From c1da37d17061153a2416da834048224e1dd6e65c Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 6 Dec 2019 18:28:04 +0100 Subject: [PATCH] update la8 --- ws2019/la/uebungen/la8.pdf | Bin 57035 -> 105250 bytes ws2019/la/uebungen/la8.tex | 371 +++++++++++++++++++++++++++++++++++-- 2 files changed, 356 insertions(+), 15 deletions(-) diff --git a/ws2019/la/uebungen/la8.pdf b/ws2019/la/uebungen/la8.pdf index bc191ea8431d312d4b6475c3f802a8910aacf089..95f580755d62b2b2526611ea0fb5e4d9fc307b6d 100644 GIT binary patch delta 90649 zcmZsCV|XTAux)JHwr$%J+qOMP-dGb%^2WBEiETTX*tRv%o$tHn-gECc=U;XA^K^Bs z+I#I)wRcZ+ffqDG{Un26XXW4p=3~amcli^PfdY2)5S@R)ZAKt(;>@* zubWMs&sSAB50{?qHNy>ttE>L&#mVgHKIdwynVQ>m`$oR7RW82{J;s}JKH^8U_BQ!t zqXW}^PAwY|_Ghjx2NH#5xeW*1KU)*gx0#Jz0Ls z3fu{u(j{`JmV1nu`fH1^^}Pxp|4WHAKHf_PLzm_7nT#A2%g?mQY}hkuCyE6jl%4|R zx7VZ(PUu2MF3nWRIpQUMw?7o-e5izB(H_#X=h)R^{&CM9MIip^3?A4q~+Gdm|Y(W;$1JfW9?F-?s zmHWlD&64?w_08CnUiAhJ_XY1)7H8WP~)IqBoZK2c%Ebd 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8} +\author{Christian Merten, Mert Biyikli} \begin{document} +\punkte + \begin{aufgabe} - + +\begin{enumerate}[(a)] + \item Beh.: $\underline{w}$ ist eine Basis von $W = K[X]_{\le 3}$ + \begin{proof} + Zz.: $\underline{w}$ ist linear unabhängig + + Seien $a, b, c, d \in K$ mit + \begin{align*} + a X^{0} + b (X^{0} + X^{1}) + c (X^{1} - X^{2} + X^{3} + d (X^{3} + X^{0}) &= 0 \\ + \implies X^{0}(a + b + d) + X^{1} (b + c) + X^{2} (-c) + X^{3}(c + d) = 0 + .\end{align*} Wegen $\underline{v}$ linear unabhängig, folgt: + \begin{align*} + c = 0 \implies d = 0 \implies b = 0 \implies a = 0 + .\end{align*} + + Zz.: $\underline{w}$ ist Erzeugendensystem + + Sei $v \in K[X]_{\le 3}$ beliebig, dann ex. $a, b, c, d \in K$ wegen $\underline{v}$ Basis + s.d. $v = a X^{0} + b X^{1} + c X^{2} + d X^{3} $. + + Wähle nun $\alpha := a - b - 2c - d, \beta := b + c, \gamma := -c, \delta := c+d$. + Damit folgt direkt: + \begin{align*} + v &= \alpha X^{0} + \beta (X^{0} + x^{1}) + \gamma (X^{1} - X^{2} + X^{3}) + + \delta (X^{3} + X^{0}) \\ + &= a X^{0} + b X^{1} + c X^{2} + d X^{3} + .\end{align*} + \end{proof} + \item Seien $\phi_{\underline{v}}\colon K^{4} \to K[X]_{\le 3}$ und + $\phi_{\underline{w}}\colon K^{4} \to K[X]_{\le 3}$ die kanonischen Isomorphismen. + \begin{enumerate}[(i)] + \item + \[ + M_{\underline{v}}^{\underline{v}}(\partial) = A := + \begin{pmatrix} + 0 & 1 & 0 & 0 \\ + 0 & 0 & 2 & 0 \\ + 0 & 0 & 0 & 3 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} + .\] + \begin{proof} + Zu zeigen.: $F_{\underline{v}}^{\underline{v}}(A) = \partial$. + + Zu überprüfen für die vier Basisvektoren von $K[X]_{\le 3}$ aus $\underline{v}$. + + \[ + F_{\underline{v}}^{\underline{v}}(A) = + \phi_{\underline{v}} \circ F_{4,4}(A) \circ \phi_{\underline{v}}^{-1} + .