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zi&u$1xk{{CCH}lBv1FA3z-m?6Z-vLdlhypCC!PNTBa4D~3T19&b98cLVQmU!ZkKl4 S2oMZ4H8C?X3MC~)Peuw(`;CDB diff --git a/docs/loesungen.tex b/docs/loesungen.tex index e1c79cd..5a47efb 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -59,6 +59,8 @@ %% | %% — body/uebung/ueb7.tex; %% | +%% — body/uebung/ueb8.tex; +%% | %% — body/ska/ska4.tex; %% | %% — body/ska/ska5.tex; @@ -1342,14 +1344,15 @@ \def\vectorspacespan{\mathop{\text{\upshape Lin}}} \def\dim{\mathop{\text{\upshape dim}}} -\def\rank{\mathop{\text{\upshape Rank}}} +\def\rank{\mathop{\text{\upshape Rang}}} \def\onematrix{\text{\upshape\bfseries I}} \def\zeromatrix{\text{\upshape\bfseries 0}} \def\zerovector{\text{\upshape\bfseries 0}} \def\graph{\mathop{\text{\textup Gph}}} \def\domain{\mathop{\text{\textup dom}}} -\def\range{\mathop{\text{\textup ran}}} +\def\range{\mathop{\text{\textup Bild}}} +\def\ker{\mathop{\text{\upshape Kern}}} \def\functionspace{\mathop{\text{\textup Abb}}} \def\id{\text{\textup id}} \def\modfn{\mathop{\text{\textup mod}}} @@ -5162,6 +5165,706 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$. \end{proof} \end{enumerate} +%% ******************************************************************************** +%% FILE: body/uebung/ueb8.tex +%% ******************************************************************************** + +\setcounternach{chapter}{8} +\chapter[Woche 8]{Woche 8} + \label{ueb:8} + +%% AUFGABE 8-1 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 1]{} + \label{ueb:8:ex:1} +\let\sectionname\altsectionname + + In dieser Aufgabe arbeiten wir im Vektorraum $V=\reell^{4}$. + Seien + + \begin{mathe}[mc]{rcl} + U_{1} &= &\{\mathbf{x}\in V\mid x_{1}+2x_{4}=x_{2}+2x_{3}\},\\ + U_{2} &= &\{\mathbf{x}\in V\mid x_{1}=x_{2}+x_{3}+x_{4}\}.\ + \end{mathe} + + Zu bestimmen sind Basen für + $U_{1}$, + $U_{2}$, + $U_{1}\cap U_{2}$, + und + $U_{1}+U_{2}$. + Wir beachten vorab + + \begin{longmathe}[mc]{RCLCL} + \mathbf{x}\in U_{1} + &\Longleftrightarrow + &x_{1}-x_{2}-2x_{3}+2x_{4}=0 + &\Longleftrightarrow + &\underbrace{ + \begin{smatrix} +1&-1&-2&2\\ +\end{smatrix} + }_{=:A_{1}} + \mathbf{x}=\zerovector\\ + \mathbf{x}\in U_{2} + &\Longleftrightarrow + &x_{1}-x_{2}-x_{3}-x_{4}=0 + &\Longleftrightarrow + &\underbrace{ + \begin{smatrix} +1&-1&-1&-1\\ +\end{smatrix} + }_{=:A_{2}} + \mathbf{x}=\zerovector\\ + \mathbf{x}\in U_{1}\cap U_{2} + &\Longleftrightarrow + &\begin{array}[t]{0l} + x_{1}-x_{2}-2x_{3}+2x_{4} = 0\\ + \text{und}\,x_{1}-x_{2}-x_{3}-x_{4} = 0\\ + \end{array} + &\Longleftrightarrow + &\underbrace{ + \begin{smatrix} +1&-1&-2&2\\ +1&-1&-1&-1\\ +\end{smatrix} + }_{=:A_{3}} + \mathbf{x}=\zerovector\\ + \end{longmathe} + + für alle $\mathbf{x}\in V$. + Folglich ist $U_{1}$ die Menge der Nullvektoren von $A_{1}$, + $U_{2}$ die Menge der Nullvektoren von $A_{2}$, + $U_{1}\cap U_{2}$ die Menge der Nullvektoren von $A_{3}$. + Um diese zu bestimmen, bringen wir diese Matrizen in Zeilenstufenform, + und bestimmen mithilfe von \cite[Satz 5.3.8]{sinn2020} + eine Basis des Lösungsraums. + + \begin{algorithm}[\rtab][\rtab] + Zeilenstufenform für $A_{1}$: + + \begin{mathe}[mc]{rcl} + A_{1} &= &\begin{smatrix} +1&-1&-2&2\\ +\end{smatrix}\\ + \end{mathe} + + Darum sind $x_{2}$, $x_{3}$, $x_{4}$ frei und $x_{1}$ wird durch diese bestimmt. + Der Zeilenstufenform zufolge gilt + + \begin{mathe}[mc]{rcl} + x_{1} &= &x_{2}+2x_{3}-2x_{4}.