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a/ws2019/la/uebungen/la6.tex b/ws2019/la/uebungen/la6.tex index 430ebfe..7695fa0 100644 --- a/ws2019/la/uebungen/la6.tex +++ b/ws2019/la/uebungen/la6.tex @@ -16,13 +16,13 @@ \begin{aufgabe} \begin{enumerate}[(a)] - \item Beh.: Der UVR $V = \{(x_1, x_2) \in Q^{2} \mid x_1-x_2 = 0\} $ hat die Basis $\{(1,1)\} $. + \item Beh.: Der UVR $V = \{(x_1, x_2) \in \Q^{2} \mid x_1-x_2 = 0\} $ hat die Basis $\{(1,1)\} $. \begin{proof} $\{(1,1)\} $ ist offensichtlich linear unabhängig. Außerdem: Sei $v \in V$ beliebig, dann ex. ein $p \in \Q$, s.d. $v = (p, p)$. Damit: $v = p \cdot (1,1)$. \end{proof} - \item Beh.: Der UVR $V = \{(x_1,x_2,\ldots,x_n) \mid 2x_1 + x_2 = 0\} $ hat die Basis\\ + \item Beh.: Der UVR $V = \{(x_1,x_2,\ldots,x_n) \in \Q^{n} \mid 2x_1 + x_2 = 0\} $ hat die Basis\\ $(v_i)_{i\in I} = \{(1, -2, 0, \ldots, 0), (0, 0, 1, \ldots, 0), \ldots, (0, \ldots, 1)\} $ mit $I = \{1, 3, 4, \ldots, n\} $. @@ -42,39 +42,43 @@ $\implies (v_i)_{i\in I}$ minimal und damit Basis. \end{proof} - \item Beh.: Der UVR ker $\partial$ hat falls char $K \not\in \{2, \ldots, n+1\} $, die Basis - $\psi^{-1}(\{(1, 0, \ldots, 0)\}) $. Falls char $K \in \{2, \ldots, n+1\} $, hat - ker $\partial$ die - Basis $(b_i)_{i \in I} = \psi^{-1}(\{(1, 0, \ldots, 0), - (0, 0, \ldots, 0, \underbrace{1}_{\text{char $K$-te Stelle}}, 0, \ldots, 0)\}) $ - mit $I = \{1, 2\} $. + \item Beh.: Der UVR ker $\partial$ hat die Basis $(b_i)_{i \in I}$ mit $I = \{0, \ldots, \hat{i}\} $. + mit $\hat{i} = 0$ für $\text{char }K = 0$ und + $\hat{i} = \left\lfloor \frac{n+1}{\text{char }K}\right\rfloor $ für $\text{char } K > 0$ + mit + \[ + b_i(k) = \begin{cases} + 1 & \text{char }K \cdot i = k \\ + 0 & \text{sonst} + \end{cases} + .\] \begin{proof} - Falls char $K \not\in \{2, \ldots, n+1\} $: - \begin{align*} - \text{ker } \partial &= \{f \in V \mid f(k) = 0 \qquad \forall k \in \{1, \ldots, n +1\} \} \\ - &= \psi^{-1}\left( \{(a, 0, \ldots, 0) \mid a \in K\} \right) - .\end{align*} - $\psi^{-1}(\{1, 0, \ldots, 0\}) $ ist offensichtlich - linear unabhängig. - Sei $v \in \text{ker } \partial$ beliebig, dann ex. $a \in K$ s.d. - $v = \psi^{-1}((a, \ldots, 0)) = a \cdot \psi^{-1}((1, \ldots, 0))$. + Zunächst: $b_i$ ist wohldefiniert, da $\text{char }K \cdot i$ eindeutig $\forall i \in I$. + Es gilt außerdem $\forall i \in I\colon i \cdot \text{char }K \le n+1$, da + für $\text{char }K = 0 \implies \hat{i} = i = 0 \le n+1$ und für + $\text{char }K > 0 \implies i \le i_0 \le \frac{n+1}{\text{char }K} + \implies i \cdot \text{char }K \le n+1$. - Falls char $K \in \{2, \ldots, n+1\}$: - \begin{align*} - \text{ker } \partial &= \{f \in V \mid f(k) = 0 \qquad \forall k \in \{1, \ldots, n +1\} \setminus \{\text{char }K\} \} \\ - &= \psi^{-1}( \{(a, 0, \ldots, 0, \underbrace{b}_{\text{char $K$-te Stelle}}, 0, \ldots, 0) \mid a \in K, b \in K\} ) - .