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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-04-10 19:57:13 +0200 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-04-10 19:57:13 +0200 |
commit | 1fb743df08b0734932d510c6b11405d0a2dbbe47 (patch) | |
tree | f21ae2cd1a5318585df6a644d47328feccf073d7 /vorlesungen/slides/7/semi.tex | |
parent | new slides (diff) | |
download | SeminarMatrizen-1fb743df08b0734932d510c6b11405d0a2dbbe47.tar.gz SeminarMatrizen-1fb743df08b0734932d510c6b11405d0a2dbbe47.zip |
new slides
Diffstat (limited to 'vorlesungen/slides/7/semi.tex')
-rw-r--r-- | vorlesungen/slides/7/semi.tex | 109 |
1 files changed, 109 insertions, 0 deletions
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex new file mode 100644 index 0000000..46f6d03 --- /dev/null +++ b/vorlesungen/slides/7/semi.tex @@ -0,0 +1,109 @@ +% +% semi.tex -- Beispiele: semidirekte Produkte +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehung/Skalierung und Verschiebung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Skalierung und Verschiebung} +Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$ +\\ +Wirkung auf $\mathbb{R}$: +\[ +x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Drehung und Verschiebung} +Gruppe +$G= +\{ (\alpha,\vec{t}) +\;|\; +\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2 +\}$ +Wirkung auf $\mathbb{R}^2$: +\[ +\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} +\] +\end{block} +\end{column} +\end{columns} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(e^{s_1},t_1)(e^{s_2},t_2)x +&= +(e^{s_1},t_1)(e^{s_2}x+t_2) +\\ +&= +e^{s_1+s_2}x + e^{s_1}t_2+t_1 +\\ +(e^{s_1},t_1)(e^{s_2},t_2) +&= +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2) +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +\vec{x} +&= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2) +\\ +&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1 +\\ +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +&= +(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) +\end{align*} +\end{block} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Matrixschreibweise} +\vspace{-12pt} +\[ +g=(e^s,t) = +\begin{pmatrix} +e^s&t\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}x\\1\end{pmatrix} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Matrixschreibweise} +\vspace{-12pt} +\[ +g=(\alpha,\vec{t}) = +\begin{pmatrix} +D_{\alpha}&\vec{t}\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}\vec{x}\\1\end{pmatrix} +\] +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |