\vspace{-15pt}
+\begin{align*}
+B(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante $j$ endet in Knoten $i$}\\
+A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dy{0.48}
+\def\dx{0.54}
+
+
+\begin{scope}
+\uncover<3->{
+\fill[color=red!20] (1.8,1.8) rectangle (4.75,2.15);
+}
+\uncover<2->{
+\fill[color=darkgreen!40,opacity=0.5] (4.46,0.36) rectangle (4.79,2.65);
+}
+\foreach \y in {1,...,5}{
+	\node[color=gray] at (5.3,{2.45-(\y-1)*\dy}) {\tiny $\y$};
+}
+\foreach \y in {1,...,6}{
+	\node[color=gray] at ({1.92+(\y-1)*\dx},2.90) {\tiny $\y$};
+}
+\draw[color=gray] (1.8,2.75) -- (4.7,2.75);
+\draw[color=gray] (5.2,2.55) -- (5.2,0.45);
+\node[color=gray] at ({1.92+2.5*\dx},3.1) {\tiny Kanten};
+\node[color=gray] at (5.3,{2.45-2*\dy}) [above,rotate=-90] {\tiny Knoten};
+\end{scope}
+
+\uncover<4->{
+\begin{scope}
+\uncover<3->{
+\fill[color=red!20] (1.8,-1.16) rectangle (4.25,-0.77);
+\fill[color=red!20] (2.3,-2.6) rectangle (2.63,-0.29);
+}
+\foreach \y in {1,...,5}{
+	\node[color=gray] at (4.7,{-0.5-(\y-1)*\dy}) {\tiny $\y$};
+	\node[color=gray] at ({1.92+(\y-1)*\dx},-0.1) {\tiny $\y$};
+}
+\draw[color=gray] (1.8,-0.22) -- (4.2,-0.22);
+\draw[color=gray] (4.6,-0.4) -- (4.6,-2.55);
+\node[color=gray] at ({1.92+2*\dx},0.1) {\tiny Knoten};
+\node[color=gray] at (4.7,{-0.5-2*\dy}) [above,rotate=-90] {\tiny Knoten};
+\end{scope}
+}
+
+\node (0,0) [right] {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix}
+1&0&0&1&1&0\\
+1&1&0&0&0&1\\
+0&1&1&0&1&0\\
+0&0&1&0&0&1\\
+0&0&0&1&0&0
+\end{pmatrix}
+\\[12pt]
+\uncover<4->{
+A(G)
+&=
+\begin{pmatrix}
+0&1&1&0&1\\
+1&0&1&1&0\\
+1&1&0&1&0\\
+0&1&1&0&0\\
+1&0&0&0&0
+\end{pmatrix}
+\end{aligned}}$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex
new file mode 100644
index 0000000..5f2f51a
--- /dev/null
+++ b/vorlesungen/slides/8/inzidenzd.tex
@@ -0,0 +1,164 @@
+%
+% inzidenzd.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Adjazenz-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+{\color<2->{darkgreen}
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+}
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\only<-2>{
+\fill[color=white] (B) circle[radius=0.2];
+}
+\only<3->{
+\fill[color=red!20] (B) circle[radius=0.2];
+}
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+{\color<2->{darkgreen}
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-15pt}
+\uncover<5->{%
+\begin{block}{Definition}
+\vspace{-20pt}
+\begin{align*}
+B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\
+B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\
+A(G)_{ij}&=\phantom{-}1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dx{0.84}
+\def\dy{0.48}
+
+\begin{scope}[xshift=4cm,yshift=3cm]
+\uncover<3->{
+\fill[color=red!20] 
+({-0.67-(7-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2})
+rectangle
+({-0.67-(7-7)*\dx+0.2},{-0.38-(2-1)*\dy+0.16});
+}
+\uncover<2->{
+\fill[color=darkgreen!40,opacity=0.5] 
+({-0.67-(7-7)*\dx-0.4},{-0.38-(5-1)*\dy-0.2})
+rectangle
+({-0.67-(7-7)*\dx+0.2},{-0.38-(1-1)*\dy+0.16});
+}
+%\draw (0,0) circle[radius=0.05];
+\foreach \x in {1,...,7}{
+	\node[color=gray] at ({-0.67-(7-\x)*\dx},0.0) {\tiny $\x$};
+}
+\draw[color=gray] ({-0.72-6*\dx},-0.1) -- (-0.6,-0.1);
+\foreach \y in {1,...,5}{
+	\node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$};
+}
+\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4);
+\node[color=gray] at ({-0.67-(7-4)*\dx},0.04) [above] {\tiny Kanten};
+\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy})
+	[above,rotate=-90] {\tiny Knoten};
+\end{scope}
+
+\uncover<4->{
+\begin{scope}[xshift=2.32cm,yshift=-0.24cm]
+%\draw (0,0) circle[radius=0.05];
+\fill[color=red!20] 
+({-0.67-(5-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2})
+rectangle
+({-0.67-(5-5)*\dx+0.2},{-0.38-(2-1)*\dy+0.16});
+\fill[color=red!20] 
+({-0.67-(5-2)*\dx-0.4},{-0.38-(5-1)*\dy-0.2})
+rectangle
+({-0.67-(5-2)*\dx+0.2},{-0.38-(1-1)*\dy+0.16});
+\foreach \x in {1,...,5}{
+	\node[color=gray] at ({-0.67-(5-\x)*\dx},0.0) {\tiny $\x$};
+}
+\draw[color=gray] ({-0.72-4*\dx},-0.1) -- (-0.6,-0.1);
+\foreach \y in {1,...,5}{
+	\node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$};
+}
+\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4);
+\node[color=gray] at ({-0.67-(5-3)*\dx},0.04) [above] {\tiny Knoten};
+\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy})
+	[above,rotate=-90] {\tiny Knoten};
+\end{scope}
+}
+
+\node at (0,0) {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix*}[r]
+-1& 0& 0& 0&+1&-1& 0\\
++1&-1& 0& 0& 0& 0&-1\\
+ 0&+1&-1& 0& 0&+1& 0\\
+ 0& 0&+1&-1& 0& 0&+1\\
+ 0& 0& 0&+1&-1& 0& 0
+\end{pmatrix*}
+\\[20pt]
+\uncover<4->{
+A(G)
+&=
+\begin{pmatrix*}[r]
+ 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\
+\phantom{-}1& 0&\phantom{-}1&\phantom{-}1& 0\\
+\phantom{-}1&\phantom{-}1& 0&\phantom{-}1& 0\\
+ 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\
+\phantom{-}1& 0& 0&\phantom{-}1& 0
+\end{pmatrix*}}
+\end{aligned}$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/laplace.tex b/vorlesungen/slides/8/laplace.tex
new file mode 100644
index 0000000..a1c364d
--- /dev/null
+++ b/vorlesungen/slides/8/laplace.tex
@@ -0,0 +1,213 @@
+%
+% laplace.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Laplace-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D);
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A);
+
+\uncover<2-4>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+}
+
+\uncover<3-7>{
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(A) -- (C);
+}
+
+\uncover<4-13>{
+\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm]
+	(A) -- (E);
+}
+
+\uncover<5->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+}
+
+\uncover<6-8>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+}
+
+\uncover<7-10>{
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(B) -- (D);
+}
+
+\uncover<8->{
+\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm]
+	(A) -- (C);
+}
