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HcmV?d00001 diff --git a/sose2020/la/uebungen/la8.tex b/sose2020/la/uebungen/la8.tex new file mode 100644 index 0000000..44868e9 --- /dev/null +++ b/sose2020/la/uebungen/la8.tex @@ -0,0 +1,196 @@ +\documentclass[uebung]{../../../lecture} + +\title{Lineare Algebra II: Übungsblatt 8} +\author{Miriam Philipp, Dominik Daniel, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + Seien $R$ ein Ring, $M$ und $N$ zwei $R$-Moduln und $f\colon M \to N$ $R$-Modulhom. + Beh.: Folgende Aussagen sind äquivalent + \begin{enumerate}[(i)] + \item $f$ ist $R$-Mod.iso + \item Für alle $R$-Moduln $L$ ist die Abbildung + \begin{align*} + \text{Hom}_R(L,M) &\to \text{Hom}_R(L,N) \\ + g &\mapsto f \circ g + .\end{align*} + bijektiv. + \end{enumerate} + \begin{proof} + \begin{itemize} + \item (i) $\implies$ (ii): $\varphi$ bezeichne die gegebene Abbildung. + Sei $f$ $R$-Mod.iso. und $g_1, g_2 \in \text{Hom}_R(L,M)$ + mit $\varphi(g_1) = \varphi(g_2)$. Da $f$ Iso. ex. ein $f^{-1}\colon N \to M$. Damit folgt + \[ + \varphi(g_1) = \varphi(g_2) \implies f \circ g_1 = f \circ g_2 + \implies f^{-1} \circ f \circ g_1 = f^{-1} \circ f \circ g_2 \implies g_1 = g_2 + .\] Also $\varphi$ injektiv. + + Sei nun $h \in \text{Hom}_R(L,N)$ beliebig. Wähle $g = f^{-1} \circ h$. Damit folgt + \[ + \varphi(g) = \varphi(f^{-1} \circ h) = f \circ f^{-1} \circ h = h + .\] Also $\varphi$ surjektiv und damit bijektiv. + \item (ii) $\implies$ (i): Mit $L = N$ folgt + $\varphi \colon \text{Hom}_R(N,M) \to \text{Hom}_R(N,N)$ bijektiv. Also ex. + $\varphi^{-1}$. Definiere $h \coloneqq \varphi^{-1}(\text{id}_N)$. Damit folgt + $\varphi(h) = f \circ h = \text{id}_N$. + + Setze nun $L = M$. Dann folgt $\psi \colon \text{Hom}_R(M,M) \to \text{Hom}_R(M,N)$ + bijektiv. Es ist dann $\psi(\text{id}_M) = f \circ \text{id}_M = f$. Damit folgt + \[ + \psi(h \circ f) = \underbrace{f \circ h}_{\text{id}_N} \circ f + = f = \psi(\text{id}_M) + .\] Da $\psi$ injektiv, folgt $h \circ f = \text{id}_M$. + + Insgesamt folgt mit $f^{-1} \coloneqq h$, $f$ bijektiv und damit Iso. + \end{itemize} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\Q \otimes_\Z \Z / 2 \Z = 0$. + \begin{proof} + Es ist nach VL: + \[ + \Q \otimes_\Z \Z / 2 \Z \stackrel{\sim }{=} \Z / 2 \Z \otimes_\Z \Q + \stackrel{\sim }{=} \Q / (2 \Z) \Q + .\] Aber $(2 \Z) \Q = \Q$, denn $(2 \Z) \Q \subseteq \Q$ klar und $\Q \subseteq (2 \Z) \Q$, denn + für $q \in \Q$ ist $q = 2 \cdot \frac{1}{2} q \in (2 \Z) \Q$. Also + $\Q / (2 \Z) \Q = 0$. Damit folgt die Beh. + \end{proof} + \item Beh.: $2 \Z \otimes_\Z \Z / 2 \Z \stackrel{\sim }{=} \Z / 2 \Z$. + \begin{proof} + Es ist nach VL: + \[ + 2 \Z \otimes_\Z \Z / 2 \Z \stackrel{\sim }{=} \Z / 2 \Z \otimes_\Z 2 \Z + \stackrel{\sim }{=} 2 \Z / (2 \Z 2 \Z) + \] + Es ist $2\Z 2 \Z = (4) = 4 \Z$ also $2 \Z / (2 \Z 2 \Z) = 2 \Z / 4 \Z$. Weiter ist + $\Z / 2 \Z \stackrel{\sim }{=} 2 \Z / 4 \Z$, denn + \[ + \#\{2 \Z / 4 \Z\} = \# \{ \ldots, 0 + 4 \Z, 2 + 4 \Z, \underbrace{4 + 4 \Z}_{0 + 4\Z}, \ldots \} = 2 = \#\{ \Z / 2 \Z\} + .