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author | LordMcFungus <mceagle117@gmail.com> | 2021-03-22 18:05:11 +0100 |
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committer | GitHub <noreply@github.com> | 2021-03-22 18:05:11 +0100 |
commit | 76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7 (patch) | |
tree | 11b2d41955ee4bfa0ae5873307c143f6b4d55d26 /vorlesungen/slides/5/kernbild.tex | |
parent | more chapter structure (diff) | |
parent | add title image (diff) | |
download | SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.tar.gz SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.zip |
Merge pull request #1 from AndreasFMueller/master
update
Diffstat (limited to '')
-rw-r--r-- | vorlesungen/slides/5/kernbild.tex | 86 |
1 files changed, 86 insertions, 0 deletions
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} |