+
+\uncover<9->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+}
+
+\uncover<10-11>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+}
+
+\uncover<11->{
+\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm]
+	(B) -- (D);
+}
+
+\uncover<12->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+}
+
+\uncover<13-14>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+}
+
+\uncover<14->{
+\draw[<->,color=darkgreen,line width=8pt,shorten <= 0cm,shorten >= 0cm]
+	(A) -- (E);
+}
+
+\uncover<15->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+}
+
+\fill[color=red] (A) circle[radius=0.3];
+\fill[color=red!50] (B) circle[radius=0.3];
+\fill[color=white] (C) circle[radius=0.3];
+\fill[color=blue!50] (D) circle[radius=0.3];
+\fill[color=blue] (E) circle[radius=0.3];
+
+\draw (A) circle[radius=0.3];
+\draw (B) circle[radius=0.3];
+\draw (C) circle[radius=0.3];
+\draw (D) circle[radius=0.3];
+\draw (E) circle[radius=0.3];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+
+\end{tikzpicture}
+\end{center}
+\uncover<16->{%
+\begin{block}{Definition}
+Laplace-Matrix
+\[
+L(G) = D(G) - A(G)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{align*}
+f
+&=
+\begin{pmatrix}
+f(1)\\
+f(2)\\
+f(3)\\
+f(4)\\
+f(5)
+\end{pmatrix}
+\\
+\frac{df}{dt}
+&=
+-\kappa
+\begin{pmatrix*}[r]
+\only<1>{\phantom{-0}}
+	\only<2>{\phantom{-}1}
+	\only<3>{\phantom{-}2}
+	\only<4->{\phantom{-}3}
+	&\only<1>{\phantom{-0}}\only<2->{{-1}}%
+	&\only<-2>{\phantom{-0}}\only<3->{{-1}}%
+	&\uncover<16->{ 0}
+	&\only<-3>{\phantom{-0}}\only<4->{{-1}}\\
+\only<-4>{\phantom{-0}}\only<5->{{-1}}
+	&\only<-4>{\phantom{-0}}
+	\only<5>{\phantom{-}1}
+	\only<6>{\phantom{-}2}
+	\only<7->{\phantom{-}3}
+	&\only<-5>{\phantom{-0}}\only<6->{{-1}}
+	&\only<-6>{\phantom{-0}}\only<7->{{-1}}
+	&\uncover<16->{ 0}\\
+\only<-7>{\phantom{-0}}\only<8->{{-1}}
+	&\only<-8>{\phantom{-0}}\only<9->{{-1}}
+	&\only<-7>{\phantom{-0}}
+	\only<8>{\phantom{-}1}
+	\only<9>{\phantom{-}2}
+	\only<10->{\phantom{-}3}
+	&\only<-9>{\phantom{-0}}\only<10->{{-1}}
+	&\uncover<16->{ 0}\\
+\uncover<16->{ 0}
+	&\only<-10>{\phantom{-0}}\only<11->{{-1}}
+	&\only<-11>{\phantom{-0}}\only<12->{{-1}}
+	&\only<-10>{\phantom{-0}}
+	\only<11>{\phantom{-}1}
+	\only<12>{\phantom{-}2}
+	\only<13->{\phantom{-}3}
+	&\only<-12>{\phantom{-0}}\only<13->{{-1}}\\
+\only<-13>{\phantom{-0}}\only<14->{{-1}}
+	&\uncover<16->{ 0}
+	&\uncover<16->{ 0}
+	&\only<-14>{\phantom{-0}}\only<15->{{-1}}
+	&\only<-13>{\phantom{-0}}
+	\only<14>{\phantom{-}1}
+	\only<15->{\phantom{-}2}
+\end{pmatrix*}
+\begin{pmatrix}
+f(1)\\
+f(2)\\
+f(3)\\
+f(4)\\
+f(5)
+\end{pmatrix}
+\\
+\uncover<17->{
+&=
+-\kappa L f}
+\end{align*}
+\vspace{-20pt}
+\uncover<18->{%
+\begin{block}{Rekonstruktion}
+Der Graph lässt sich aus $L$ rekonstruieren
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+
+\egroup
+
+
diff --git a/vorlesungen/slides/8/pfade/adjazenz.tex b/vorlesungen/slides/8/pfade/adjazenz.tex
new file mode 100644
index 0000000..f923262
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/adjazenz.tex
@@ -0,0 +1,97 @@
+%
+% adjazenz.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.5,0,0}
+\begin{frame}[fragile]
+\newboolean{pfeilspitzen}
+\setboolean{pfeilspitzen}{true}
+\frametitle{Adjazenz-Matrix}
+
+\begin{columns}[t]
+\begin{column}{0.48\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\r{2.2}
+\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)});
+\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)});
+\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)});
+\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)});
+\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)});
+
+\def\knoten#1#2{
+        \fill[color=white] #1 circle[radius=0.3];
+        \draw[line width=1pt] #1 circle[radius=0.3];
+        \node at #1 {$#2$};
+}
+
+\def\kante#1#2#3{
+        \ifthenelse{\boolean{pfeilspitzen}}{
+                \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }{
+                \draw[line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }
+%        \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22];
+%        \node at ($0.5*#1+0.5*#2$) {$#3$};
+}
+
+\kante{(A)}{(E)}{1}
+\kante{(B)}{(C)}{2}
+\kante{(B)}{(D)}{13}
+\kante{(C)}{(A)}{3}
+\kante{(D)}{(C)}{6}
+\kante{(E)}{(B)}{5}
+\kante{(E)}{(D)}{6}
+
+\knoten{(A)}{1}
+\knoten{(B)}{2}
+\knoten{(C)}{3}
+\knoten{(D)}{4}
+\knoten{(E)}{5}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\hsize}
+\[
+a_{{\color{darkred}i}{\color{blue}j}}
+=
+\begin{cases}
+1&\quad\text{\# Kanten von ${\color{blue}j}$ nach ${\color{darkred}i}$}\\
+0&\quad\text{sonst}
+\end{cases}
+\]
+\begin{center}
+\begin{tikzpicture}[>=latex]
+\node at (0,0) {$\displaystyle
+A=
+\begin{pmatrix}
+0&0&1&0&0\\
+0&0&0&0&1\\
+0&1&0&1&0\\
+0&1&0&0&1\\
+1&0&0&0&0
+\end{pmatrix}
+$};
+\def\s{0.54}
+\foreach \x in {1,...,5}{
+	\node[color=blue] at ({-0.71+(\x-1)*\s},1.4) {\tiny $\x$};
+}
+\node[color=blue] at ({-0.71+2*\s},1.7) {von};
+\def\r{0.48}
+\foreach \y in {1,...,5}{
+	\node[color=darkred] at ({-0.71+5*\s},{0.02+(3-\y)*\r}) {\tiny $\y$};
+}
+\node[color=darkred] at ({-0.4+5*\s},{0.02}) [rotate=90] {nach};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/pfade/beispiel.tex b/vorlesungen/slides/8/pfade/beispiel.tex
new file mode 100644
index 0000000..43685f3
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/beispiel.tex
@@ -0,0 +1,404 @@
+%
+% beispiel.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\newboolean{pfeilspitzen}
+
+\def\knoten#1#2{
+        \fill[color=white] #1 circle[radius=0.3];
+        \draw[line width=1pt] #1 circle[radius=0.3];
+        \node at #1 {$#2$};
+}
+
+\def\kante#1#2#3{
+        \ifthenelse{\boolean{pfeilspitzen}}{
+                \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }{
+                \draw[line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }
+%        \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22];