\] + \end{proof} + \item Beh.: $2 \otimes 1 = 0$ in $\Z \otimes_\Z \Z / 2 \Z$, aber $2 \otimes 1 \neq 0$ in + $2 \Z \otimes_\Z \Z / 2 \Z$. + \begin{proof} + Es ist $\Z \otimes_\Z \Z / 2 \Z \ni 2 \otimes \overline{1} = (2 \cdot 1) \otimes \overline{1} = + 1 \otimes (2 \cdot \overline{1}) = 1 \otimes \overline{0} = 0$. + + Definiere $\beta\colon 2 \Z \otimes \Z / 2\Z \to \Z / 2\Z$, $(a, \overline{b}) + \mapsto \overline{\frac{a}{2} b}$. + $\beta$ ist wohldefiniert, da $\forall a \in 2 \Z$ ist $\frac{a}{2} \in \Z$. Außerdem + $\beta$ bilinear. Wende (UT) auf $\Z / 2\Z$ und $\beta$ an. Erhalte + einen $\Z$-Mod.hom $f\colon 2 \Z \otimes_\Z \Z / 2 \Z \to \Z / 2\Z$ mit + $f \circ \tau = \beta$. Ang.: $2 \Z \otimes_\Z \Z / 2 \Z \ni 2 \otimes \overline{1} = 0$. Dann + folgt + \[ + f(\tau(2, \overline{1})) = f(2 \otimes \overline{1}) = f(0) = \overline{0} + \neq \overline{1} = \beta(2, \overline{1}) \quad \contr + .\] Also $2 \Z \otimes_\Z \Z / 2 \Z \ni 2 \otimes \overline{1} \neq 0$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $R$ ein Ring, $I \subseteq R$ Ideal und $M$ $R$-Modul. + \begin{enumerate}[(a)] + \item Beh.: Es gibt einen eindeutigen surjektiven $R$-Mod.hom. $f\colon I \otimes_R M \to IM$ + mit $f(a \otimes m) = am$ für $a \in I, m \in M$. + \begin{proof} + Definiere $\beta\colon I \times M \to IM$, $(a,m) \mapsto am$. $\beta$ bilinear. Mit + (UT) angewendet auf $IM$ und $\beta$, existiert genau ein $R$-Mod.hom. + $f\colon I \otimes_R M \to IM$ + mit $f \circ \tau = \beta$. Damit folgt für $a \in I$, $m \in M$: + \[ + f(a \otimes m) = f(\tau(a,m)) = \beta(a, m) = am + .\] + + Sei nun $m \in IM$ beliebig. Dann ex. $a_i \in I$, $m_i \in M$ mit + \begin{salign*} + m &= \sum_{i=1}^{n} a_i m_i \\ + &= \sum_{i=1}^{n} f(a_i \otimes m_i) \\ + &\stackrel{f \ R\text{-Hom.}}{=} + f\underbrace{\left(\sum_{i=1}^{n} a_i \otimes m_i \right)}_{\coloneqq \xi \in I \otimes_R M} \\ + &= f(\xi) + .\end{salign*} + Also $f$ surjektiv. + \end{proof} + \item Beh.: $f$ aus Teil (a) ist i.A. nicht injektiv. + \begin{proof} + Mit $R = \Z$, $I = 2 \Z$ und $M = \Z / 2 \Z$ folgt + \[ + f \colon 2 \Z \otimes_{\Z} \Z / 2 \Z \to 2 \Z \left( \Z / 2\Z \right) = 0 + ,\] aber mit 29(b) ist $2 \Z \otimes_\Z \Z / 2 \Z \stackrel{\sim }{=} \Z / 2 \Z \neq 0$. + Damit ist $f$ nicht injektiv. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $K$ Kp, $V$ e.d. $K$-VR. und $f, g \in \text{End}_K(V)$. Seien $\lambda \in K$ EW von $f$ + und $\mu \in K$ EW von $g$. + \begin{enumerate}[(a)] + \item Beh.: Für $v, w \in V \setminus \{0\}$ gilt $v \otimes w \neq 0$ in $V \otimes_K V$. + \begin{proof} + Seien $v, w \in V \setminus \{0\} $. Da $V$ VR. und $v \neq 0 \neq w$, + ergänze $v$ zu Basis $(v_i)_{i \in I}$ und + w zu Basis $(w_i)_{i \in I}$ mit $v = v_1$ und $w = w_1$. + Dann definiere + \begin{align*} + \beta\colon V \times V &\to K \\ + (x,y) = \left( \sum_{i \in I} \alpha_i v_i, \sum_{i \in I} \beta_i w_i\right) + &\mapsto \sum_{i \in I} \alpha_i \cdot \sum_{i \in I} \beta_i + .\end{align*} + Da $(v_i)_{i \in I}$ und $(w_i)_{i \in I}$ Basen, sind die Darstellungen + eindeutig und damit $\beta$ wohldefiniert. Außerdem $\beta$ bilinear, denn + $\forall x, y, z \in V$ und $\lambda \in K$ gilt + \begin{salign*} + \beta(\lambda x + y, z) &= \beta\left( \sum_{i \in I} (\lambda \alpha_i + \beta_i) v_i, + \sum_{i \in I}^{} \gamma_i w_i\right) \\ + &= \sum_{i \in I}^{} (\lambda \alpha_i + \beta_i) \cdot \sum_{i \in I} \gamma_i \\ + &= \lambda \sum_{i \in I}^{} \alpha_i \sum_{i \in I} \gamma_i + + \sum_{i \in I} \beta_i \sum_{i \in I}^{} \gamma_i \\ + &= \lambda \beta(x, z) + \beta(y,z) + .\end{salign*} + Für zweites Argument analog. Weiter ist + $v = 1 \cdot v_1$ und $w = 1\cdot w_1$, also $\beta(v,w) = 1 \cdot 1 = 1 \neq 0$. + + Mit (UT) angewendet auf $\beta$ und $K$, existiert ein $R$-Mod.hom. + $f\colon V \otimes_K V \to K$. mit $f \circ \tau = \beta$. Ang.: $v \otimes w = 0$. Dann + ist $f(\tau(v \otimes w)) = f(v \otimes w) = f(0) = 0 \neq 1 = \beta(v, w)$. + Also $v \otimes w \neq 0$. + \end{proof} + \item Beh.: $\lambda \mu$ ist EW von $f \otimes g \in \text{End}_K(V \otimes_K V)$. + \begin{proof} + Sei $v$ EV von $f$ bezügl. $\lambda$ und $w$ EV von $g$ bezügl. $\mu$. Dann gilt + \begin{salign*} + (f \otimes g)(v \otimes w) &= f(v) \otimes g(w) \\ + &= \lambda v \otimes \mu w \\ + &= \lambda \mu (v \otimes w) + .\end{salign*} + Da $v, w$ EV sind $v \neq 0 \neq w$, also wegen (a) auch $v \otimes w \neq 0$ und damit EV von + $f \otimes g$ zu $\lambda \mu$. Also insbes. $\lambda \mu$ EW von $f \otimes g$. + \end{proof} + \item Beh.: $\lambda + \mu$ ist EW von $f \otimes \text{id}_V + \text{id}_V \otimes g \in \text{End}_K(V \otimes_K V)$. + \begin{proof} + Sei $v$ EV von $f$ bezügl. $\lambda$ und $w$ EV von $g$ bezügl. $\mu$. Dann gilt + \begin{salign*} + (f \otimes \text{id}_V + \text{id}_V \otimes g)(v \otimes w) + &= (f \otimes \text{id}_V)( v \otimes w) + (\text{id}_V \otimes g)( v \otimes w) \\ + &= f(v) \otimes \text{id}_V(w) + \text{id}_V(v) \otimes g(w) \\ + &= \lambda v \otimes w + v \otimes \mu w \\ + &= (\lambda + \mu) (v \otimes w) + .\end{salign*} + Da $v, w$ EV sind $v \neq 0 \neq w$, also wegen (a) auch $v \otimes w \neq 0$ und damit EV von + $f \otimes \text{id}_V + \text{id}_V \otimes g$ zu $\lambda + \mu$. + Also insbes. $\lambda + \mu$ EW von $f \otimes \text{id}_V + \text{id}_V \otimes g$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}