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z98)F3K6E)8SuV_|%&E-Di5o>VXBS86GBXm<;K`^y3zyImWX19Sl7AE(_2?-d5G))1 z7{(GfB$N<52&<)HU1`nrZ#|#qY4LfmO}U?QL6ijd8m7vvNH_=6u_#E(*tP$bWbA}x zGQr@hSRv6&!o~IV4t@hsnAxzdtA#gi+m1+uDW>`^&bj9RvQ*U6(rPqhp;?29yvdrtIU3@ptz*G5Eh*e^;YHgRwpl+1CQ zVl0;7A+DwZx{nQtD7|r)mEV{WGaY=6i}k^rr-&l-A0+D?{>=aE&+dE2eG#F1l2w+r z0J>gf|IqZ-oXCs~%G#7CO}>BnDbeSor}vnc-o|J`JdE}w6?(F1GbbE;m&oYFjLwSi zW(=+L&$jQ3?FjTO&jnF!LmX_2m@A03PKqCnS$Yc4w(O6yRkrS=&yNb`*c0kBzu9y# zrT2SHGqvaLu!XC**^Oiqj$g~fN*0_%DsHR$rh(ZN%L@(+ ogSWdsOyi6Im2dv%y>xLkad!1^ez=HWFh2w&fXB=%qb!T}Z_$w;egFUf literal 0 HcmV?d00001 diff --git a/ws2019/la/uebungen/la11.tex b/ws2019/la/uebungen/la11.tex new file mode 100644 index 0000000..2912547 --- /dev/null +++ b/ws2019/la/uebungen/la11.tex @@ -0,0 +1,162 @@ +\documentclass[uebung]{../../../lecture} + +\title{Lineare Algebra I: Übungsblatt 11} +\author{Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + Die Adjunkte zu $A$ erhalten wir nach Berechnung zahlreicher Unterdeterminanten und + lästigem Ausrechnen. Damit ergibt sich + \begin{align*} + \widetilde{A} = \begin{pmatrix} + 8 & 8 & -32 & -8 \\ + 3 & 3 & -12 & -3 \\ + 1 & 1 & -4 & -1 \\ + -5 & -5 & 20 & 5 + \end{pmatrix} + .\end{align*} + Weiter folgt + \begin{align*} + A \cdot \widetilde{A} = \begin{pmatrix} + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} + .\end{align*} + Damit folgt mit der ersten Cramerschen Regel $\text{det}(A) = 0$. +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item + \begin{align*} + \intertext{(1)} + \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ + 4 & 3 & 2 & 1 & 5 \end{pmatrix} + &= \begin{pmatrix} 1 & 4 \end{pmatrix} + \circ \begin{pmatrix} 3 & 2 \end{pmatrix} \\ + \intertext{Permutationsmatrix} + \begin{pmatrix} + 0 & 0 & 0 & 1 & 0\\ + 0 & 0 & 1 & 0 & 0 \\ + 0 & 1 & 0 & 0 & 0 \\ + 1 & 0 & 0 & 0 & 0\\ + 0 & 0 & 0 & 0 & 1 + \end{pmatrix} + \intertext{(2)} + \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ + 2 & 4 & 5 & 1 & 3\end{pmatrix} + &= \begin{pmatrix} 3 & 5 \end{pmatrix} \circ \begin{pmatrix} 1 & 4 \end{pmatrix} + \circ \begin{pmatrix} 1 & 2 \end{pmatrix} + \intertext{Permutationsmatrix} + \begin{pmatrix} + 0 & 0 & 0 & 1 & 0\\ + 1 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 1 \\ + 0 & 1 & 0 & 0 & 0\\ + 0 & 0 & 1 & 0 & 0 + \end{pmatrix} + .\end{align*} + \item Beh.: siehe Aufgabe + \begin{proof} + Sei $F := \{ i \in \{1, \ldots, n\} \mid \sigma(i) = i \} $ + \begin{align*} + \text{Sp}(\varphi(\sigma)) &= \sum_{i=1}^{n} \varphi(\sigma)_{ii} \\ + &= \sum_{i=1}^{n} \left( e_{\sigma(i)} \right)_i \\ + &= \sum_{i = 1, i \in F}^{n} (e_i)_i + + \sum \{ (e_j)_i \mid i \in \{1, \ldots, n\} \setminus F, \; j = \sigma(i) + \implies j \neq i\} \\ + &= \sum_{i = 1, i \in F}^{n} 1 + \sum_{i = 1, i \not\in F}^{n} 0 = (\# F)_K + .