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-\frac{1}{2} - \frac{5}{2}k \\ -1 \\ k \\ \end{pmatrix} \in \R^{3}$. + +\begin{proof} +\[ +\begin{pmatrix} -\frac{1}{2} - \frac{5}{2}k \\ -1 \\ k \end{pmatrix} +\times +\begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix} += +\begin{pmatrix} 2 - 0k \\ 5k - \left(2\left(\frac{1}{2} + \frac{5}{2}k\right)\right) \\ 5 \end{pmatrix} += +\begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} +.\] +\end{proof} + +Es existiert kein $y \in \R^{3}$ mit: +\[ +y \times \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix} += +\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} +.\] + +\begin{proof} + +Die obenstehende Gleichung ergibt folgendes LGS: +\begin{align*} + 2 y_2 - y_3 &= 1 \\ + 2 y_3 - 2 y_1 &= 2 \\ + y_1 - 2 y_2 &= 0 \\ +.\end{align*} + +Aus (I) folgt: +\[ + y_3 = 2 y_2 - 1 +.\] +Damit ergibt sich aus (II): +\begin{align*} + && 2 y_3 - 2y_1 = 2 (2y_2 - 1) - 2y_1 &= 2 \\ + \implies&& y_1 &= 2y_2 - 2 \\ +.\end{align*} +Daraus entsteht ein Widerspruch in (III): +\begin{align*} + y_1 - 2y_2 = 2y_2 - 2 - 2y_2 = -2 \neq 0 +.\end{align*} + +\end{proof} +\end{aufgabe} + +\begin{aufgabe} +Wir definieren die Abbildung $-^{\bot}: \R^{2} \to \R^{2}$ durch +\[ + \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}^{\bot} = + \begin{pmatrix} -x_2 \\ x_1 \end{pmatrix} +.\] +\end{aufgabe} + +\textbf{a)} Zu zeigen: $x$ $\bot$ $x^{\bot}$ und +$ \|x\| = \|x^{\bot}\|$ $\forall x \in R^{2}$ + +\begin{proof} + Sei $x \in R^{2}$ beliebig. + $x$ $\bot$ $x^{\bot}$: + \[ + \left = \left< \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + , \begin{pmatrix} -x_2 \\ x_1 \end{pmatrix} + \right> = -x_1 x_2 + x_2 x_1 = 0 + .\] + $\|x\| = \|x^{\bot}\|$: + \[ + \|x\| = \sqrt{\left} = \sqrt{x_1^2 + x_2^2} = \sqrt{\left( -x_2 \right) ^2 + x_1^2} = \|x^{\bot}\| + .\] +\end{proof} + +\textbf{b)} Ist $x \in \R^{2} \setminus \{0\}$ und $y \in \R^{2}$ mit $x$ $\bot$ +$y$, so existiert ein $a \in \R$ derart, dass $y = a \cdot x^{\bot}$. + +\begin{proof} Sei $x \in \R^{2} \setminus $ mit $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} $ +und $y \in \R^{2}$ mit $y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$ + +Wegen $x$ $\bot$ $y$ folgt: +\begin{align*} + x_1y_1 + x_2y_2 = 0 \implies x_1y_1 = -x_2y_2 +.\end{align*} + +\underline{Fall 1:} $x_1 = 0$. $\implies x_2 \neq 0$. +\[ + x_2 y_2 = 0 \implies y_2 = 0 +.\] Wähle nun $a := -\frac{y_1}{x_2} \in \R$: +\[ +y = a \cdot x^{\bot} \begin{pmatrix} -a x_2 \\ ax_1 \end{pmatrix} += \begin{pmatrix} \frac{y_1}{x_2} x_2 \\ 0 \end{pmatrix} += \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} +.\] + +\underline{Fall 2:} $x_2 = 0$. $\implies x_1 \neq 0$. +\[ + x_1 y_1 = 0 \implies y_1 = 0 +.\] Wähle nun $a := \frac{y_2}{x_1} \in \R$: +\[ +y = a \cdot x^{\bot} \begin{pmatrix} -a x_2 \\ ax_1 \end{pmatrix} += \begin{pmatrix} 0 \\ \frac{y_2}{x_1} x_1 \end{pmatrix} += \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} +.\] + +\underline{Fall 3:} $x_1 \neq 0$ und $x_2 \neq 0$. +\[ +-\frac{y_1}{x_2} = \frac{y_2}{x_1} +.\] Wähle nun $a := \frac{y_2}{x_1} = -\frac{y_1}{x_2} \in \R$: +\[ +y = a \cdot x^{\bot} = \begin{pmatrix} -a x_2 \\ a x_1 \end{pmatrix} += \begin{pmatrix} \frac{y_1}{x_2} x_2 \\ \frac{y_2}{x_1} x_1 \end{pmatrix} += \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} +.\] +\end{proof} + +\textbf{c)} Für $x \in \R^{2} \setminus \{0\} $ sei $f_x: \R^{2} \to \R^{2}$ +Abbildung mit: +\[ + y \mapsto \frac{\left}{\left} \cdot x ++ \frac{\left}{\left} \cdot x^{\bot} +.\] +Zu zeigen: Für beliebiges $x \in \R \setminus \{0\} $ ist $f_x$ gleich der +Identität des $\R^{2}$. + +\begin{proof} + Zz: $f_x(y) = y$ $\forall y \in \R^{2}$ + + Seien $x \in R^{2} \setminus \{0\}$ mit $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ + und $y \in R^{2}$ mit $y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$. + + Nun: + \begin{align*} + f_x(y) &= \frac{\left}{\left} \cdot x + + \frac{\left}{\left} \cdot x^{\bot} \\ + &= \frac{1}{\left} \left( + \begin{pmatrix} x_1^2y_1 + x_1x_2y_2 \\ x_1x_2y_1 + x_2^2y_2 \end{pmatrix} + + + \begin{pmatrix} y_1x_2^2 - x_1x_2y_2 \\ -x_1x_2y_1 + x_1^2y_2 \end{pmatrix} + \right) \\ + &= \frac{1}{\left} \begin{pmatrix} y_1x_1^2 + y_1x_2^2 \\ y_2x_2^2 + y_2x_1^2 \end{pmatrix} \\ + &= \frac{1}{x_1^2 + x_2^2} \begin{pmatrix} y_1(x_1^2 + x_2^2) \\ y_2(x_1^2 + x_2^2) \end{pmatrix} \\ + &= \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \\ + &= y + .\end{align*} +\end{proof} + +\end{document}