Aufgabe:
Welche der folgenden Abbildungen sind linear?
(i) \( \mathbb{R}^{2} \rightarrow \mathbb{R}, \quad\left(\begin{array}{l}x \\ y\end{array}\right) \mapsto x+2 y \)
(v) \( \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad\left(\begin{array}{l}x \\ y\end{array}\right) \mapsto\left(\begin{array}{l}x+1 \\ y-1\end{array}\right) \)
(ii) \( \mathbb{R}^{2} \rightarrow \mathbb{R}, \quad\left(\begin{array}{l}x \\ y\end{array}\right) \mapsto x+y^{2} \)
(vi) \( \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \quad\left(\begin{array}{l}x \\ y\end{array}\right) \mapsto\left(\begin{array}{c}x-y \\ x+2 y\end{array}\right) \)
(iii) \( \mathbb{R}^{2} \rightarrow \mathbb{R}, \quad\left(\begin{array}{l}x \\ y\end{array}\right) \mapsto x y \)
(vii) \( \mathbb{R}_{n}[x] \rightarrow \mathbb{R}, \quad p(x) \mapsto p(1) \)
(iv) \( \mathbb{C} \rightarrow \mathbb{C}, \quad z \mapsto \bar{z} \)
(viii) \( \mathbb{R}_{n}[x] \rightarrow \mathbb{R}_{n+2}[x], \quad p(x) \mapsto x^{2} p(x) \)
