From b87c960654ecda9b85a1f80a0008c7bb30291a23 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 19 Nov 2019 21:00:43 +0100 Subject: [PATCH] finish la5, improve lecture class --- lecture.cls | 8 ++ ws2019/la/uebungen/la5.pdf | Bin 104227 -> 104558 bytes ws2019/la/uebungen/la5.tex | 174 ++++++++++++++++++++++++++----------- 3 files changed, 133 insertions(+), 49 deletions(-) diff --git a/lecture.cls b/lecture.cls index f663181..4fde8c7 100644 --- a/lecture.cls +++ b/lecture.cls @@ -14,6 +14,14 @@ \RequirePackage{transparent} \RequirePackage{xcolor} +\DeclareOption{uebung}{ + \makeatletter + \lhead{\@title} + \rhead{\@author} + \makeatother +} +\ProcessOptions\relax + % PAGE GEOMETRY \geometry{ left=15mm, diff --git a/ws2019/la/uebungen/la5.pdf b/ws2019/la/uebungen/la5.pdf index 63d413a6164c0265c54eaa7d5f187ff7c1f59b71..7b8a62ad0b732f8c213b9128650d7aef9a0f1c97 100644 GIT binary patch delta 83533 zcmZttWmFwq6Sj-4g=?^d1$TFc;I6^lEy3M_EZn^b!7UI7Zo%E%-JReBx4_Qx?r)zx 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zxB?R2L6VoLKVWDBq~C$zH(+EDj3&VKB)Aa)H#U&?HIV@0e36Gy&7{1mxuFi}kYfyn`o50U?%z+*fwn62u_ zS>icRc`Iq b^i1{vBF0bKmqnNX9||=yF$yImMNdWw-&ojD diff --git a/ws2019/la/uebungen/la5.tex b/ws2019/la/uebungen/la5.tex index 2a74975..2c6f887 100644 --- a/ws2019/la/uebungen/la5.tex +++ b/ws2019/la/uebungen/la5.tex @@ -1,9 +1,21 @@ -\documentclass{../../../lecture} +\documentclass[uebung]{../../../lecture} \usepackage{enumerate} +\usepackage{array} + +\title{Übungsblatt Nr. 5} +\author{Christian Merten, Mert Biyikli} \begin{document} +\begin{tabular}{|c|m{1cm}|m{1cm}|m{1cm}|m{1cm}|m{1cm}|@{}m{0cm}@{}} + \hline + Aufgabe & \centering A1 & \centering A2 & \centering A3 & \centering A4 & \centering $\sum$ & \\[5mm] \hline + Punkte & & & & & & \\[5mm] \hline +\end{tabular} + +\vspace{5mm} + \begin{aufgabe} Es sei $K$ Körper, $M$ eine Menge und $m_0 \in M$ ein fest gewähltes @@ -11,11 +23,23 @@ Element. In \\$V = \text{Abb}(M, K)$ betrachten wir die Teilmengen $U = \{f \in V \mid f(m_0) = 0\} $ und \\$W = \{f \in V \mid \forall x, y \in M \colon f(x) = f(y)\} $ -Zunächst: $K$ ist K-Vektorraum mit $(K, +, 0)$. Damit wird -$V = \text{Abb}(M, K)$ zum Vektorraum. - \begin{enumerate}[a)] - \item Beh.: $U \subset V$ ist Untervektorraum. + \item Zunächst: $K$ ist K-Vektorraum. Damit wird + $V = \text{Abb}(M, K)$ mit $0_V(m) = 0 \text{ } \forall m \in M$ zum K-Vektorraum. + + Damit eine Teilmenge $M \subset V$ zum Untervektorraum von $V$ wird, muss gelten: + \[ + m_1 + m_2 \in M \text{ } \forall m_1,m_2 \in M + .\] und + \[ + a m_1 \in M \text{ } \forall m_1 \in M, a \in K + .\] Die Inversen der zugehörigen Untergruppe sind gegeben durch + \[ + m^{-1} = (-1)_K m \in M \text{ } \forall m \in M + .\] + + + Beh.: $U \subset V$ ist Untervektorraum. \begin{proof} Seien $f_1, f_2 \in U$, $a \in K$ beliebig. Zu zeigen: @@ -48,11 +72,15 @@ $V = \text{Abb}(M, K)$ zum Vektorraum. \item Beh.: $U \cap W = \{0\} $ \begin{proof} + Zunächst: $0_V(m_0) = 0 \implies 0_V \in U$ und + $0_V(x) = 0 = 0_V(y) \text{ } \forall x,y \in M \implies 0_V \in W$. Daraus folgt + $0_V \in U \cap W \implies U \cap W \neq \emptyset$. + Sei $f \in U \cap W$ beliebig: \begin{align*} &\forall m \in M \colon f(m) = f(m_0) \land f(m_0) = 0 \\ \implies &\forall m \in M \colon f(m) = 0 \\ - \implies &f = 0 + \implies &f = 0_V .