From 4d16eab7a381a8379bc139ed2744912683661bab Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Tue, 2 Mar 2021 21:25:06 +0100
Subject: add slides

---
 vorlesungen/slides/5/kernbild.tex | 68 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 68 insertions(+)
 create mode 100644 vorlesungen/slides/5/kernbild.tex

(limited to 'vorlesungen/slides/5/kernbild.tex')

diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex
new file mode 100644
index 0000000..f0bd6fa
--- /dev/null
+++ b/vorlesungen/slides/5/kernbild.tex
@@ -0,0 +1,68 @@
+%
+% 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}
+\begin{block}{Kern}
+Lineare Abbildung $f\colon V\to V$
+\[
+\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\}
+\]
+\end{block}
+\begin{block}{Kern von $A^k$}
+\[
+\mathcal{K}^k(f) = \operatorname{ker} f^k
+\]
+\begin{align*}
+\mathcal{K}^k(f)
+&=
+\{v\in V\;|\; f^{k}(v)=0\}
+\\
+&\subset
+\{v\in V\;|\; f^{k+1}(v)=0\}
+\\
+&=\mathcal{K}^{k+1}(f)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Bild}
+Lineare Abbildung $f\colon V\to V$
+\[
+\operatorname{im}f
+=
+\mathcal{J}(f)
+=
+\{f(v)\;|\; v\in V\}
+\]
+\end{block}
+\begin{block}{Bild von $A^k$}
+\[
+\mathcal{J}^k(f) = \operatorname{im}f^k
+\]
+\begin{align*}
+\mathcal{J}^k(f)
+&=
+\operatorname{im}f^k
+=
+\operatorname{im}(f^{k}\circ f)
+\\
+&=
+\{f^{k-1} w\;|\; w = f(v)\}
+\\
+&\subset
+\{f^{k-1} w\;|\; w \in V\}
+\\
+&\mathcal{J}^{k-1}(f)
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
-- 
cgit v1.2.1


From 38f326c35243942285c43310751025d2c16801dd Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 4 Mar 2021 20:30:33 +0100
Subject: Spektraltheorie 1. Teil

---
 vorlesungen/slides/5/kernbild.tex | 28 +++++++++++++++++++++++-----
 1 file changed, 23 insertions(+), 5 deletions(-)

(limited to 'vorlesungen/slides/5/kernbild.tex')

diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex
index f0bd6fa..3890717 100644
--- a/vorlesungen/slides/5/kernbild.tex
+++ b/vorlesungen/slides/5/kernbild.tex
@@ -10,29 +10,38 @@
 \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}
+\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{block}}
 \end{column}
 \begin{column}{0.48\textwidth}
+\uncover<2->{%
 \begin{block}{Bild}
 Lineare Abbildung $f\colon V\to V$
 \[
@@ -42,27 +51,36 @@ Lineare Abbildung $f\colon V\to V$
 =
 \{f(v)\;|\; v\in V\}
 \]
-\end{block}
+\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\}
+}
 \\
-&\mathcal{J}^{k-1}(f)
+\uncover<11->{
+&=\mathcal{J}^{k-1}(f)
+}
 \end{align*}
-\end{block}
+\end{block}}
 \end{column}
 \end{columns}
 \end{frame}
-- 
cgit v1.2.1

