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zt^GK=POVm9cuKv&9VLLaJQ8~1ps{m_(gy`S&%wOCb9HQUYeK7JiC+@lwpyc`sf(?rOzXG$5V!$ zpzsgZQ`{Oerc8LN`$BSO)mgS2V*e!MZ9B8Ttt|4k-b7wdXHG!oial}O-B~bt$gK@Z zacs~DXt^vBx(JB<{NBi|U&Z7@AF#;tq6wm>_uhbn;NeFHkNwP(3COFrjEdP09 zy`+lGgnp-nICbb*^%J2lKK{oCPBRdnWe@>tZg1w|>TGUo_kRXrSs0s{a4t diff --git a/ws2019/la/uebungen/la9.tex b/ws2019/la/uebungen/la9.tex index 7572256..dc8e414 100644 --- a/ws2019/la/uebungen/la9.tex +++ b/ws2019/la/uebungen/la9.tex @@ -12,7 +12,7 @@ \begin{aufgabe} \begin{enumerate}[(a)] \item Es seien $f\colon V \to W$ und $g: W \to V$ lineare Abbildungen - zwischen $V$ und $W$ Vektorräumen. + zwischen Vektorräumen $V$ und $W$. Beh.: Es existiert genau dann ein $v \in V \setminus \{0\} $ mit $(g \circ f)(v) = v$, wenn es ein $w \in W \setminus \{0\} $ gibt mit $(f \circ g)(w) = w$. @@ -37,10 +37,10 @@ \begin{align*} &\text{ker}(id_{K^{n}} - a \circ b) = \{0\} \\ \implies & id_{K^{n}}(v) - a(b(v)) \neq 0 \quad \forall v \in V \setminus \{0\} \\ - \implies &v \neq a(b(v)) \quad \forall v \in V \setminus \{0\} + \implies &v \neq a(b(v)) \quad \forall v \in V \setminus \{0\} \quad (*) \intertext{Sei nun $w \in K^{m}$ mit $id_{K^{m}} - b(a(w)) = 0$.} \implies & w = b(a(w)) \\ - \stackrel{\text{1a)}}{\implies} &w = 0 + \stackrel{\text{(a) und } * }{\implies} &w = 0 .\end{align*} Damit ist $id_{K^{m}} - b \circ a$ ein injektiver Endomorphismus, also auch bijektiv, also Automorphismus.\\ @@ -55,18 +55,20 @@ \begin{enumerate}[(a)] \item Beh.: $\text{ker } A = \{x - BAx \mid x \in K^{m}\} $ \begin{proof} - Zz.: $\text{ker } A \subset \{x - BAx \mid x \in K^{m}\} $ + Zz.: $\text{ker } A \subset \{x' - BAx' \mid x' \in K^{m}\} $ Sei $x \in \text{ker } A$, d.h. $Ax = 0$, damit: - \[ - x - BAx = x - B\cdot 0 = x - .\] + \begin{align*} + &x - BAx = x - B\cdot 0 = x \\ + \implies & x \in \{x' - BAx' \mid x' \in K^{m}\} + .\end{align*} Zz.: $\{x - BAx \mid x \in K^{m}\} \subset \text{ker } A$ Sei $r \in K^{m}$, dann $x := r - BAr$. Damit folgt: \begin{align*} - Ax = Ar - ABAr \stackrel{ABA=A}{=} Ar - Ar = 0 + &Ax = Ar - ABAr \stackrel{ABA=A}{=} Ar - Ar = 0\\ + \implies & x \in \text{ker } A .\end{align*} \end{proof} \item Beh.: $Ax = b$ hat eine Lösung $\iff ABb = b$ @@ -232,7 +234,7 @@ &a \cdot \binom{1}{2} + b \binom{0}{-1} = 0 \\ \implies & a = 0 \land 2a -b = 0 \implies b = 0 .\end{align*} - $\implies$ $\underline{v}$ ist linear unabhängig wegen $\text{dim } \Q^{2} = 2$ eine Basis + $\implies$ $\underline{v}$ ist linear unabhängig und wegen $\text{dim } \Q^{2} = 2$ eine Basis von $\Q^{2}$. \end{proof} Beh.: $\underline{w} = \left( (1,1)^{t}, (3,2)^{t} \right) $ ist Basis von $\Q^{2}$ @@ -245,7 +247,7 @@ \implies & a + 3b = 0 \land a +2b = 0 \implies b = a = 0 .\end{align*} - $\implies$ $\underline{v}$ ist linear unabhängig wegen $\text{dim } \Q^{2} = 2$ eine Basis + $\implies$ $\underline{v}$ ist linear unabhängig und wegen $\text{dim } \Q^{2} = 2$ eine Basis von $\Q^{2}$. \end{proof}