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author | Joshua Baer <the.baer.joshua@gmail.ch> | 2021-04-12 21:51:55 +0200 |
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committer | Joshua Baer <the.baer.joshua@gmail.ch> | 2021-04-12 21:51:55 +0200 |
commit | 2db90bfe4b174570424c408f04000902411d8755 (patch) | |
tree | e297a6274ff748de27257bffd7097c6b362ba12d /vorlesungen/slides/7/semi.tex | |
parent | add new files (diff) | |
download | SeminarMatrizen-2db90bfe4b174570424c408f04000902411d8755.tar.gz SeminarMatrizen-2db90bfe4b174570424c408f04000902411d8755.zip |
update to current state of book
Diffstat (limited to 'vorlesungen/slides/7/semi.tex')
-rw-r--r-- | vorlesungen/slides/7/semi.tex | 234 |
1 files changed, 117 insertions, 117 deletions
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 66b8d27..d74b7d0 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -1,117 +1,117 @@ -% -% 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} -\uncover<2->{% -\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} -\uncover<3->{% -\begin{block}{Verknüpfung} -\vspace{-15pt} -\begin{align*} -(e^{s_1},t_1)(e^{s_2},t_2)x -&\uncover<4->{= -(e^{s_1},t_1)(e^{s_2}x+t_2)} -\\ -&\uncover<5->{= -e^{s_1+s_2}x + e^{s_1}t_2+t_1} -\\ -\uncover<6->{ -(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} -\uncover<7->{% -\begin{block}{Verknüpfung} -\vspace{-15pt} -\begin{align*} -(\alpha_1,\vec{t}_1) -(\alpha_2,\vec{t}_2) -\vec{x} -&\uncover<8->{= -(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} -\\ -&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1} -\\ -\uncover<10->{ -(\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} -\uncover<11->{% -\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} -\uncover<12->{% -\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 +%
+% 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}
+\uncover<2->{%
+\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}
+\uncover<3->{%
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(e^{s_1},t_1)(e^{s_2},t_2)x
+&\uncover<4->{=
+(e^{s_1},t_1)(e^{s_2}x+t_2)}
+\\
+&\uncover<5->{=
+e^{s_1+s_2}x + e^{s_1}t_2+t_1}
+\\
+\uncover<6->{
+(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}
+\uncover<7->{%
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+\vec{x}
+&\uncover<8->{=
+(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
+\\
+&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1}
+\\
+\uncover<10->{
+(\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}
+\uncover<11->{%
+\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}
+\uncover<12->{%
+\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
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