+%        \node at ($0.5*#1+0.5*#2$) {$#3$};
+}
+
+\begin{frame}
+\setboolean{pfeilspitzen}{true}
+\frametitle{Beispiel}
+\begin{columns}[t]
+\begin{column}{0.37\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\r{2.2}
+\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)});
+\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)});
+\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)});
+\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)});
+\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)});
+
+\kante{(A)}{(E)}{1}
+\kante{(B)}{(C)}{2}
+\kante{(B)}{(D)}{13}
+\kante{(C)}{(A)}{3}
+\kante{(D)}{(C)}{6}
+\kante{(E)}{(B)}{5}
+\kante{(E)}{(D)}{6}
+
+\knoten{(A)}{1}
+\knoten{(B)}{2}
+\knoten{(C)}{3}
+\knoten{(D)}{4}
+\knoten{(E)}{5}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.59\hsize}
+
+\only<1>{
+\begin{block}{Pfade der Länge 1}
+\[
+A=
+\begin{pmatrix}
+0&0&1&0&0\\
+0&0&0&0&1\\
+0&1&0&1&0\\
+0&1&0&0&1\\
+1&0&0&0&0
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<2>{
+\begin{block}{Pfade der Länge 2}
+\[
+A^2=\begin{pmatrix}
+   0 & 1 & 0 & 1 & 0 \\
+   1 & 0 & 0 & 0 & 0 \\
+   0 & 1 & 0 & 0 & 2 \\
+   1 & 0 & 0 & 0 & 1 \\
+   0 & 0 & 1 & 0 & 0
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<3>{
+\begin{block}{Pfade der Länge 3}
+\[
+A^3=\begin{pmatrix}
+   0 & 1 & 0 & 0 & 2 \\
+   0 & 0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 \\
+   2 & 0 & 0 & 0 & 1 \\
+   1 & 0 & 1 & 0 & 0 \\
+   0 & 1 & 0 & 1 & 0
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<4>{
+\begin{block}{Pfade der Länge 4}
+\[
+A^4=\begin{pmatrix}
+   2%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 & 0 & 1 \\
+   0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 1 & 0 \\
+   1 & 0 & 2%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 \\
+   0 & 1 & 1 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 \\
+   0 & 1 & 0 & 0 & 2%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<5>{
+\begin{block}{Pfade der Länge 5}
+\[
+A^5=\begin{pmatrix}
+   1 & 0 & 2 & 0 & 0 \\
+   0 & 1 & 0 & 0 & 2 \\
+   0 & 2 & 1 & 2 & 0 \\
+   0 & 2 & 0 & 1 & 2 \\
+   2 & 0 & 0 & 0 & 1
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<6>{
+\begin{block}{Pfade der Länge 6}
+\[
+A^6=\begin{pmatrix}
+   0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 2 & 1 & 2 & 0 \\
+   2 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 & 1 \\
+   0 & 3 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 1 & 4 \\
+   2 & 1 & 0 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 3 \\
+   1 & 0 & 2 & 0 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<7>{
+\begin{block}{Pfade der Länge 7}
+\[
+A^7=\begin{pmatrix}
+   0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 3 & 0 & 1 & 4 \\
+   1 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 2 & 0 & 0 \\
+   4 & 1 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 4 \\
+   3 & 0 & 2 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 1 \\
+   0 & 2 & 1 & 2 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<8>{
+\begin{block}{Pfade der Länge 8}
+\[
+A^8=\begin{pmatrix}
+   4 & 1 & 0 & 0 & 4 \\
+   0 & 2 & 1 & 2 & 0 \\
+   4 & 0 & 4 & 0 & 1 \\
+   1 & 2 & 3 & 2 & 0 \\
+   0 & 3 & 0 & 1 & 4
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<9>{
+\begin{block}{Pfade der Länge 9}
+\[
+A^9=\begin{pmatrix}
+   4 & 0 & 4 & 0 & 1 \\
+   0 & 3 & 0 & 1 & 4 \\
+   1 & 4 & 4 & 4 & 0 \\
+   0 & 5 & 1 & 3 & 4 \\
+   4 & 1 & 0 & 0 & 4
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<10>{
+\begin{block}{Pfade der Länge 10}
+\[
+A^{10}=\begin{pmatrix}
+   1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 4 & 4 & 4 & 0 \\
+   4 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 & 4 \\
+   0 & 8 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 4 & 8 \\
+   4 & 4 & 0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 8 \\
+   4 & 0 & 4 & 0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<11>{
+\begin{block}{Pfade der Länge 15}
+\[
+A^{15}=\begin{pmatrix}
+    1 & 20 &  6 & 12 & 16 \\
+   12 &  1 &  8 &  0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+&  6 \\
+   16 & 18 &  1 &  6 & 32 \\
+   20 &  6 &  8 &  1 & 18 \\
+    6 &  8 & 12 &  8 &  1
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<12>{
+\begin{block}{Pfade der Länge 20}
+\[
+A^{20}=\begin{pmatrix}
+   33 & 56 &  8 & 24 & 80 \\
+   24 & 17 & 32 & 16 &  8 \\
+   80 & 32 & 33 &  8 & 80 \\
+   56 & 24 & 48 & 17 & 32 \\
+    8 & 48 & 24 & 32 & 33
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<13>{
+\begin{block}{Pfade der Länge 25}
+\[
+A^{25}=\begin{pmatrix}
+   193 & 120 &  74 &  40 & 240 \\
+    40 & 113 &  80 &  80 &  74 \\
+   240 & 114 & 193 &  74 & 160 \\
+   120 & 154 & 160 & 113 & 114 \\
+    74 & 160 &  40 &  80 & 193
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<14>{
+\begin{block}{Pfade der Länge 30}
+\[
+A^{30}=\begin{pmatrix}
+   673 & 348 & 460 & 188 & 560 \\
+   188 & 433 & 160 & 240 & 460 \\
+   560 & 648 & 673 & 460 & 536 \\
+   348 & 700 & 400 & 433 & 648 \\
+   460 & 400 & 188 & 160 & 673
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<15>{
+\begin{block}{Pfade der Länge 35}
+\[
+A^{35}=\begin{pmatrix}
+   1793%
+\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3)
+& 1644 & 1806 & 1108 & 1632 \\
+   1108 & 1233 &  536 &  560 & 1806 \\
+   1632 & 2914 & 1793%
+\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3)
+& 1806 & 2752 \\
+   1644 & 2366 & 1096 & 1233 & 2914 \\
+   1806 & 1096 & 1108 &  536 & 1793%
+\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3)
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\end{column}
+\end{columns}
+\vbox to2cm{
+\vfill
+\only<3>{
+	\begin{block}{Kürzester Verbindung von 3 nach 2}
+	Der Weg 3---1---6---2 ist die kürzeste Verbindung von 3 nach 2
+	\end{block}
+}
+\only<4>{
+	\begin{block}{Kürzeste Zyklen}
+	Jeder Knoten liegt auf einem Zyklus der Länge 4,
+	dies sind die kürzesten Zyklen.
