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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-14 07:26:10 +0200 |
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committer | GitHub <noreply@github.com> | 2021-06-14 07:26:10 +0200 |
commit | 114633b43a0f1ebedbc5dfd85f75ede9841f26fd (patch) | |
tree | 18e61c7d69883a1c9b69098b7d36856abaed5c1e /vorlesungen/slides/6/permutationen | |
parent | Delete buch.pdf (diff) | |
parent | Fix references.bib (diff) | |
download | SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.tar.gz SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.zip |
Merge branch 'master' into master
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-rw-r--r-- | vorlesungen/slides/6/permutationen/matrizen.tex | 79 |
1 files changed, 79 insertions, 0 deletions
diff --git a/vorlesungen/slides/6/permutationen/matrizen.tex b/vorlesungen/slides/6/permutationen/matrizen.tex new file mode 100644 index 0000000..d40c396 --- /dev/null +++ b/vorlesungen/slides/6/permutationen/matrizen.tex @@ -0,0 +1,79 @@ +% +% matrizen.tex -- Darstellung der Permutationen als Matrizen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Permutationsmatrizen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Permutationsabbildung} +$\sigma\in S_n$ eine Permutation, definiere +\[ +f +\colon +e_i \mapsto e_{\sigma(i)} +\] +($e_i$ Standardbasisvektor) +\end{block} +\uncover<2->{% +\begin{block}{Lineare Abbildung} +$f$ kann erweitert werden zu einer linearen Abbildung +\[ +\tilde{f} +\colon +\Bbbk^n \to \Bbbk^n +: +\sum_{k=1}^n a_i e_i +\mapsto +\sum_{k=1}^n a_i f(e_i) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Permutationsmatrix} +Matrix $A_{\tilde{f}}$ der linearen Abbildung $\tilde{f}$ +hat die Matrixelemente +\[ +a_{ij} += +\begin{cases} +1&\qquad i=\sigma(j)\\ +0&\qquad\text{sonst} +\end{cases} +\] +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Beispiel} +\vspace{-10pt} +\[ +\begin{pmatrix} +1&2&3&4\\ +3&2&4&1 +\end{pmatrix} +\mapsto +\begin{pmatrix} +0&0&0&1\\ +0&1&0&0\\ +1&0&0&0\\ +0&0&1&0 +\end{pmatrix} +\] +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Homomorphismus} +Die Abbildung +$S_n\to\operatorname{GL}(\Bbbk)\colon \sigma \mapsto A_{\tilde{f}}$ +ist ein Homomorphismus +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |