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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-03-02 21:25:06 +0100 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-03-02 21:25:06 +0100 |
commit | 4d16eab7a381a8379bc139ed2744912683661bab (patch) | |
tree | 46bef3ffb8ce26d5950e99b849b3b539d5bc61f4 /vorlesungen/slides/5/kernbild.tex | |
parent | new presentation for spectral theory (diff) | |
download | SeminarMatrizen-4d16eab7a381a8379bc139ed2744912683661bab.tar.gz SeminarMatrizen-4d16eab7a381a8379bc139ed2744912683661bab.zip |
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-rw-r--r-- | vorlesungen/slides/5/kernbild.tex | 68 |
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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} |