\end{align*} \end{proof} \item Beh.: $V = U + W$ @@ -123,13 +151,13 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \begin{align*} \partial(v_1+v_2)(i) &= (i+1) \cdot (v_1 + v_2)(i+1) \\ &= (i+1) \cdot (v_1(i+1) + v_2(i+1)) \\ - &= (i+1)v_1(i+1) + (i+1)v_2(i+1) \\ + &= (i+1) \cdot v_1(i+1) + (i+1) \cdot v_2(i+1) \\ &= \partial(v_1)(i) + \partial(v_2)(i) .\end{align*} \begin{align*} - \partial(a v_1)(i) &= (i + 1)(a v_1)(i+1) \\ - &= a (i+1) v_1 (i+1) \\ - &= a \partial(v_1)(i) + \partial(a v_1)(i) &= (i + 1) \cdot (a v_1)(i+1) \\ + &= a (i+1) \cdot v_1 (i+1) \\ + &= a \cdot \partial(v_1)(i) .\end{align*} \end{proof} @@ -139,17 +167,17 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Seien $v_1, v_2 \in V$ mit $\psi(v_1) = \psi(v_2)$. Dann \begin{align*} - &\psi(v_1) = \left( f_1(0), f_1(1), \ldots, f_1(n+1) \right) - = \left( f_2(0), f_2(1), \ldots, f_2(n+1) \right) = \psi(v_2)\\ - \implies& f_1(k) = f_2(k) \text{ }\forall k \in \{0, \ldots, n+1\} \\ - \implies& f_1 = f_2 + &\psi(v_1) = \left( v_1(0), v_1(1), \ldots, v_1(n+1) \right) + = \left( v_2(0), v_2(1), \ldots, v_2(n+1) \right) = \psi(v_2)\\ + \implies& v_1(k) = v_2(k) \text{ }\forall k \in \{0, \ldots, n+1\} \\ + \implies& v_1 = v_2 .\end{align*} $\implies \psi$ ist injektiv. - Sei $c = (c_0, \ldots, c_{n+1}) \in K^{n+2}$, dann ex. ein $f \in V$, s.d. + Sei $c = (c_0, \ldots, c_{n+1}) \in K^{n+2}$, dann ex. ein $v \in V$, s.d. \begin{align*} - &f(k) = c_k \text{ } \forall k \in \{0, \ldots, n+1\} \\ - \implies &\psi(f) = c + &v(k) = c_k \text{ } \forall k \in \{0, \ldots, n+1\} \\ + \implies &\psi(v) = c .\end{align*} $\implies \psi$ ist surjektiv. \end{proof} @@ -175,32 +203,63 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. k+1 \neq 0$ $\forall k \in \{0, \ldots, n\} $. \begin{align*} &k + 1 \neq 0 \\ - \stackrel{k > 0}{\iff} & k + 1 \neq \text{char}K \\ - \stackrel{0 \le k \le n}\iff & \text{char}K = 0 \lor \text{char}K > n + 1 \\ + \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\} .\end{align*} \end{proof} - \item Beh.: $\psi(\text{ker }\partial) = - \left\{ (c, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\right\} $ - - \begin{proof} - Zunächst: $\text{ker }\partial$. - - Damit $r \in V$ im Kern von $\partial$ liegt, muss gelten: - $\partial(r)(k) = 0$ $\forall k \in \{0, \ldots, n\}$ + \item Bestimmen Sie $\psi(\text{ker }K) \subset K^{n+2}$. + \begin{proof}[Lösung] \begin{align*} - &\partial(r)(k) = (k+1) \cdot r(k+1) \\ - \stackrel{k+1 \neq 0}{\implies} &r(k+1) = 0 + &\ker \partial = + \{f \in V \mid \left( \partial(f) \right)(k) = 0 \text{ } \forall k \in \{0, 1, \ldots n\} \} .\end{align*} - Damit: $r(k) = 0$ $\forall k \in \{1, \ldots, n+1\} $. + Damit $f \in \text{ker } \partial$, muss folglich gelten: \begin{align*} - \psi(r) &= \left( r(0), r(1), \ldots, r(n+1) \right) \\ - &= (c, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \text{ } \forall c \in K + &(\partial(f))k = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} .\end{align*} - Das heißt: - \[ - \psi(\text{ker }\partial) = \left\{ (c, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\right\} - .\] + $\iff$ + \begin{align*} + (k+1) \cdot f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} + .\end{align*} + $\stackrel{K \text{ Körper}}{\iff}$ + \begin{align*} + k+1 = 0 \lor f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} + .\end{align*} + + Aus (c) folgt: $k+1 \neq 0 \iff \text{char K} \not\in \{2, \ldots, n+1\} $. + + \begin{enumerate}[(i)] + \item $\text{char }K \not\in \{2, \ldots, n+1\} $. Dann ist $k + 1 \neq 0$, d.h. + \begin{align*} + &f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} \\ + \implies &f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \\ + \implies & \text{ker } \partial = \{f \in V \mid f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \} + .