\\ + \end{mathe} + + Mithilfe von \cite[Satz 5.3.8]{sinn2020} + erhalten wir eine Basis des Lösungsraums, + indem wir jeweils eine freie Unbekannte auf $1$ + und alle anderen auf $0$ setzen: + + \begin{mathe}[mc]{rcl} + x_{2}:=1,\,x_{3}:=0,\,x_{4}:=0 + &\Longrightarrow + &\mathbf{x}=\begin{svector}1\\1\\0\\0\\\end{svector},\\ + x_{2}:=0,\,x_{3}:=1,\,x_{4}:=0 + &\Longrightarrow + &\mathbf{x}=\begin{svector}2\\0\\1\\0\\\end{svector},\\ + x_{2}:=0,\,x_{3}:=0,\,x_{4}:=1 + &\Longrightarrow + &\mathbf{x}=\begin{svector}-2\\0\\0\\1\\\end{svector}.\\ + \end{mathe} + \end{algorithm} + + Darum gilt + + \begin{mathe}[mc]{rclcl} + U_{1} + &= &\{\mathbf{x}\in V\mid A_{1}\mathbf{x}=\zerovector\} + &= &\vectorspacespan\underbrace{ + \{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}2\\0\\1\\0\\\end{svector}, + \begin{svector}-2\\0\\0\\1\\\end{svector} + \} + }_{=:B}\\ + \end{mathe} + + und \fbox{$B_{1}$ bildet eine Basis für $U_{1}$}. + + \begin{algorithm}[\rtab][\rtab] + Zeilenstufenform für $A_{2}$: + + \begin{mathe}[mc]{rcl} + A_{2} &= &\begin{smatrix} +1&-1&-2&2\\ +\end{smatrix}\\ + \end{mathe} + + Darum sind $x_{2}$, $x_{3}$, $x_{4}$ frei und $x_{1}$ wird durch diese bestimmt. + Der Zeilenstufenform zufolge gilt + + \begin{mathe}[mc]{rcl} + x_{1} &= &x_{2}+x_{3}+x_{4}.\\ + \end{mathe} + + Mithilfe von \cite[Satz 5.3.8]{sinn2020} + erhalten wir eine Basis des Lösungsraums wie oben: + + \begin{mathe}[mc]{rcl} + x_{2}:=1,\,x_{3}:=0,\,x_{4}:=0 + &\Longrightarrow + &\mathbf{x}=\begin{svector}1\\1\\0\\0\\\end{svector},\\ + x_{2}:=0,\,x_{3}:=1,\,x_{4}:=0 + &\Longrightarrow + &\mathbf{x}=\begin{svector}1\\0\\1\\0\\\end{svector},\\ + x_{2}:=0,\,x_{3}:=0,\,x_{4}:=1 + &\Longrightarrow + &\mathbf{x}=\begin{svector}1\\0\\0\\1\\\end{svector}.\\ + \end{mathe} + \end{algorithm} + + Darum gilt + + \begin{mathe}[mc]{rclcl} + U_{2} + &= &\{\mathbf{x}\in V\mid A_{2}\mathbf{x}=\zerovector\} + &= &\vectorspacespan\underbrace{ + \{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}1\\0\\1\\0\\\end{svector}, + \begin{svector}1\\0\\0\\1\\\end{svector} + \} + }_{=:B_{2}}\\ + \end{mathe} + + und \fbox{$B_{2}$ bildet eine Basis für $U_{2}$}. + + \begin{algorithm}[\rtab][\rtab] + Zeilenstufenform für $A_{3}$ ($\text{Zeile}_{2} \rightsquigarrow \text{Zeile}_{2}-\text{Zeile}_{1}$): + + \begin{mathe}[mc]{rclcl} + A_{3} + &= &\begin{smatrix} +1&-1&-2&2\\ +1&-1&-1&-1\\ +\end{smatrix} + &\rightsquigarrow &\begin{smatrix} +1&-1&-2&2\\ +0&0&1&-3\\ +\end{smatrix}\\ + \end{mathe} + + Darum sind $x_{2}$, $x_{4}$, frei und $x_{1}$, $x_{3}$ werden durch diese bestimmt. + Der Zeilenstufenform zufolge gilt + + \begin{mathe}[mc]{rcl} + x_{3} &= &3x_{4}\\ + x_{1} &= &x_{2}+2x_{3}-2x_{4} + = x_{2}+2(3x_{4})-2x_{4} + = x_{2}+4x_{4}.\\ + \end{mathe} + + Mithilfe von \cite[Satz 5.3.8]{sinn2020} + erhalten wir eine Basis des Lösungsraums wie oben: + + \begin{mathe}[mc]{rcl} + x_{2}:=1,\,x_{4}:=0 + &\Longrightarrow + &\mathbf{x}=\begin{svector}1\\1\\0\\0\\\end{svector},\\ + x_{2}:=0,\,x_{4}:=1 + &\Longrightarrow + &\mathbf{x}=\begin{svector}4\\0\\3\\1\\\end{svector}.