+	1, 3 und 5 liegen auf beiden Zyklen, 2 und 4 nur auf einem.
+	\end{block}
+}
+\only<6>{
+	\begin{block}{Zyklen der Länge 6}
+	{\em Keine} Zyklen der Länge 6
+	\end{block}
+}
+\only<7>{
+	\begin{block}{Zyklen der Länge 7}
+	{\em Keine} Zyklen der Länge 7
+	\end{block}
+}
+\only<10>{
+	\begin{block}{Zyklen der Länge 10}
+	Genau ein Zyklus der Länge 10
+	\end{block}
+}
+\only<11>{
+	\begin{block}{Verbindung von 4 nach 2}
+	{\em Keine} Verbindung der Länge 15 von 4 nach 2
+	\end{block}
+}
+\only<15>{
+	\begin{block}{Zyklen der Länge 35}
+	Es gibt 1793 Zyklen, die 1 enthalten, und sie enthalten alle auch 3 und 5
+	\end{block}
+}
+}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/pfade/gf.tex b/vorlesungen/slides/8/pfade/gf.tex
new file mode 100644
index 0000000..e89a1fb
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/gf.tex
@@ -0,0 +1,54 @@
+%
+% gf.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\frametitle{Erzeugende Funktion}
+Alle Weglängen zusammen:
+\[
+\uncover<7->{f({\color{darkred}t})=}
+\uncover<4->{E+}
+A
+\uncover<3->{{\color{darkred}t}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{2!}}
+A^2
+\uncover<3->{{\color{darkred}t^2}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{3!}}
+A^3
+\uncover<3->{{\color{darkred}t^3}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{4!}}
+A^4
+\uncover<3->{{\color{darkred}t^4}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{5!}}
+A^5
+\uncover<3->{{\color{darkred}t^5}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{6!}}
+A^6
+\uncover<3->{{\color{darkred}t^6}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{7!}}
+A^7
+\uncover<3->{{\color{darkred}t^7}}
+\dots
+\uncover<6->{= e^{A{\color{darkred}t}}}
+\]
+\uncover<4->{%
+heisst {\em\usebeamercolor[fg]{title} \only<5->{exponentiell} erzeugende Funktion}
+der Wege-Anzahlen}
+
+\begin{itemize}
+\item<8->
+Begriff der Entropie auf einem Graphen
+\item<9->
+Wahrscheinlichkeit, dass ein Zufallsspaziergänger auf einem Graphen an
+einem bestimmten Knoten vorbeikommt
+\end{itemize}
+
+\end{frame}
diff --git a/vorlesungen/slides/8/pfade/langepfade.tex b/vorlesungen/slides/8/pfade/langepfade.tex
new file mode 100644
index 0000000..8c0dd0d
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/langepfade.tex
@@ -0,0 +1,59 @@
+%
+% langepfade.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.5,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}
+\frametitle{Wieviele Pfade der Länge $k$?}
+\begin{definition}
+Anzahl Pfade der Länge $k$ zwischen zwei Knoten
+\[
+a_{{\color{darkred}i}{\color{blue}j}}^{(k)} 
+=
+\#\{\text{Pfade der Länge $k$ von $\color{blue}j$ nach $\color{darkred}i$}\},
+\qquad
+A^{(k)}
+=
+\left(
+a_{{\color{darkred}i}{\color{blue}j}}^{(k)}
+\right)
+\]
+\end{definition}
+\uncover<2->{
+{\usebeamercolor[fg]{title}Spezialfall:} $A^{(1)}=A$.
+}
+
+\uncover<3->{
+\begin{block}{Rekursionsformel}
+\vspace{-25pt}
+\begin{align*}
+a_{{\color{darkred}i}{\color{blue}{\color{blue}j}}}^{(k)}
+&\uncover<4->{=
+\sum_{{\color{darkgreen}l}=1}^n
+\#\{\text{Pfade der Länge $1$ von $\color{darkgreen}l$ nach $\color{darkred}i$}\}}
+\\[-11pt]
+&\uncover<4->{\qquad\qquad\times
+\#\{\text{Pfade der Länge $k-1$ von $\color{blue}j$ nach $\color{darkgreen}l$}\}}
+\\
+&\uncover<5->{=
+\sum_{{\color{darkgreen}l}=1}^n
+a_{{\color{darkred}i}{\color{darkgreen}l}}^{(1)}
+\cdot
+a_{{\color{darkgreen}l}{\color{blue}j}}^{(k-1)}}
+\\
+\uncover<6->{
+\Rightarrow\qquad
+A^{(k)}}
+&\uncover<6->{=
+A\;A^{(k-1)}}
+\uncover<7->{
+\qquad\Rightarrow\qquad
+A^{(k)} = A^k}
+\end{align*}
+\end{block} }
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex
new file mode 100644
index 0000000..1d8b725
--- /dev/null
+++ b/vorlesungen/slides/8/produkt.tex
@@ -0,0 +1,100 @@
+%
+% produkt.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Laplace-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+
+\end{tikzpicture}
+\end{center}
+\vspace{-15pt}
+\begin{block}{Berechne}
+\vspace{-20pt}
+\begin{align*}
+\uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dx{0.84}
+\def\dy{0.48}
+
+\node at (0,0) {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix*}[r]
+-1& 0& 0& 0&+1&-1& 0\\
++1&-1& 0& 0& 0& 0&-1\\
+ 0&+1&-1& 0& 0&+1& 0\\
+ 0& 0&+1&-1& 0& 0&+1\\
+ 0& 0& 0&+1&-1& 0& 0
+\end{pmatrix*}
+\\[20pt]
+\uncover<2->{
+L(G)
+&=
+\begin{pmatrix*}[r]
+ 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{-1}\\
+\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}\\
+\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{ 0}\\
+\uncover<3->{ 0}&\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}\\
+\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{ 0}&\uncover<3->{-1}& 2
+\end{pmatrix*}}
+\end{aligned}$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex
new file mode 100644
index 0000000..425fe1c
--- /dev/null
+++ b/vorlesungen/slides/8/spanningtree.tex
@@ -0,0 +1,164 @@
+%
+% spanningtree.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{Spannbäume}
+
+\vspace{-16pt}
+
+\begin{columns}[t]
+
+\begin{column}{0.40\hsize}
+\begin{block}{Netzwerk}
+Alle Knoten erreichen, Schleifen vermeiden $\Rightarrow$ Spannbaum
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,scale=0.18]
+
+\coordinate (A) at ( 1.2927,-15.0076);
+\coordinate (B) at ( 5.0261,- 7.7143);
+\coordinate (C) at ( 4.9260,-13.0335);
+\coordinate (D) at (12.2094,-22.9960);
+\coordinate (F) at (17.8334,-13.4687);
+\coordinate (G) at ( 6.4208,-10.2438);
+\coordinate (H) at (17.2367,- 3.1047);
+\coordinate (K) at (24.3760,- 3.0293);
+\coordinate (L) at (23.2834,- 1.3563);
+\coordinate (M) at (28.7093,- 4.0627);
+
+\fill (A) circle[radius=0.5];
+\fill (B) circle[radius=0.5];
+\fill (C) circle[radius=0.5];
