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-19,7 +19,7 @@ \subsection*{1.1.1} \subsection*{a)} \begin{itemize} -\item Matrix Multiplication $\equiv$ transformation +\item Matrix Multiplication $\widehat{=}$ transformation \item Inverse of a transformation $\tilde{T}^{-1}$ \\ \begin{align*} \Rightarrow && \tilde{T}*\tilde{T}^{-1} &= \tilde{T}^{-1}*\tilde{T} = \tilde{E} && \left |\ *\tilde{p} \right. \\ @@ -29,21 +29,101 @@ \end{itemize} \subsection*{b)} +For $R_1 \times R_2$\\ +\begin{gather*} +\begin{pmatrix} a & b & 0 \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} e & f & 0 \\ g& h& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times e + b \times g& a \times f + b \times h & 0 \\ c \times e + d \times g& c \times f + d \times h & 0\\ 0&0&1 \\ \end{pmatrix} +\end{gather*} +For $R_2 \times R_1$\\ +\begin{gather*} +\begin{pmatrix} e & f & 0 \\ g& h& 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 + c \times f& b \times e + d \times f & 0 \\ a \times g + c \times h& b \times g + d \times h & 0\\ 0&0&1 \\ \end{pmatrix} +\end{gather*} + +$\Rightarrow R_1 \times R_2 \neq R_2 \times R_1 \Rightarrow$ does not commute\\ + +\newpage + +\hspace{-5.5em} +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} +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} +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} +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} +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} +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} +\end{gather*} + +$\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\ + + \subsection*{1.1.2} \subsection*{a)} \begin{gather*} -\begin{pmatrix} 1 & 0 & 0 \\ 0&1&0\\ 0&0&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} =\begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ z_1&z_2&\cdots&z_n\\ y_1&y_2&\cdots&y_n \\ \end{pmatrix} +\begin{pmatrix} 1 & 0 & 0 \\ 0&1&0\\ 0&0&1 \\ \end{pmatrix} \times \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} =\begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ z_1&z_2&\cdots&z_n\\ y_1&y_2&\cdots&y_n \\ \end{pmatrix} \end{gather*} + \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} * \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 \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*} \subsection*{c)} \begin{align*} L(p) &= length(p_x - ((p_x \cdot (\vec{e} + \vec{d} * t)) *\vec{d} * t) - \vec{e}) \\ -Vector &= ( L(p_x), L(p_{x+1}), ... , L(p_{x+n})) \hspace{6em} n \equiv Anz. Punkte \\ +Vector &= ( L(p_x), L(p_{x+1}), ... , L(p_{x+n})) \hspace{6em} n \widehat{=} Anz. Punkte \\ \end{align*} Berechnung: