From 4d16eab7a381a8379bc139ed2744912683661bab Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= 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?= 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