+\fill (D) circle[radius=0.5];
+\fill (F) circle[radius=0.5];
+\fill (G) circle[radius=0.5];
+\fill (H) circle[radius=0.5];
+\fill (K) circle[radius=0.5];
+\fill (L) circle[radius=0.5];
+\fill (M) circle[radius=0.5];
+
+%\uncover<1-4>{
+%\node at (A) [above] {$A$};
+%\node at (B) [above] {$B$};
+%\node at (C) [below] {$C$};
+%\node at (D) [below] {$D$};
+%\node at (F) [below right] {$F$};
+%\node at (G) [above] {$G$};
+%\node at (H) [above] {$H$};
+%\node at (K) [above right] {$K$};
+%\node at (L) [above] {$L$};
+%\node at (M) [above] {$M$};
+%}
+
+\uncover<5->{
+\node at (A) [above] {$1$};
+\node at (B) [above] {$2$};
+\node at (C) [below] {$3$};
+\node at (D) [below] {$4$};
+\node at (F) [below right] {$5$};
+\node at (G) [above] {$6$};
+\node at (H) [above] {$7$};
+\node at (K) [above right] {$8$};
+\node at (L) [above] {$9$};
+\node at (M) [above] {$10$};
+}
+
+\draw (L)--(H);
+\draw (L)--(K);
+\draw (L)--(M);
+
+\draw (H)--(B);
+\draw (H)--(G);
+\draw (H)--(F);
+\draw (H)--(K);
+
+\draw (K)--(F);
+\draw (K)--(M);
+
+\draw (M)--(F);
+\draw (M)--(D);
+
+\draw (B)--(A);
+\draw (B)--(C);
+\draw (B)--(G);
+
+\draw (G)--(C);
+\draw (G)--(F);
+
+\draw (F)--(D);
+
+\draw (C)--(F);
+\draw (C)--(A);
+\draw (C)--(D);
+
+\draw (A)--(D);
+
+\uncover<2>{
+\draw[line width=2pt,join=round]
+	(A)--(D)--(C)--(F)--(G)--(B)--(H)--(K)--(L)--(M);
+}
+
+\uncover<3>{
+\draw[line width=2pt,join=round]
+	(M)--(D)--(A)--(C)--(G)--(B)--(H)--(L)--(K)--(F);
+}
+
+\uncover<4->{
+\draw[line width=2pt] (M)--(K)--(L)--(H)--(F)--(D);
+\draw[line width=2pt] (F)--(G)--(C)--(A);
+\draw[line width=2pt] (G)--(B);
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+Wieviele Spannbäume gibt es?
+\end{block}
+\end{column}
+
+\begin{column}{0.56\hsize}
+\uncover<5->{%
+\begin{block}{Laplace-Matrix}
+\vspace{-15pt}
+\[
+L=
+\tiny
+\begin{pmatrix}
+ 3&-1&-1&-1& 0& 0& 0& 0& 0& 0\\
+-1& 4&-1& 0& 0&-1&-1& 0& 0& 0\\
+-1&-1& 5&-1&-1&-1& 0& 0& 0& 0\\
+-1& 0&-1& 4&-1& 0& 0& 0& 0&-1\\
+ 0& 0&-1&-1& 6&-1&-1&-1& 0&-1\\
+ 0&-1&-1& 0&-1& 4&-1& 0& 0& 0\\
+ 0&-1& 0& 0&-1&-1& 5&-1&-1& 0\\
+ 0& 0& 0& 0&-1& 0&-1& 4&-1&-1\\
+ 0& 0& 0& 0& 0& 0&-1&-1& 3&-1\\
+ 0& 0& 0&-1&-1& 0& 0&-1&-1& 4\\
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<6->{%
+\begin{block}{Satz von Kirchhoff}
+Die Anzahl der Spannbäume eines Netzwerkes ist ein Kofaktor
+des Laplaceoperators
+\vspace{-5pt}
+\[
+\det L_{ij} = 
+\left|
+L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{array}\right.
+\right|
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<7->{%
+{\usebeamercolor[fg]{title}Beispiel:} 41524
+}
+
+\end{column}
+
+\end{columns}
+
+\end{frame}
diff --git a/vorlesungen/slides/8/tokyo/bahn0.tex b/vorlesungen/slides/8/tokyo/bahn0.tex
new file mode 100644
index 0000000..9c39712
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/bahn0.tex
@@ -0,0 +1,11 @@
+%
+% bahn.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/tokyo/tokyosubway.pdf}
+\end{center}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/bahn1.tex b/vorlesungen/slides/8/tokyo/bahn1.tex
new file mode 100644
index 0000000..6ac3344
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/bahn1.tex
@@ -0,0 +1,28 @@
+%
+% bahn.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Tokyo Bahn-Netz}
+\begin{center}
+\begin{tabular}{rl}
+882&Bahnstationen\\
+108&Bahnlinien\\
+29&Bahngesellschaften\\
+20\,000\,000&Passagiere täglich\\
+7\,000\,000&Passagiere alleine in Shinjuku\\
+\end{tabular}
+\end{center}
+\uncover<2->{
+\begin{block}{Dirichlet-Zerlegung und Bahnhöfe}
+\begin{center}
+\uncover<3->{Passagiere wählen den nächsten Bahnhöfe}\\
+\uncover<4->{$\Downarrow$}\\
+\uncover<5->{Bahnhöfe definieren eine Dirichletzerlegung der Stadt}
+\end{center}
+\end{block}
+}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/bahn2.tex b/vorlesungen/slides/8/tokyo/bahn2.tex
new file mode 100644
index 0000000..4adc1bf
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/bahn2.tex
@@ -0,0 +1,12 @@
+%
+% bahn.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/tokyo/shinjuku-subway-map.jpg}
+\end{center}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/google.tex b/vorlesungen/slides/8/tokyo/google.tex
new file mode 100644
index 0000000..d37c98d
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/google.tex
@@ -0,0 +1,11 @@
+%
+% google.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/tokyo/transportnetworkgraph.png}
+\end{center}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg
new file mode 100644
index 0000000..1c513da
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg
diff --git a/vorlesungen/slides/8/tokyo/tokyosubway.pdf b/vorlesungen/slides/8/tokyo/tokyosubway.pdf
new file mode 100644
index 0000000..6b84a8d
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/tokyosubway.pdf
diff --git a/vorlesungen/slides/8/tokyo/transportnetworkgraph.png b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png
new file mode 100644
index 0000000..4a11183
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png
diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc
new file mode 100644
index 0000000..fa6c29b
--- /dev/null
+++ b/vorlesungen/slides/9/Makefile.inc
@@ -0,0 +1,14 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter9 =								\
+	../slides/9/google.tex						\
+	../slides/9/markov.tex						\
+	../slides/9/irreduzibel.tex					\
+	../slides/9/stationaer.tex					\
+	../slides/9/pf.tex						\
+	../slides/9/chapter.tex
+
diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex
new file mode 100644
index 0000000..9e26587
--- /dev/null
+++ b/vorlesungen/slides/9/chapter.tex
@@ -0,0 +1,14 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+
