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\begin{enumerate}[(a)] \item Beh.: $\varphi$ ist linear. + Zunächst: $\varphi$ ist wohldefiniert, da jedem $v_1$ eindeutig die Äquivalenzklasse von + $v_1 = v_1 + 0 \in (V_1 + V_2)$ zugeordnet wird. + Seien $v_1, v_2 \in V_1$ und $a \in K$ beliebig. \begin{proof} Homomorphismus \[ @@ -98,7 +101,7 @@ \varphi(a v_1) = (a v_1) + V_2 = [a v_1] = a \cdot [v_1] = a \cdot \varphi(v_1) .\] \end{proof} - \item Beh.: $\partial$ ist surjektiv. + \item Beh.: $\varphi$ ist surjektiv. \begin{proof} Sei $v \in (V_1 + V_2)$ beliebig. Dann ist $[v] = v + V_2$ und es ex. $v_1 \in V_1$ und $v_2 \in V_2$ mit $v = v_1 + v_2$. @@ -108,7 +111,7 @@ v - v_1 = v_1 + v_2 - v_1 = v_2 \in V_2 \implies v_1 \sim v_2 .\] \end{proof} - \item Beh.: ker $\partial = V_1 \cap V_2$ + \item Beh.: ker $\varphi= V_1 \cap V_2$ \begin{proof} Das neutrale Element von $(V_1 + V_2) / V_2))$ ist $V_2$. Sei $v \in V$ beliebig. \begin{align*}