\] + \begin{enumerate} + \item $v_1 = X_0$, $\phi_{\underline{v}}^{-1}(X^{0}) = (1, 0, 0, 0)$ + \[ + \begin{pmatrix} + 0 & 1 & 0 & 0 \\ + 0 & 0 & 2 & 0 \\ + 0 & 0 & 0 & 3 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} + .\] + $\implies \phi_{\underline{v}}(0, 0,0,0) = 0 = \partial(X_0)$ + \item $v_2 = X_1$, $\phi_{\underline{v}}^{-1}(X^{1}) = (0, 1, 0, 0)$ + \[ + \begin{pmatrix} + 0 & 1 & 0 & 0 \\ + 0 & 0 & 2 & 0 \\ + 0 & 0 & 0 & 3 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} + .\] + $\implies \phi_{\underline{v}}(1, 0, 0, 0) = X^{0} = \partial(X_1)$ + \item $v_3 = X_2$, $\phi_{\underline{v}}^{-1}(X^{2}) = (0, 0, 1, 0)$ + \[ + \begin{pmatrix} + 0 & 1 & 0 & 0 \\ + 0 & 0 & 2 & 0 \\ + 0 & 0 & 0 & 3 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix} 0 \\ 2 \\ 0 \\ 0 \end{pmatrix} + .\] + $\implies \phi_{\underline{v}}(0, 2, 0, 0) = 2X^{1} = \partial(X_2)$ + \item $v_4 = X_3$, $\phi_{\underline{v}}^{-1}(X^{3}) = (0, 0, 0, 1)$ + \[ + \begin{pmatrix} + 0 & 1 & 0 & 0 \\ + 0 & 0 & 2 & 0 \\ + 0 & 0 & 0 & 3 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 3 \\ 0 \end{pmatrix} + .\] + $\implies \phi_{\underline{v}}(0, 0, 3, 0) = 3X^{2} = \partial(X_3)$ + \end{enumerate} + $\implies F_{\underline{v}}^{\underline{v}}(A) = \partial$ + \end{proof} + \item + \[ + M_{\underline{w}}^{\underline{v}}(id_W) = A := + \begin{pmatrix} + 1 & -1 & -2 & -1 \\ + 0 & 1 & 1 & 0 \\ + 0 & 0 & -1 & 0 \\ + 0 & 0 & 1 & 1 \\ + \end{pmatrix} + .\] + \begin{proof} + Zu zeigen.: $F_{\underline{w}}^{\underline{v}}(A) = id$ + + Zu überprüfen für die vier Basisvektoren von $K[X]_{\le 3}$ aus $\underline{v}$. + + \[ + F_{\underline{w}}^{\underline{v}}(A) = + \phi_{\underline{w}} \circ F_{4,4}(A) \circ \phi_{\underline{v}}^{-1} + .\] + \begin{enumerate} + \item $v_1 = X_0$, $\phi_{\underline{v}}^{-1}(X^{0}) = (1, 0, 0, 0)$ + \[ + \begin{pmatrix} + 1 & -1 & -2 & -1 \\ + 0 & 1 & 1 & 0 \\ + 0 & 0 & -1 & 0 \\ + 0 & 0 & 1 & 1 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} + .\] + $\implies \phi_{\underline{w}}(1, 0,0,0) = X^{0} = id_W(X^{0})$ + \item $v_2 = X_1$, $\phi_{\underline{v}}^{-1}(X^{1}) = (0, 1, 0, 0)$ + \[ + \begin{pmatrix} + 1 & -1 & -2 & -1 \\ + 0 & 1 & 1 & 0 \\ + 0 & 0 & -1 & 0 \\ + 0 & 0 & 1 & 1 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 0 \\ 0 \end{pmatrix} + .\] + $\implies \phi_{\underline{w}}(-1, 1, 0, 0) = -X^{0} + X^{0} + X^{1} = X^{1} = id_W(X^{1})$ + \item $v_3 = X_2$, $\phi_{\underline{v}}^{-1}(X^{2}) = (0, 0, 1, 0)$ + \[ + \begin{pmatrix} + 1 & -1 & -2 & -1 \\ + 0 & 1 & 1 & 0 \\ + 0 & 0 & -1 & 0 \\ + 0 & 0 & 1 & 1 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix} -2 \\ 1 \\ -1 \\ 1 \end{pmatrix} + .