+
+\folie{9/google.tex}
+\folie{9/markov.tex}
+\folie{9/stationaer.tex}
+\folie{9/irreduzibel.tex}
+\folie{9/pf.tex}
+
+
diff --git a/vorlesungen/slides/9/google.tex b/vorlesungen/slides/9/google.tex
new file mode 100644
index 0000000..d1ec31d
--- /dev/null
+++ b/vorlesungen/slides/9/google.tex
@@ -0,0 +1,123 @@
+%
+% google.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Google-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.4}
+\coordinate (A) at (0,0);
+\coordinate (B) at (0:\r);
+\coordinate (C) at (60:\r);
+\coordinate (D) at (120:\r);
+\coordinate (E) at (180:\r);
+
+\foreach \a in {2,...,5}{
+	\fill[color=white] ({60*(\a-2)}:\r) circle[radius=0.2];
+	\draw ({60*(\a-2)}:\r) circle[radius=0.2];
+	\node at ({60*(\a-2)}:\r) {$\a$};
+}
+\fill[color=white] (A) circle[radius=0.2];
+\draw (A) circle[radius=0.2];
+\node at (A) {$1$};
+
+{\color<6>{red}
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+}
+
+{\color<7>{red}
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) to[out=-150,in=-30] (E);
+}
+
+{\color<8>{red}
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-90,in=30] (A);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-30,in=90] (B);
+}
+
+{\color<9>{red}
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (C);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (A);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+}
+
+{\color<10>{red}
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) to[out=90,in=-150] (D);
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\renewcommand{\arraystretch}{1.1}
+\uncover<5->{
+\begin{align*}
+H&=\begin{pmatrix}
+\uncover<6->{0          }
+	&\uncover<7->{0          }
+	&\uncover<8->{{\color<8>{red}\frac{1}{2}}}
+	&\uncover<9->{{\color<9>{red}\frac{1}{3}}}
+	&\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\
+\uncover<6->{{\color<6>{red}\frac{1}{2}}}
+	&\uncover<7->{0          }
+	&\uncover<8->{{\color<8>{red}\frac{1}{2}}}
+	&\uncover<9->{0          }
+	&\uncover<10->{0          }\\
+\uncover<6->{{\color<6>{red}\frac{1}{2}}}
+	&\uncover<7->{{\color<7>{red}\frac{1}{2}}}
+	&\uncover<8->{0          }
+	&\uncover<9->{{\color<9>{red}\frac{1}{3}}}
+	&\uncover<10->{0          }\\
+\uncover<6->{0          }
+	&\uncover<7->{0          }
+	&\uncover<8->{0          }
+	&\uncover<9->{0          }
+	&\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\
+\uncover<6->{0          }
+	&\uncover<7->{{\color<7>{red}\frac{1}{2}}}
+	&\uncover<8->{0          }
+	&\uncover<9->{{\color<9>{red}\frac{1}{3}}}
+	&\uncover<10->{0          }
+\end{pmatrix}
+\\
+\uncover<11->{
+h_{ij}
+&=
+\frac{1}{\text{Anzahl Links ausgehend von $j$}}
+}
+\end{align*}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Bestimme die Wahrscheinlichkeit $p(i)$, mit der sich ein Surfer
+auf der Website $i$ befindet
+\end{block}
+\uncover<2->{
+\begin{block}{Navigation}
+$p(i) = P(i,\text{vor Navigation})$,
+\uncover<3->{$p'(i)=P(i,\text{nach Navigation})$}
+\uncover<4->{
+\[
+p'(i) = \sum_{j=1}^n h_{ij} p(j)
+\]}
+\end{block}}
+\vspace{-15pt}
+\begin{block}{Freier Wille}
+\vspace{-12pt}
+\[
+G = \alpha H + (1-\alpha)\frac{UU^t}{n}
+\]
+Google-Matrix
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/9/irreduzibel.tex b/vorlesungen/slides/9/irreduzibel.tex
new file mode 100644
index 0000000..87e90e4
--- /dev/null
+++ b/vorlesungen/slides/9/irreduzibel.tex
@@ -0,0 +1,136 @@
+%
+% irreduzibel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Irreduzible Markovkette}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2}
+\coordinate (A) at ({\r*cos(0*60)},{\r*sin(0*60)});
+\coordinate (B) at ({\r*cos(1*60)},{\r*sin(1*60)});
+\coordinate (C) at ({\r*cos(2*60)},{\r*sin(2*60)});
+\coordinate (D) at ({\r*cos(3*60)},{\r*sin(3*60)});
+\coordinate (E) at ({\r*cos(4*60)},{\r*sin(4*60)});
+\coordinate (F) at ({\r*cos(5*60)},{\r*sin(5*60)});
+
+\uncover<-2>{
+\draw (A) -- (B);
+\draw (A) -- (C);
+\draw (A) -- (D);
+\draw (A) -- (E);
+\draw (A) -- (F);
+
+\draw (B) -- (A);
+\draw (B) -- (C);
+\draw (B) -- (D);
+\draw (B) -- (E);
+\draw (B) -- (F);
+
+\draw (C) -- (A);
+\draw (C) -- (B);
+\draw (C) -- (D);
+\draw (C) -- (E);
+\draw (C) -- (F);
+
+\draw (D) -- (A);
+\draw (D) -- (B);
+\draw (D) -- (C);
+\draw (D) -- (E);
+\draw (D) -- (F);
+
+\draw (E) -- (A);
+\draw (E) -- (B);
+\draw (E) -- (C);
+\draw (E) -- (D);
+\draw (E) -- (F);
+
+\draw (F) -- (A);
+\draw (F) -- (B);
+\draw (F) -- (C);
+\draw (F) -- (D);
+\draw (F) -- (E);
+}
+
+\uncover<3->{
+
+\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] (A) to[out=90,in=-30] (B);
+\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] (A) -- (C);
+\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) -- (A);
+\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] (B) to[out=150,in=30] (C);
+\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) to[out=-90,in=-150,distance=1cm] (B);
+\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) to[out=-60,in=180] (A);
+\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) -- (B);
+
+\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
+	(D) to[out=-90,in=150] (E);
+\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt]
+	(E) -- (D);
+\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt]
+	(E) to[out=-30,in=-150] (F);
+\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
+	(F) -- (E);
+\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
+	(F) to[out=120,in=0] (D);
+\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
+	(D) -- (F);
+}
+
+\fill[color=white] (A) circle[radius=0.2];
+\fill[color=white] (B) circle[radius=0.2];
+\fill[color=white] (C) circle[radius=0.2];
+\fill[color=white] (D) circle[radius=0.2];