\end{align*} + Damit folgt: + \[ + \psi(\text{ker }\partial) = + \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\} + .\] + \item $\text{char }K \in \{2, \ldots, n+1\} $. Dann gilt für $k = \text{char }K-1$: + \[ + k + 1 = \text{char } K - 1 + 1 = \text{char } K = 0_K + .\] + Für alle $k \in \{0, 1, \ldots, n\}, k \neq \text{char } K - 1$, folgt analog zu (i): + \[ + f(k + 1) = 0 + .\] + Damit folgt: + \[ + \text{ker } \partial + = \left\{ f \in V \mid f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} + \setminus \{\text{char } K - 1\} \right\} + .\] Damit ergibt sich: + \[ + \psi(\text{ker } \partial) = \{ (a_0, a_1, \ldots, a_{n+1}) \in K^{n+2} + \mid a_k = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \setminus \{\text{char }K - 1\} \} + .\] + \end{enumerate} \end{proof} \end{enumerate} @@ -221,8 +280,8 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. = f^{*}(\varphi_1) + f^{*}(\varphi_2) \\ f^{*}(a \varphi_1) &= (a \varphi_1) \circ f - \stackrel{\varphi_1 \text{ linear}}{=} - a ((\varphi_1) \circ f) + = + a (\varphi_1 \circ f) = a f^{*}(\varphi_1) .\end{align*} \end{proof} @@ -234,14 +293,14 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \begin{proof} Seien $u_1, u_2 \in U$, $a \in K$ und $f \in U^{*}$ beliebig. \begin{align*} - \text{ev}(u_1 + u_2)(f) &= + (\text{ev}(u_1 + u_2))(f) &= f(u_1 + u_2) \stackrel{f \text{ linear}} {=} f(u_1) + f(u_2) - = \text{ev}(u_1)(f) + \text{ev}(u_2)(f) \\ - \text{ev}(a u_1)(f) &= + = (\text{ev}(u_1))(f) + (\text{ev}(u_2))(f) \\ + \left(\text{ev}(a u_1)\right)(f) &= f(a u_1) \stackrel{f \text{ linear}} {=} a f(u_1) - = a \cdot \text{ev}(u_1)(f) + = a \cdot (\text{ev}(u_1))(f) .\end{align*} \end{proof} \end{enumerate} @@ -251,7 +310,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Es sei $K$ ein Körper und $U, V$ zwei $K$-Vektorräume. \begin{enumerate}[a)] - \item Die Abbildung $*$: + \item Beh.: Die Abbildung $*$: $\text{Hom}_K(U,V) \to \text{Hom}_K(V^{*}, U^{*})$ ist linear. @@ -260,19 +319,36 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. $\varphi \in V^{*}$ und $a \in K$ beliebig. \begin{align*} - *(f_1 + f_2)(\varphi) &= (f_1 + f_2)^{*}(\varphi) + (*(f_1 + f_2))(\varphi) &= ((f_1 + f_2)^{*})(\varphi) = \varphi \circ (f_1 + f_2) = \varphi \circ f_1 + \varphi \circ f_2 - = *(f_1)(\varphi) + *(f_2)(\varphi) \\ - *(a f_1)(\varphi) - &= (a f_1)*(\varphi) + = (*(f_1))(\varphi) + (*(f_2))(\varphi) \\ + (*(a f_1))(\varphi) + &= ((a f_1)^{*})(\varphi) = \varphi \circ (a f_1) = a (\varphi \circ f_1) - = a*(f_1)(\varphi) + = a\cdot (*(f_1))(\varphi) .\end{align*} \end{proof} - \item Ist $f\colon U \to V$ linear und surjektiv, so ist + \item Beh.: Ist $f\colon U \to V$ linear und surjektiv, so ist $f^{*}\colon V^{*} \to U^{*}$ injektiv. + + \begin{proof} + Seien $\varphi_1, \varphi_2 \in \text{Hom}_K(V,K) = V^{*}$ mit + $f^{*}(\varphi_1) = f^{*}(\varphi_2)$. Dann folgt: + \begin{align*} + \varphi_1 \circ f = \varphi_2 \circ f + .\end{align*} + das heißt: + \begin{align*} + \forall u \in U\colon \varphi_1(f(u)) = \varphi_2(f(u)) + .\end{align*} + Wegen $f$ surjektiv gilt: $V = f(U)$ und damit: + \begin{align*} + \forall v \in V\colon \varphi_1(v) = \varphi_2(v) + .\end{align*} + $\implies \varphi_1 = \varphi_2$ + \end{proof} \end{enumerate} \end{aufgabe}