Aufgabe:
(1) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad f\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}2 x \\ 3 y\end{array}\right) \)
(2) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad f\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}y \\ x\end{array}\right) \)
(3) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad f\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}\cos (\alpha) x-\sin (\alpha) y \\ \sin (\alpha) x+\cos (\alpha) y\end{array}\right) \)
(4) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad f\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}\sqrt{\frac{1}{2}} x-\sqrt{\frac{1}{2}} y \\ \sqrt{\frac{1}{2}} x+\sqrt{\frac{1}{2}} y\end{array}\right) \)
(5) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad f\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+2 \\ y+3\end{array}\right) \)
(6) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad f\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}\frac{1}{2}(x+y) \\ \frac{1}{2}(x+y)\end{array}\right) \)
(7) \( f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}, \quad f\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{c}\cos (\alpha) x-\sin (\alpha) y \\ \sin (\alpha) x+\cos (\alpha) y \\ z\end{array}\right) \)
e) Welche der Abbildungen sind injektiv/surjektiv/bijektiv? Im Falle der Bijktivität geben
Sie auch die Umkehrabbildung an!
Problem:
