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diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex new file mode 100644 index 0000000..1116a83 --- /dev/null +++ b/vorlesungen/slides/5/ketten.tex @@ -0,0 +1,79 @@ +% +% ketten.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ketten von Unterräumen} +\begin{block}{Schachtelung} +Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt: +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcccc} +0 &=&\mathcal{K}^0(f) + &\subset&\mathcal{K}^1(f) + &\subset&\dots + &\subset&\mathcal{K}^k(f) + &\subset&\mathcal{K}^{k+1}(f) + &\subset&\dots + &\subset&\displaystyle\bigcup_{k=0}^\infty \mathcal{K}^k(f) + &=:&\mathcal{K}(f) +\\[14pt] +\Bbbk^n &=&\mathcal{J}^0(f) + &\supset&\mathcal{J}^1(f) + &\supset&\dots + &\supset&\mathcal{J}^{k}(f) + &\supset&\mathcal{J}^{k+1}(f) + &\supset&\dots + &\supset&\displaystyle\bigcap_{k=0}^\infty \mathcal{J}^k(f) + &=:&\mathcal{J}(f) +\end{array} +\] +\end{block} +\vspace{-20pt} +\uncover<2->{% +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Abildung der Kerne} +\vspace{-10pt} +\begin{align*} +f \mathcal{K}^k(f) +&= +\{f(v)\;|\; f^k(v) = 0\} +\\ +&\subset +\{ v\;|\; f^{k+1}(v)=0\} +\\ +&= +\mathcal{K}^{k+1}(f) +\\ +\Rightarrow +f\mathcal{K}(f)&= f\mathcal{K}(f) +\quad\text{invariant} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Abbildung der Bild} +\vspace{-10pt} +\begin{align*} +f\mathcal{J}^k(f) +&= +\{f(f^{k}(v))\;|\; v\in V\} +\\ +&= +\{f^{k+1}(v)\;|\; v\in V\} +\\ +&= +\mathcal{J}^{k+1}(f) +\\ +\Rightarrow +f\mathcal{J}(f)&= \mathcal{J}(f) +\quad\text{invariant} +\end{align*} +\end{block} +\end{column} +\end{columns}} +\end{frame} |