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zTm>zq(3%71^WlQ-*LEG+-$2KE=v2Ln@1QFhx|96;CWwh*lDW$vzm{z7UYSoR=Dzm$ zG}&DLW1mvZ(t3TGVpj0~r%5xb{_N9Kk!9BY#hG30a{mDQ*l9zTV3`3P3o|$|ISM5u HMNdWw6W8{o diff --git a/ws2019/la/uebungen/la5.tex b/ws2019/la/uebungen/la5.tex index 2c6f887..e7ccc00 100644 --- a/ws2019/la/uebungen/la5.tex +++ b/ws2019/la/uebungen/la5.tex @@ -2,6 +2,7 @@ \usepackage{enumerate} \usepackage{array} +\usepackage{mathtools} \title{Übungsblatt Nr. 5} \author{Christian Merten, Mert Biyikli} @@ -202,10 +203,10 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Bleibt zu zeigen: char$K \not\in \{2, \ldots, n+1\} \iff k+1 \neq 0$ $\forall k \in \{0, \ldots, n\} $. \begin{align*} - &k + 1 \neq 0 \\ - \stackrel{k \ge 0}{\iff} & k + 1 \neq \text{char}K \\ - \stackrel{1 \le k + 1 \le n + 1}\iff & \text{char}K = 0 \lor \text{char}K > n + 1 \\ - \iff & \text{char}K \not\in \{2, \ldots, n+1\} + &k + 1 \neq 0 \\[-2mm] + \stackrel{\mathclap{\strut k \ge 0}}{\qquad \iff \qquad} &k + 1 \neq \text{char}K \\[-2mm] + \stackrel{\mathclap{\strut 1 \le k + 1 \le n + 1}}{\qquad \iff \qquad} &\text{char}K = 0 \lor \text{char}K > n + 1 \\[1mm] + \qquad \iff \qquad & \text{char}K \not\in \{2, \ldots, n+1\} .\end{align*} \end{proof} \item Bestimmen Sie $\psi(\text{ker }K) \subset K^{n+2}$. @@ -239,7 +240,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Damit folgt: \[ \psi(\text{ker }\partial) = - \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\} + \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid a \in K\} .\] \item $\text{char }K \in \{2, \ldots, n+1\} $. Dann gilt für $k = \text{char }K-1$: \[ @@ -321,12 +322,12 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \begin{align*} (*(f_1 + f_2))(\varphi) &= ((f_1 + f_2)^{*})(\varphi) = \varphi \circ (f_1 + f_2) - = \varphi \circ f_1 + \varphi \circ f_2 + \stackrel{\varphi \text{ linear}}{=} \varphi \circ f_1 + \varphi \circ f_2 = (*(f_1))(\varphi) + (*(f_2))(\varphi) \\ (*(a f_1))(\varphi) &= ((a f_1)^{*})(\varphi) = \varphi \circ (a f_1) - = a (\varphi \circ f_1) + \stackrel{\varphi \text{ linear}}{=} a \cdot (\varphi \circ f_1) = a\cdot (*(f_1))(\varphi) .\end{align*} \end{proof}