+\fill[color=white] (E) circle[radius=0.2];
+\fill[color=white] (F) circle[radius=0.2];
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+\draw (F) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (F) {$6$};
+
+\end{tikzpicture}
+\end{center}
+\uncover<2->{%
+\begin{block}{Irreduzibel}
+Graph zusammenhängend $\Rightarrow$
+Keine Zerlegung in Teilgraphen möglich
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Reduzibel}
+Die Zustandsmenge zerfällt in zwei disjunkte Teilmengen $V=V_1\cup V_2$
+und es gibt keine Übergängen zwischen den Mengen:
+\uncover<4->{%
+\begin{align*}
+P
+&=
+\begin{pmatrix*}[l]
+0  &0.2&0.5&   &   &   \\
+0.3&0.2&0.5&   &   &   \\
+0.7&0.6&0  &   &   &   \\
+   &   &   &0  &0.3&0.4\\
+   &   &   &0.4&0  &0.6\\
+   &   &   &0.6&0.7&0  
+\end{pmatrix*}
+\end{align*}}%
+\uncover<5->{%
+$P$ zerfällt in zwei Blöcke die unabhängig voneinander analysiert werden können
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/9/markov.tex b/vorlesungen/slides/9/markov.tex
new file mode 100644
index 0000000..e92ff0f
--- /dev/null
+++ b/vorlesungen/slides/9/markov.tex
@@ -0,0 +1,111 @@
+%
+% markov.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{frame}[t]
+\frametitle{Markovketten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!40]
+	(A) -- (C);
+\draw[color=white,line width=8pt] (B) -- (D);
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!80]
+	(B) -- (D);
+
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!60]
+	(A) -- (B);
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!20]
+	(B) -- (C);
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black]
+	(C) -- (D);
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black]
+	(D) -- (E);
+\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black]
+	(E) -- (A);
+
+\fill[color=white] (A) circle[radius=0.2];
+\fill[color=white] (B) circle[radius=0.2];
+\fill[color=white] (C) circle[radius=0.2];
+\fill[color=white] (D) circle[radius=0.2];
+\fill[color=white] (E) circle[radius=0.2];
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 0.6$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 0.2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 1$};
+\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 1$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 1$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 0.4$};
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 0.8$};
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Verteilung}
+\begin{itemize}
+\item<8->
+Welche stationäre Verteilung auf den Knoten stellt sich ein?
+\item<9->
+$P(i)=?$
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{\strut\mbox{Übergang\only<3->{s-/Wahrscheinlichkeit}smatrix}}
+$P_{ij} = P(i | j)$, Wahrscheinlichkeit, in den Zustand $i$ überzugehen,
+\begin{align*}
+P
+&=
+\begin{pmatrix}
+   &   & & &1\phantom{.0}\\
+0.6&   & & & \\
+0.4&0.2& & & \\
+   &0.8&1\phantom{.0}& & \\
+   &   & &1\phantom{.0}& 
+\end{pmatrix}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<4->{%
+\begin{block}{Eigenschaften}
+\begin{itemize}
+\item<5-> $P_{ij}\ge 0\;\forall i,j$
+\item<6-> Spaltensumme:
+\(
+\displaystyle
+\sum_{i=1}^n P_{ij} = 1\;\forall j
+\)
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/9/pf.tex b/vorlesungen/slides/9/pf.tex
new file mode 100644
index 0000000..da2ef2b
--- /dev/null
+++ b/vorlesungen/slides/9/pf.tex
@@ -0,0 +1,53 @@
+%
+% pf.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Perron-Frobenius-Theorie}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Positive Matrizen und Vektoren}
+$P\in M_{m\times n}(\mathbb{R})$ 
+\begin{itemize}
+\item<2->
+$P$ heisst positiv, $P>0$, wenn $p_{ij}>0\;\forall i,j$
+\item<3->
+$P\ge 0$, wenn $p_{ij}\ge 0\;\forall i,j$
+\end{itemize}
+\end{block}
+\uncover<4->{%
+\begin{block}{Beispiele}
+\begin{itemize}
+\item<5->
+Adjazenzmatrix $A(G)$
+\item<6->
+Gradmatrix $D(G)$
+\item<7->
+Wahrscheinlichkeitsmatrizen
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Satz}
+Es gibt einen positiven Eigenvektor $p$ von $P$ zum Eigenwert $1$
+\end{block}}
+\uncover<9->{%
+\begin{block}{Satz}
+$P$ irreduzible Matrix, $P\ge 0$, hat einen Eigenvektor $p$, $p\ge 0$,
+zum Eigenwert $1$
+\end{block}}
+\uncover<10->{%
+\begin{block}{Potenzmethode}
+Falls $P\ge 0$ einen eindeutigen Eigenvektor $p$ hat\uncover<11->{,
+dann konveriert die rekursiv definierte Folge
+\[
+p_{n+1}=\frac{Pp_n}{\|Pp_n\|}, p_0 \ge 0, p_0\ne 0
+\]}%
+\uncover<12->{$\displaystyle\lim_{n\to\infty} p_n = p$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/9/stationaer.tex b/vorlesungen/slides/9/stationaer.tex
new file mode 100644
index 0000000..92fab16
--- /dev/null
+++ b/vorlesungen/slides/9/stationaer.tex
@@ -0,0 +1,57 @@
+%
+% stationaer.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Stationäre Verteilung}
+%\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zeitentwicklung}
+\begin{itemize}
+\item<2->
+$P$ eine Wahrscheinlichkeitsmatrix
+\item<3->
+$p_0\in\mathbb{R}^n$ Verteilung zur Zeit $t=0$ bekannt
+\item<4->
+$p_k\in\mathbb{R}^n$ Verteilung zur Zeit $t=k$
+\end{itemize}
+\uncover<5->{%
+Entwicklungsgesetz
+\begin{align*}
+P(i,t=k)
+&=
+\sum_{j=1}^n P_{ij} P(j,t=k-1)
+\\
+\uncover<6->{
+p_k &= Pp_{k-1}
+}
+\end{align*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Stationär}
+Bedingung: $p_{k\mathstrut} = p_{k-1}$
+\uncover<8->{
+\begin{align*}
+\Rightarrow
+Pp &= p
+\end{align*}}\uncover<9->{%
+Eigenvektor zum Eigenwert $1$}
+\end{block}}
+\uncover<10->{%
+\begin{block}{Fragen}
+\begin{enumerate}
+\item<11->
+Gibt es eine stationäre Verteilung?
+\item<12->
+Gibt es einen Eigenvektor mit Einträgen $\ge 0$?
+\item<13->
+Gibt es mehr als eine Verteilung?
