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exc 1 formating theorie

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fruitstaa 2018-11-12 11:37:08 +01:00
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@ -1,4 +1,4 @@
\documentclass[12pt, a4paper]{scrbook} \documentclass[12pt, a4paper]{article}
%packages %packages
\usepackage[ngerman]{babel} \usepackage[ngerman]{babel}
@ -41,65 +41,60 @@ For $R_2 \times R_1$\\
$\Rightarrow R_1 \times R_2 \neq R_2 \times R_1 \Rightarrow$ does not commute\\ $\Rightarrow R_1 \times R_2 \neq R_2 \times R_1 \Rightarrow$ does not commute\\
\newpage \newpage
\hspace{-2em}
\hspace{-5.5em}
For $T_1 \times T_2$\\ For $T_1 \times T_2$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} 1 & 0 & a \\ 0&1&b\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & c \\ 0& 1& d\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} 1& 0 & a+c \\ 0& 1 & b+d\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & a \\ 0&1&b\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & c \\ 0& 1& d\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} 1& 0 & a+c \\ 0& 1 & b+d\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
\hspace{-4.5em} \hspace{-0.5em}
For $T_2 \times T_1$\\ For $T_2 \times T_1$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} 1 & 0 & c \\ 0&1&d\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & a \\ 0& 1& b\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} 1& 0 & a+c \\ 0& 1 & b+d\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & c \\ 0&1&d\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & a \\ 0& 1& b\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} 1& 0 & a+c \\ 0& 1 & b+d\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
$\Rightarrow T_1 \times T_2 = T_2 \times T_1 \Rightarrow$ does commute\\ $\Rightarrow T_1 \times T_2 = T_2 \times T_1 \Rightarrow$ does commute\\
\\
\hspace{-5.5em} \hspace{-1.5em}
For $S_1 \times S_2$\\ For $S_1 \times S_2$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} a & 0 & 0 \\ 0&b&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} c & 0 & 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times c & 0 & 0 \\ b \times d& 0 & 0\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} a & 0 & 0 \\ 0&b&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} c & 0 & 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times c & 0 & 0 \\ b \times d& 0 & 0\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
\hspace{-4.5em}
For $S_2 \times S_1$\\ For $S_2 \times S_1$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} c & 0& 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & 0 & 0 \\ 0&b&0\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} a \times c & 0 & 0 \\ b \times d& 0 & 0\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} c & 0& 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & 0 & 0 \\ 0&b&0\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} a \times c & 0 & 0 \\ b \times d& 0 & 0\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
$\Rightarrow S_1 \times S_2 = S_2 \times S_1 \Rightarrow$ does commute\\ $\Rightarrow S_1 \times S_2 = S_2 \times S_1 \Rightarrow$ does commute\\
\\
\hspace{-5.5em} \hspace{-1.5em}
For $R \times T$\\ For $R \times T$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & e \\ 0& 1& f\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a& b & a \times e + b \times f \\ c& d & c\times e + d \times f\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & e \\ 0& 1& f\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a& b & a \times e + b \times f \\ c& d & c\times e + d \times f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
\hspace{-4.5em}
For $T \times R$\\ For $T \times R$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} 1 & 0 & e \\ 0& 1& f\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} a& b & e \\ c& d & f\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & e \\ 0& 1& f\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} a& b & e \\ c& d & f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
$\Rightarrow R \times T \neq T \times R \Rightarrow$ does not commute\\ $\Rightarrow R \times T \neq T \times R \Rightarrow$ does not commute\\
\\
\hspace{-1.25em} \hspace{-1.5em}
For $R \times S$\\ For $R \times S$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} e & 0 & 0 \\ 0& f& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times e& b \times f & 0 \\ c \times f & d \times f & 0\times e + d \times f\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} e & 0 & 0 \\ 0& f& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times e& b \times f & 0 \\ c \times f & d \times f & 0\times e + d \times f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
\hspace{-0.25em}
For $S \times R$\\ For $S \times R$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} e & 0 & 0 \\ 0& f& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} a \times e& b \times f & 0 \\ c \times f & d \times f & 0\times e + d \times f\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} e & 0 & 0 \\ 0& f& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b& 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} a \times e& b \times f & 0 \\ c \times f & d \times f & 0\times e + d \times f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
$\Rightarrow R \times S = S \times R \Rightarrow$ does commute\\ $\Rightarrow R \times S = S \times R \Rightarrow$ does commute\\
\\
\hspace{-1.25em} \hspace{-1.5em}
For $S \times T$\\ For $S \times T$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} 1 & 0 & a \\ 0&0&b\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} c & 0 & 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} c & 0 & a \\ 0& d & b\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & a \\ 0&0&b\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} c & 0 & 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} c & 0 & a \\ 0& d & b\\ 0&0&1 \\ \end{pmatrix}
\end{gather*} \end{gather*}
\hspace{-0.25em}
For $T \times S$\\ For $T \times S$\\
\begin{gather*} \begin{gather*}
\begin{pmatrix} c & 0 & 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & a \\ 0&0&b\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} c & 0 & a \times c \\ d& 0 & b \times d\\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} c & 0 & 0 \\ 0& d& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & a \\ 0&0&b\\ 0&0&1 \\ \end{pmatrix} =\begin{pmatrix} c & 0 & a \times c \\ d& 0 & b \times d\\ 0&0&1 \\ \end{pmatrix}
@ -116,10 +111,9 @@ $\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\
\subsection*{b)} \subsection*{b)}
\begin{gather*} \begin{gather*}
\begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ y_1&y_2&\cdots&y_n\\ z_1&z_2&\cdots&z_n \\ \end{pmatrix} \times \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \\ \end{pmatrix} = \begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ y_1&y_2&\cdots&y_n\\ z_1&z_2&\cdots&z_n \\ \end{pmatrix} \times n \downarrow \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \\ \end{pmatrix} =
\begin{pmatrix} \sum\limits_{i=1} x_i \\ \sum\limits_{i=1} y_i \\ \sum\limits_{i=1} z_i \end{pmatrix} \begin{pmatrix} \sum\limits_{i=1} x_i \\ \sum\limits_{i=1} y_i \\ \sum\limits_{i=1} z_i \end{pmatrix}
\end{gather*} \end{gather*}
\newpage
\subsection*{c)} \subsection*{c)}
\begin{gather*} \begin{gather*}
@ -133,6 +127,7 @@ y_\phi = \frac {\begin{pmatrix} 0\\1\\0\\ \end{pmatrix} \times p_x} {1 \times |p
x_\phi = \frac {\begin{pmatrix} 1\\0\\0\\ \end{pmatrix} \times p_x} {1 \times |p_x|} \Rightarrow \begin{pmatrix} 0 & 0 & 0 \\ 0 &cos(x_\phi)&-sin(x_\phi)\\ 0&sin(x_\phi)&cos(x_\phi) \\ \end{pmatrix}=R_x\\ x_\phi = \frac {\begin{pmatrix} 1\\0\\0\\ \end{pmatrix} \times p_x} {1 \times |p_x|} \Rightarrow \begin{pmatrix} 0 & 0 & 0 \\ 0 &cos(x_\phi)&-sin(x_\phi)\\ 0&sin(x_\phi)&cos(x_\phi) \\ \end{pmatrix}=R_x\\
M = R_x \times R_y \times R_z \\ M = R_x \times R_y \times R_z \\
\end{gather*} \end{gather*}
\newpage
Berechnung für: \\ Berechnung für: \\
$p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$ $p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$
\begin{gather*} \begin{gather*}