(a) Entscheiden Sie, welche der folgenden Abbildungen \( \beta \) bilinear sind.
(i) \( \beta: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R}, \beta\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right):=x_{1} y_{2}+x_{2} \)
(ii) \( \beta: \mathbb{C}^{2} \times \mathbb{C}^{2} \rightarrow \mathbb{C}, \beta\left(\left(z_{1}, z_{2}\right),\left(w_{1}, w_{2}\right)\right):=z_{1} w_{1}+z_{2} w_{2} \)
(iii) Sei \( V \) ein \( K \) -Vektorraum und \( \beta: \operatorname{Hom}_{K}(V, K) \times V \rightarrow K, \beta(\varphi, v):=\varphi(v) \).
(b) Welche der folgenden Abbildungen (,\( ): V \times V \rightarrow \mathbb{C} \) definieren ein Skalarprodukt auf dem C-Vektorraum \( V \) ?
(i) \( V=\mathbb{C}^{2},\left(\left(z_{1}, z_{2}\right),\left(w_{1}, w_{2}\right)\right):=z_{1} w_{1}+z_{2} w_{2} \)
(ii) \( V=\mathbb{C}^{2},\left(\left(z_{1}, z_{2}\right),\left(w_{1}, w_{2}\right)\right):=\left(2 z_{1}-i z_{2}\right) \bar{w}_{1}+\left(i z_{1}+5 z_{2}\right) \bar{w}_{2} \)
(iii) \( V=\mathbb{C}^{3},\left(\left(z_{1}, z_{2}, z_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right):=\left(z_{1}, z_{2}, z_{3}\right)\left(\begin{array}{rrr}0 & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{l}\bar{w}_{1} \\ \bar{w}_{2} \\ \bar{w}_{3}\end{array}\right) \).
