add new slides

13 files changed, 475 insertions, 0 deletions

diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc
index 34dadf6..c857fec 100644
--- a/vorlesungen/slides/2/Makefile.inc
+++ b/vorlesungen/slides/2/Makefile.inc
@@ -15,5 +15,7 @@ chapter2 =								\
 	../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/chapter.tex b/vorlesungen/slides/2/chapter.tex
index 38eb21f..49e656a 100644
--- a/vorlesungen/slides/2/chapter.tex
+++ b/vorlesungen/slides/2/chapter.tex
@@ -13,3 +13,5 @@
 \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/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/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