\] + $\implies \phi_{\underline{v}}(-2, 1, -1, 1) = -2X^{0} + X^{0} + X^{1} - X^{1} + X^{2} - X^{3} + X^{3} + X^{0} = X^{2} = id_W(X^{2})$ + \item $v_4 = X_3$, $\phi_{\underline{v}}^{-1}(X^{3}) = (0, 0, 0, 1)$ + \[ + \begin{pmatrix} + 1 & -1 & -2 & -1 \\ + 0 & 1 & 1 & 0 \\ + 0 & 0 & -1 & 0 \\ + 0 & 0 & 1 & 1 \\ + \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix} -1 \\ 0 \\ 0 \\ 1 \end{pmatrix} + .\] + $\implies \phi_{\underline{v}}(-1, 0, 0, 1) = -X^{0} + X^{3} + X^{0} = X^{3} = id_W(X^{3})$ + \end{enumerate} + $\implies F_{\underline{w}}^{\underline{v}}(A) = id_W$ + \end{proof} + \item + \[ + M_{\underline{v}}^{\underline{w}}(id_W) = A := + \begin{pmatrix} + 1 & 1 & 0 & 1 \\ + 0 & 1 & 1 & 0 \\ + 0 & 0 & -1 & 0 \\ + 0 & 0 & 1 & 1 \\ + \end{pmatrix} + .\] + \begin{proof} + Erfolgt analog zu (ii). + \end{proof} + \item + \[ + M_{\underline{w}}^{\underline{w}}(\partial) = M_{\underline{w}}^{\underline{v}}(id_W) \cdot + M_{\underline{v}}^{\underline{v}}(\partial) \cdot M_{\underline{v}}^{\underline{w}}(id_W)= + \begin{pmatrix} + 0 & 1 & -3 & -6 \\ + 0 & 0 & 1 & 3 \\ + 0 & 0 & -3 & -3 \\ + 0 & 0 & 3 & 3 \\ + \end{pmatrix} + .\] + \item + \[ + M_{\underline{w}}^{\underline{v}}(\partial) = M_{\underline{w}}^{\underline{v}}(id_W) \cdot + M_{\underline{v}}^{\underline{v}}(\partial) = + \begin{pmatrix} + 0 & 1 & -2 & -6 \\ + 0 & 0 & 2 & 3 \\ + 0 & 0 & 0 & -3 \\ + 0 & 0 & 0 & 3 \\ + \end{pmatrix} + .\] + \item + \[ + M_{\underline{v}}^{\underline{w}}(\partial) = M_{\underline{v}}^{\underline{v}}(\partial) \cdot + M_{\underline{v}}^{\underline{w}}(id_W) = + \begin{pmatrix} + 0 & 1 & 1 & 0 \\ + 0 & 0 & -2 & 0 \\ + 0 & 0 & 3 & 3 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} + .\] + \end{enumerate} +\end{enumerate} + \end{aufgabe} -\begin{aufgabe} - +\begin{aufgabe} Sei $f\colon U \to V$ und $g\colon V \to W$ lineare Abbildungen + zwischen endlich dimensionalen Vektorräumen. + \begin{enumerate}[(a)] + \item Beh.: $\text{dim } \text{ker}(g\circ f) \le \text{dim } \text{ker} g + \text{dim } \text{ker }f$ + \begin{proof} + Schränke $g$ auf $\text{Bild}(f)$ ein durch $g'\colon \text{Bild}(f) \to W$ mit + $v \mapsto g(v)$. + \begin{align*} + \text{ker }(g \circ f) &= \text{dim } U - \text{dim}(\text{Bild}(g \circ f)) \\ + &= \text{dim }U - \text{Rg}(g')\\ + &= \text{Rg}(f) - \text{Rg}(g') + \text{dim }U - \text{Rg}(f) \\ + &= \text{ker }g' + \text{ker }f \\ + &\le \text{ker }g + \text{ker }f + .