\end{align*} - - $\{(1, 0, \ldots, 0), (0, \ldots, 0, 1,0, \ldots, 0)\} $ - ist Teilmenge der kanonischen Basis des $K^{n+2}$ und damit linear unabhängig und - insbesondere $(b_i)_{i\in I}$ linear unabhängig. + Zz.: $(b_i)_{i \in I}$ ist Erzeugendensystem. Sei $f \in \text{ker }\partial$ beliebig, dann + wähle $\alpha_i = f(i \cdot \text{char }K) \quad \forall i \in I$. Da $I$ endlich, ist + $(\alpha_i)_{i\in I} \in K^{(I)}$. Sei nun $k \in \{0, \ldots, n+1\} $ beliebig: + + Falls $\text{char }K \nmid k$: $\forall i \in I\colon b_i(k) = 0 = f(k)$. + + Falls $\text{char }K \mid k$: $\exists! j \in I\colon \text{char }K \cdot j = k$. + \[ + \sum_{i \in I} \alpha_i b_i = \alpha_j = f(j \cdot \text{char }K) = f(k) + .\] - Sei $v \in \text{ker } \partial$ beliebig, dann ex. $a, b \in K$, s.d. + Zz.: $(b_i)_{i \in I}$ ist minimal. + + Sei $i_0 \in I$ beliebig. Wähle + $f \in \text{ker }\partial$ mit $f(i_0 \cdot \text{char }K) = 1$, dann gilt: \begin{align*} - v &= \psi^{-1}\left( (a, 0, \ldots, 0, b, 0, \ldots, 0) \right) \\ - &= a \cdot \psi^{-1}\left( (1, 0, \ldots, 0) \right) + b \cdot \psi^{-1}\left( (0, \ldots, 0, 1, 0, \ldots, 0 \right) \\ - &= a \cdot b_1 + b \cdot b_2 + &\forall i \in I \setminus \{i_0\}\colon b_i(i_0 \cdot \text{char }K) = 0 \\ + \implies & f \not\in \text{Lin}\left( (b_i)_{i\in I \setminus \{i_0\} } \right) \\ + \implies & (b_i)_{i\in I} \text{ minimal} .\end{align*} \end{proof} \end{enumerate} @@ -237,17 +241,42 @@ $I$ ist nicht endlich. Zu zeigen: $(v^{*}_i)_{i\in I}$ ist nicht Erzeugendensystem von $V^{*}$. - Sei $f^{*} \in V^{*}$ und $v \in V$. Wegen $(v_i)_{i\in I}$ Basis, ex. - $(\beta_i)_{i \in I} \in K^{I}$, s.d. $v = \sum_{i \in I} \beta_i v_i$. - Angenommen es ex. $(\alpha_i)_{ i \in I} \in K^{I}$, s.d. + Sei $f^{*} \in V$ mit + \[ + v = \sum_{i \in I} \beta_i v_i \mapsto \sum_{i \in I} \beta_i + .\] + Wegen der Eindeutigkeit der Darstellung durch die Basis $(v_i)_{i \in I}$ ist diese + Abbildung wohldefiniert. + + Zu zeigen.: $f^{*}$ ist linear. Seien $v, w \in V$ und $k \in K$ beliebig. Wegen + $(v_i)_{i \in I}$ Basis, \\ex. $(a_i)_{i \in I}, (b_i)_{i\in I} \in K^{(I)}$, s.d. + $v = \sum_{i \in I} a_i v_i$ und $w = \sum_{i \in I} b_i v_i$. \begin{align*} - f^{*}(v) &= \sum_{i \in I} \alpha_i v^{*}_i(v) \\ - &= \sum_{i \in I} \alpha_i v^{*}_i \left( \sum_{i \in I} \beta_i v_i \right) \\ - &= \sum_{i \in I} \alpha_i \sum_{i \in I} \beta_j v^{*}_i(v_j) \\ - &= \sum_{i \in I} \alpha_i \beta_i + f^{*}(k v + w) &= f^{*}\left( \sum_{i \in I} (k (a_i v_i) + (b_i v_i)) \right) \\ + &= f^{*}\left( \sum_{i \in I} (k a_i + b_i) v_i \right) \\ + &= \sum_{i \in I} (k a_i + b_i) + = k \sum_{i \in I} a_i + \sum_{i \in I} b_i + = k \cdot f^{*}\left(\sum_{i \in I} a_i v_i\right) + f^{*}\left( \sum_{i \in I} b_i v_i\right) \\ + &= k \cdot f^{*}(v) + f^{*}(w) .\end{align*} - Wegen $\alpha_i$ und $\beta_i$ i.A. nicht endlich, ist die Summe nicht bestimmt. - Widerspruch $\implies (v^{*}_i)_{i\in I}$ ist kein Erzeugendensystem + $\implies f^{*}$ linear und damit $f^{*}\in V^{*}$.\\ + + Nun gilt insbesondere für $v_i \in (v_i)_{i \in I}$ + \[ + f^{*}(v_i) = f^{*}(1 \cdot v_i) = 1 + .\] + + Angenommen $(v^{*}_i)_{i \in I}$ ist Erzeugendensystem. + Dann ex. $(\alpha_i)_{i\in I} \in K^{(I)}$, s.d. + \[ + f^{*} = \sum_{i \in I} \alpha_i v^{*}_i + .\] + + Sei $i \in I$ nun beliebig. Dann gilt + \[ + f^{*}(v_i) = \sum_{j \in I} \alpha_j v^{*}_j(v_i) = \alpha_i = 1 + .\] Daraus folgt, dass $\forall i \in I\colon \alpha_i = 1$. Widerspruch zur Annahme, dass + $(\alpha_i)_{i \in I}$ endlich ist. \end{proof} \end{enumerate} \end{aufgabe}