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la9 changes, add ana18

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christian
2019-12-20 12:20:00 +01:00
parent 1b386c8ba7
commit effbab2688
4 changed files with 198 additions and 5 deletions
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ws2019/la/uebungen/la9.pdf
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ws2019/la/uebungen/la9.tex
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@@ -40,7 +40,10 @@
\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{(a) und } * }{\implies} &w = 0
\stackrel{\text{(a) und } * }{\implies} & \forall w \in
K^{m} \setminus \{0\} \colon w \neq (b(a(w)) \\
\implies &w = 0 \\
\implies & \text{ker}(id_{K^{m}} - b \circ a) = \{0\}
.\end{align*}
Damit ist $id_{K^{m}} - b \circ a$ ein injektiver Endomorphismus, also
auch bijektiv, also Automorphismus.\\
@@ -188,7 +191,7 @@
\to
\begin{gmatrix}[p] 1 & -1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0
\end{gmatrix}
\intertext{$\implies$ Rang 0 \vspace{2mm} \newline
\intertext{$\implies$ Rang 1 \vspace{2mm} \newline
Für $a \neq 1 \land a \neq -1 \implies 1 - a^2 \neq 0$, damit:}
&\begin{gmatrix}[p]
a & 1 & a \\ 1 & a & 1 \\ a & 1 & a
@@ -254,7 +257,7 @@
Beh.:
\[
T = M_{\underline{e}}^{\underline{v}}(\text{id}_V) =
\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix}
.\]
\begin{proof}
Zu Überprüfen für die zwei Basisvektoren aus $\underline{v}$.
@@ -262,14 +265,14 @@
\item $v_1 = (1,2)^{t}$. $\phi(v_1) = (1,0)^{t}$.
\begin{align*}
\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix}
\cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}
= \begin{pmatrix} 1 \\ 2 \end{pmatrix}
.\end{align*}
\item $v_2 = (0,-1)^{t}$. $\phi(v_2) = (0,1)^{t}$.
\begin{align*}
\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix}
\cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}
= \begin{pmatrix} 0 \\ -1 \end{pmatrix}
.\end{align*}