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/Makefile b/vorlesungen/slides/Makefile
new file mode 100644
index 0000000..d3c7a17
--- /dev/null
+++ b/vorlesungen/slides/Makefile
@@ -0,0 +1,34 @@
+#
+# Makefile -- build the slide collection
+#
+# (c) 2019 Prof Dr Andreas Müller, Hochschule Rapeprswil
+#
+test:	test-handout.pdf test-presentation.pdf
+
+slides:	slides-handout.pdf slides-presentation.pdf
+
+include Makefile.inc
+
+files = common.tex $(slides)
+
+catalog:	slides-catalog.pdf
+presentation:	slides-presentation.pdf
+handout:	slides-handout.pdf
+
+slides-handout.pdf:	slides-handout.tex slides.tex $(files)
+	pdflatex slides-handout.tex
+
+slides-catalog.pdf:	slides-handout.pdf
+	pdfjam --outfile slides-catalog.pdf \
+		--paper a4paper --nup 2x5 \
+		slides-handout.pdf
+
+slides-presentation.pdf:	slides-presentation.tex slides.tex $(files)
+	pdflatex slides-presentation.tex
+
+test-handout.pdf:	test-handout.tex test.tex $(files)
+	pdflatex test-handout.tex
+
+test-presentation.pdf:	test-presentation.tex test.tex $(files)
+	pdflatex test-presentation.tex
+
diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc
new file mode 100644
index 0000000..4bf9348
--- /dev/null
+++ b/vorlesungen/slides/Makefile.inc
@@ -0,0 +1,17 @@
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+include ../slides/0/Makefile.inc
+include ../slides/1/Makefile.inc
+include ../slides/2/Makefile.inc
+include ../slides/3/Makefile.inc
+include ../slides/4/Makefile.inc
+include ../slides/5/Makefile.inc
+include ../slides/8/Makefile.inc
+include ../slides/9/Makefile.inc
+
+slides = 								\
+	$(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4)	\
+	$(chapter5) $(chapter8) $(chapter9)
diff --git a/vorlesungen/slides/common.tex b/vorlesungen/slides/common.tex
new file mode 100644
index 0000000..866bab1
--- /dev/null
+++ b/vorlesungen/slides/common.tex
@@ -0,0 +1,25 @@
+%
+% common.tex -- gemeinsame definition
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\input{../common/packages.tex}
+\mode<beamer>{%
+\usetheme[hideothersubsections,hidetitle]{Hannover}
+}
+\beamertemplatenavigationsymbolsempty
+\title[Seminar]{Seminar}
+\subtitle{Foliensammlung}
+\author[A.~Müller]{Andreas Müller}
+\date[]{}
+\newboolean{presentation}
+
+\def\folie#1{
+%\subsection{#1}
+\begin{frame}
+\begin{center}
+\tt #1
+\end{center}
+\end{frame}
+\input{#1}
+}
diff --git a/vorlesungen/slides/slides-handout.tex b/vorlesungen/slides/slides-handout.tex
new file mode 100644
index 0000000..d834053
--- /dev/null
+++ b/vorlesungen/slides/slides-handout.tex
@@ -0,0 +1,12 @@
+%
+% slides-handout.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{slides.tex}
+\end{document}
+
diff --git a/vorlesungen/slides/slides-presentation.tex b/vorlesungen/slides/slides-presentation.tex
new file mode 100644
index 0000000..ff80a11
--- /dev/null
+++ b/vorlesungen/slides/slides-presentation.tex
@@ -0,0 +1,12 @@
+%
+% slides-presentation.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\input{slides.tex}
+\end{document}
+
diff --git a/vorlesungen/slides/slides.tex b/vorlesungen/slides/slides.tex
new file mode 100644
index 0000000..b606375
--- /dev/null
+++ b/vorlesungen/slides/slides.tex
@@ -0,0 +1,79 @@
+%
+% slides.tex collection of all slides
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\def\titel{
+% title slide for this chapter
+\begin{frame}
+\titlepage
+\end{frame}
+\ifthenelse{\boolean{presentation}}{}{
+% add an empty slide for alignment in the slide catalog
+\begin{frame}
+\end{frame}
+}
+}
+
+\title[MathSem]{Mathematisches Seminar: Matrizen}
+\section{Intro}
+\titel
+\author[]{}
+\subtitle{}
+\input{0/chapter.tex}
+
+\title[Grundlagen]{Grundlagen}
+\section{Grundlagen}
+\titel
+\input{1/chapter.tex}
+
+\title[Vektoren/Matrizen]{Vektoren und Matrizen}
+\section{Vektoren und Matrizen}
+\titel
+\input{2/chapter.tex}
+
+\title[Polynome]{Polynome}
+\section{Polynome}
+\titel
+\input{3/chapter.tex}
+
+\title[Endliche Körper]{Endliche Körper}
+\section{Endliche Körper}
+\titel
+\input{4/chapter.tex}
+
+\title[EW/EV]{Eigenwerte und Eigenvektoren}
+\section{Eigenwerte und Eigenvektoren}
+\titel
+\input{5/chapter.tex}
+
+%\title[Permutationen]{Permutationen}
+%\section{Permutationen}
+%\titel
+%\input{6/chapter.tex}
+
+%\title[Matrizengruppen]{Matrizengruppen}
+%\section{Matrizengruppen}
+%\titel
+%\input{7/chapter.tex}
+
+\title[Graphen]{Graphen}
+\section{Graphen}
+\titel
+\input{8/chapter.tex}
+
+\title[Wahrscheinlichkeit]{Wahrscheinlichkeitsmatrizen}
+\section{Wahrscheinlichkeitsmatrizen}
+\titel
+\input{9/chapter.tex}
+
+%\title[Krypto]{Anwendungen in Kryptographie und Codierungstheorie}
+%\section{Krypto}
+%\titel
+%\input{a/chapter.tex}
+
+%\title[Homologie]{Homologie}
+%\section{Homologie}
+%\titel
+%\input{b/chapter.tex}
+
diff --git a/vorlesungen/slides/test-handout.tex b/vorlesungen/slides/test-handout.tex
new file mode 100644
index 0000000..63f41e4
--- /dev/null
+++ b/vorlesungen/slides/test-handout.tex
@@ -0,0 +1,12 @@
+%
+% test-handout.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{test.tex}
+\end{document}
+
diff --git a/vorlesungen/slides/test-presentation.tex b/vorlesungen/slides/test-presentation.tex
new file mode 100644
index 0000000..2cf9816
--- /dev/null
+++ b/vorlesungen/slides/test-presentation.tex
@@ -0,0 +1,12 @@
+%
+% test-presentation.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\input{test.tex}
+\end{document}
+
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
new file mode 100644
index 0000000..e4b9ad7
--- /dev/null
+++ b/vorlesungen/slides/test.tex
@@ -0,0 +1,17 @@
+%
+% test.tex collection of all slides
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+%\folie{5/verzerrung.tex}
+ 
+% XXX Visualisierung Cayley-Hamilton-Produkte
+% XXX \folie{5/chvisual.tex}
+
+% XXX stone weierstrass incomplete
+%\folie{5/stoneweierstrass.tex}
+
+% XXX polynome auf dem spektrum
+% XXX Motiviation für *-Operation
+%\folie{5/normal.tex}
+