\end{align*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Gesucht sind zunächst die Nullstellen des charakteristischen Polynoms. Mit + \begin{align*} + \lambda E_3 - A &= + \begin{pmatrix} + \lambda - 4 & -5 & -6 \\ + 0 & \lambda - 3 & 0 \\ + 3 & 5 & \lambda + 5 + \end{pmatrix} + \intertext{gilt für die Determinante} + \text{det}(\lambda E_3 - A) &= (\lambda - 4)(\lambda - 3)(\lambda + 5) +18(\lambda -3) \\ + &= (\lambda -3) \left[ \lambda^2 + \lambda -2 \right]. + \intertext{Diese Polynom hat die Nullstellen} + \lambda_1 &= 1, \lambda_2 = -2, \lambda_3 = 3. + \intertext{Für $\lambda_1$ ergibt sich das homogene GLS} + \begin{pmatrix} + -3 & -5 & - 6\\ + 0 & -2 & 0 \\ + 3 & 5 & 6 + \end{pmatrix} + &\to + \begin{pmatrix} + 1 & 0 & 2 \\ + 0 & 1 & 0 \\ + 0 & 0 & 0 + \end{pmatrix} + \intertext{Damit folgt direkt} + \text{ker}(\lambda_1 E_3 - A) &= \text{Lin}\left[ \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix} \right] + \intertext{Für $\lambda_2$ und $\lambda_3$ analog} + \text{ker}(\lambda_2 E_3 - A) &= \text{Lin}\left[ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \right] \\ + \text{ker}(\lambda_3 E_3 - A) &= \text{Lin}\left[ \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \right] + .\end{align*} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $(-)^{t}$ ist linear. + \begin{proof} + Seien $A, B \in M_{n,n}(\Q)$ und $\lambda \in Q$. Dann folgt direkt + \[ + (A + \lambda B)^{t} = A^{t} + (\lambda B)^{t} = A^{t} + \lambda B^{t} + .\] + \end{proof} + \item Beh.: $\text{dim}_\Q\left( \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) + = \begin{cases} + \frac{n^2 + n}{2} & \lambda = 1 \\ + \frac{n^2 - n}{2} & \lambda = -1 \\ + 0 & \text{sonst} + \end{cases}$ + \begin{proof} + Sei $A \in \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t})$. + für ein $\lambda \in \Q$. + + Falls $\lambda = 0 \implies A^{t} = 0 \implies A = 0 + \implies \text{dim}_\Q\left( \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) = 0$. + + Sei $\lambda \neq 0$. + \begin{align*} + \lambda A - A^{t} &= 0 \implies \lambda a_{ij} = a_{ji} \land \lambda a_{ji} = a_{ij} + \implies \lambda^2 a_{ij} = a_{ij} + \quad \forall i, j \in \{1, \ldots, n\}. + \intertext{Für $A \neq 0$ gilt, es ex. mind. ein $a_{ij} \neq 0$. Für diese $i$ und $j$ + gilt weiter} + \lambda^2 &= 1 \implies \lambda = \pm 1 + .\end{align*} + Für $\lambda \neq \pm 1 \implies A = 0 \implies \text{dim}_\Q\left( \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) = 0$ + + Sei nun $\lambda = 1$. Dann ist $A = A^{t}$, also ist $A$ symmetrisch. + Damit bleiben $n$ Diagonaleinträge frei und die Hälfte + der restlichen Einträge, damit folgt: + \[ + \text{dim}_\Q\left( \text{ker}(\text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) + = n + \frac{n^2 - n}{2} = \frac{n^2 + n}{2} + .\] Für $\lambda = -1$ ist $A = -A^{t}$, also $A$ antisymmetrisch. Damit + sind alle Diagonaleinträge $0$. Es folgt analog + \[ + \text{dim}_\Q\left( \text{ker}(-\cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) = + \frac{n^2 - n}{2} + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}