\end{align*} + \end{proof} + \item Beh.: $\text{Rg}(f) - \text{Rg}(g \circ f) \le \text{dim } V - \text{Rg}(g)$ + \begin{proof} + Aus (a) folgt: + \begin{align*} + \text{dim } \text{ker}(g \circ f) &\le \text{dim } \text{ker}(g) + \text{dim } \text{ker}(f) \\ + \implies \text{dim } U - \text{Rg}(g \circ f) &\le \text{dim } V - \text{Rg}(g) + \text{dim } U - \text{Rg}(f) \\ + \implies \text{Rg}(f) - \text{Rg}(g \circ f) &\le \text{dim } V - \text{Rg}(g) + .\end{align*} + \end{proof} + \item Beh.: Für $A \in M_{n, m}(K)$ und $B \in M_{l,n}(K)$ gilt + $S\text{Rg}(A) - S\text{Rg}(B \cdot A) \le n - S\text{Rg}(B) $. + + \begin{proof} + Seien $A \in M_{n,m}(K)$ und $B \in M_{l,n}(K)$ beliebig, dann definiere + $f := F_{n, m}(A)$ und $g := F_{l, n}(B)$. Damit folgt: $F_{m, l}(B \cdot A) = g \circ f$. + + Dann folgt aus (b) direkt: + \[ + \text{Rg}(f) - \text{Rg}(g \circ f) \le \text{dim } K^{n} - \text{Rg}(g) + .\] Mit $\text{Rg}(f) = S\text{Rg}(A) $, $\text{Rg}(g) = S\text{Rg}(B) $ und + $\text{Rg}(g \circ f) = S\text{Rg}(A\cdot B)$ ergibt sich + \[ + S\text{Rg}(A) - S\text{Rg}(B \cdot A) \le n - S\text{Rg}(B) + .\] + \end{proof} + \end{enumerate} \end{aufgabe} -\begin{aufgabe} - \begin{enumerate} - \item - \item - \item Beh.: Für $f : V \to V$ eine lineare Abbildung gilt +\begin{aufgabe} Sei $V$ ein Vektorraum, $U$ ein Untervektorraum und $W$ ein Komplement von $U$ in $V$. + \begin{enumerate}[(a)] + \item Beh.: Es existiert eine eindeutige lineare Abbildung $\pi\colon V \to V$, welche eingeschränkt + auf $U$ die Identität und eingeschränkt auf $W$ konstant null ist. + + \begin{proof} + Sei $(v_i)_{i\in I}$ Basis von $U$ und $(v_j)_{j \in J}$ mit $J \cap U = \emptyset$ + Basis von $W$. + Damit ist $(v_i)_{i \in I \cup J}$ Basis von $V$. Definiere $\pi\colon V \to V$ linear mit + \[ + \pi(v_i) = \begin{cases} + v_i & \text{falls } i \in I \\ + 0 & \text{falls } i \in J + \end{cases} + .\] + + Schränke nun $\pi$ auf $U$ ein: Dann ex. für alle $u \in U$ ein + $(\alpha_i)_{i \in I} \in K^{(I)}$, s.d. + $v = \sum_{i \in I} v_i$. Damit: + \[ + \pi(v) = \sum_{i \in I} \alpha_i \pi(v_i) = \sum_{i \in I} \alpha_i v_i = v + .\] + + Schränke nun $\pi$ auf $W$ ein: Dann ex. für alle $w \in W$ + ein $(\alpha_j)_{j \in J} \in K^{(J)}$, s.d. + $v = \sum_{i \in J} \alpha_j v_j$. Damit + \[ + \pi(v) = \sum_{j \in J} \alpha_j \pi(v_j) = 0 + .\] + + $\pi$ ist eindeutig, da eindeutig durch die Basisvektoren definiert. + \end{proof} + \item Beh.: Für dieses $\pi$ gilt: $\pi \circ \pi = \pi$. + \begin{proof} + Seien die Basen wie in (a). + Sei $v \in V$ beliebig. Dann ex. ein $(\alpha_i)_{i\in I} \in K^{(I)}$ und ein + $(\beta_j)_{j\in J} \in K^{(J)}$, s.d. + \[ + v = \sum_{i \in I} \alpha_i v_i + \sum_{j \in J} \beta_j v_j + .