1. Betrachte \( \left(\begin{array}{l} \pi \\ \pi \end{array}\right) \mapsto\left(\begin{array}{c} \cos \left(\pi\right) \\ \sin \left(\pi\right) \end{array}\right) = \left(\begin{array}{l} -1 \\ 0 \end{array}\right) \)
Aber \( 2 \cdot \left(\begin{array}{l} \pi \\ \pi \end{array}\right) = \left(\begin{array}{l} 2\pi \\ 2\pi \end{array}\right)\)
\( \mapsto\left(\begin{array}{c} \cos \left(2\pi\right) \\ \sin \left(2\pi\right) \end{array}\right) = \left(\begin{array}{l} 1 \\ 0 \end{array}\right) \ne 2 \cdot \left(\begin{array}{l} -1 \\ 0 \end{array}\right) \)
3. Die spitzen Klammern bezeichnen wohl das Skalarprodukt:
\( \left\langle\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right),\left(\begin{array}{c}-1 \\ 2\end{array}\right)\right\rangle = -x_1 + 2x_2 \) Das Ergebnis also in ℝ.