exc 1 formating theorie

exercise1/Theorie.tex

@@ -1,4 +1,4 @@
-\documentclass[12pt, a4paper]{scrbook}
+\documentclass[12pt, a4paper]{article}
 
 %packages
 \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\\
 
 \newpage
-
-\hspace{-5.5em}
+\hspace{-2em}
 For $T_1 \times T_2$\\
 \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}
 \end{gather*}
-\hspace{-4.5em}
+\hspace{-0.5em}
 For $T_2 \times T_1$\\
 \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}
 \end{gather*}
 
 $\Rightarrow T_1 \times T_2 = T_2 \times T_1 \Rightarrow$ does commute\\
-
-\hspace{-5.5em}
+\\
+\hspace{-1.5em}
 For $S_1 \times S_2$\\
 \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}
 \end{gather*}
-\hspace{-4.5em}
 For $S_2 \times S_1$\\
 \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}
 \end{gather*}
 
 $\Rightarrow S_1 \times S_2 = S_2 \times S_1 \Rightarrow$ does commute\\
-
-\hspace{-5.5em}
+\\
+\hspace{-1.5em}
 For $R \times T$\\
 \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}
 \end{gather*}
-\hspace{-4.5em}
 For $T \times R$\\
 \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}
 \end{gather*}
 
 $\Rightarrow R \times T \neq T \times R \Rightarrow$ does not commute\\
-
-\hspace{-1.25em}
+\\
+\hspace{-1.5em}
 For $R \times S$\\
 \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}
 \end{gather*}
-\hspace{-0.25em}
 For $S \times R$\\
 \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}
 \end{gather*}
 
 $\Rightarrow R \times S = S \times R \Rightarrow$ does commute\\
-
-\hspace{-1.25em}
+\\
+\hspace{-1.5em}
 For $S \times T$\\
 \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}
 \end{gather*}
-\hspace{-0.25em}
 For $T \times S$\\
 \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}
@@ -116,10 +111,9 @@ $\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\
 
 \subsection*{b)}
 \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}
 \end{gather*}
-\newpage
 
 \subsection*{c)}
 \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\\
 M = R_x \times R_y \times R_z \\
 \end{gather*}
+\newpage
 Berechnung für: \\
 $p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$
 \begin{gather*}