\] Damit gilt + \[ + \pi(v) = \sum_{i \in I} \alpha_i \pi(v_i) + \sum_{j \in J} \beta_j \pi(v_j) + = \sum_{i \in I} \alpha_i v_i + .\] $\implies$ + \[ + \pi(\pi(v)) = \pi\left( \sum_{i \in I} \alpha_i v_i\right) = + \sum_{i \in I} \alpha_i v_i = \pi(v) + .\] $\implies \pi = \pi \circ \pi$ + \end{proof} + \item Beh.: Für $\pi : V \to V$ eine lineare Abbildung gilt \[ - V \stackrel{\sim }{=} \text{Bild}(f) \oplus \text{ker } f + V \stackrel{\sim }{=} \text{Bild}(\pi) \oplus \text{ker } \pi .\] \begin{proof} - Sei $(u_i)_{i\in I}$ Basis von $V \setminus \text{ker } f$. + Sei $U$ Komplement zu $\text{ker }\pi$ und $(u_i)_{i\in I}$ Basis von $U$. - Definiere $f: V / \text{ker } f \oplus \text{ker } f \to V$ mit + Wegen Homomorphiesatz gilt: $\text{Bild}(\pi) \stackrel{\sim }{=} V / \text{ker }(\pi)$. + Nach Blatt 6 gilt: $(u_i + \text{ker } \pi)_{i \in I}$ ist Basis von $V / \text{ker }(\pi)$. + Damit: \[ - ([u_i + \text{ker }f], k) \mapsto u_i + k - .\] Wohldefiniert, da $([u_i + \text{ker }f])_{i \in I}$ - nach Blatt 6 Basis von $V / \text{ker }f$. + \text{Bild}(\pi) \stackrel{\sim }{=} V / \text{ker }(\pi) \stackrel{\sim }{=} U + .\] Daraus folgt direkt: + \[ + \text{Bild}(\pi) \oplus \text{ker }\pi \stackrel{\sim }{=} U \oplus \text{ker }\pi + \stackrel{\sim }{=} V + .\] \end{proof} \end{enumerate} \end{aufgabe} +\begin{aufgabe} + +\[ + A_1 := \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad + A_2 := \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}, \quad + A_3 := \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} +.\] +\begin{proof} + \begin{enumerate} + \item $A_1$ ist die Einheitsmatrix $\implies A_1 \cdot A_1 = A_1$ und + $A_1 \cdot (1,1)^{t} = (1,1)^{t}$. + \item \[ + A_2 \cdot A_2 = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} + \cdot \begin{pmatrix} 0 & 1 \\ 0 & 1\end{pmatrix} + = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} = A_2 + .\] \[ + A_2 \cdot (1,1)^{t} = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} + \cdot \begin{pmatrix} 1 \\ 1 \end{pmatrix} + = \begin{pmatrix} 1 \\ 1 \end{pmatrix} = (1,1)^{t} + .\] + \item \[ + A_3 \cdot A_3 = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} + \cdot \begin{pmatrix} 1 & 0 \\ 1 & 0\end{pmatrix} + = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} = A_3 + .\] \[ + A_3 \cdot (1,1)^{t} = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} + \cdot \begin{pmatrix} 1 \\ 1 \end{pmatrix} + = \begin{pmatrix} 1 \\ 1 \end{pmatrix} = (1,1)^{t} + .\] + \end{enumerate} +\end{proof} + +\end{aufgabe} + \end{document}