\\ + \end{mathe} + + gegeben. + \end{algorithm} + + Darum gilt + + \begin{mathe}[mc]{rclcl} + U_{1}\cap U_{2} + &= &\{\mathbf{x}\in V\mid A_{3}\mathbf{x}=\zerovector\} + &= &\vectorspacespan\underbrace{ + \{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}4\\0\\3\\1\\\end{svector} + \} + }_{=:B_{3}}\\ + \end{mathe} + + und \fbox{$B_{3}$ bildet eine Basis für $U_{1}\cap U_{2}$}. + + Es bleibt, nur noch eine Basis für $U_{1}+U_{2}$ zu bestimmen. + + \textbf{ANSATZ I.}\\ + Mithilfe der oben berechneten Basen für $U_{1}$, $U_{2}$, wissen wir + + \begin{mathe}[mc]{rcl} + U_{1}+U_{2} + &= &\vectorspacespan\{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}2\\0\\1\\0\\\end{svector}, + \begin{svector}-2\\0\\0\\1\\\end{svector} + \} + +\vectorspacespan\{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}1\\0\\1\\0\\\end{svector}, + \begin{svector}1\\0\\0\\1\\\end{svector} + \}\\ + &= &\vectorspacespan\{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}2\\0\\1\\0\\\end{svector}, + \begin{svector}-2\\0\\0\\1\\\end{svector} + \begin{svector}1\\0\\1\\0\\\end{svector}, + \begin{svector}1\\0\\0\\1\\\end{svector} + \}.\\ + \end{mathe} + + Wir haben nun ein Erzeugendensystem bestimmt. + Diese Menge muss auf eine maximale linear unabhängige reduziert werden, + um eine Basis daraus zu berechnen. + Um diese zu tun, reicht es aus, + die Vektoren in ein homogenes LGS zu überführen, + die Matrix auf Zeilenstufenform zu reduzieren, + um etwa durch die Spalten entsprechend den freien Unbekannten zu bestimmen, + welche Spalten linear abhängig sind. + + \begin{algorithm}[\rtab][\rtab] + Homogenes System: + + \begin{mathe}[mc]{rcl} + \begin{smatrix} +1&2&-2&1&1\\ +1&0&0&0&0\\ +0&1&0&1&0\\ +0&0&1&0&1\\ +\end{smatrix}\\ + \end{mathe} + + Zeilenoperation + ${Z_{2}\leftsquigarrow Z_{1}-Z_{2}}$ + anwenden: + + \begin{mathe}[mc]{rcl} + \begin{smatrix} +1&2&-2&1&1\\ +0&2&-2&1&1\\ +0&1&0&1&0\\ +0&0&1&0&1\\ +\end{smatrix}\\ + \end{mathe} + + Zeilenoperation + ${Z_{3}\leftsquigarrow 2\cdot Z_{3} - Z_{2}}$ + anwenden: + + \begin{mathe}[mc]{rcl} + \begin{smatrix} +1&2&-2&1&1\\ +0&2&-2&1&1\\ +0&0&2&1&-1\\ +0&0&1&0&1\\ +\end{smatrix}\\ + \end{mathe} + + Zeilenoperation + ${Z_{4}\leftsquigarrow Z_{3} - 2\cdot Z_{4}}$ + anwenden: + + \begin{mathe}[mc]{rcl} + \begin{smatrix} +1&2&-2&1&1\\ +0&2&-2&1&1\\ +0&0&2&1&-1\\ +0&0&0&1&-3\\ +\end{smatrix}\\ + \end{mathe} + + $\Longrightarrow$ nur $x_{5}$ frei. + \end{algorithm} + + Also hängt die 5. Spalte von Spalten 1--4 ab, welche der Zeilenstufenform zufolge linear unabhängig sind. + Folglich ist + + \begin{mathe}[mc]{c} + \left\{ + \begin{svector}1\\1\\0\\0\\\end{svector}, + \begin{svector}2\\0\\1\\0\\\end{svector}, + \begin{svector}-2\\0\\0\\1\\\end{svector} + \begin{svector}1\\0\\1\\0\\\end{svector} + \right\}\\ + \end{mathe} + + eine \fbox{Basis für $U_{1}+U_{2}$}. + + \textbf{ANSATZ II.}\\ + Mithilfe der Dimensionsformel (siehe \cite[Satz~5.4.3~(2)]{sinn2020}) wissen wir + + \begin{mathe}[mc]{rcl} + \dim(U_{1}+U_{2}) + &= &\dim(U_{1}) + \dim(U_{2}) - \dim(U_{1}\cap U_{2})\\ + &= &3+3-2 = 4 = \dim(V).\\ + \end{mathe} + + Da $V$ endlich dimensional ist, + und $\dim(U_{1}+U_{2})=\dim(V)$ + für den linearen Unterraum $U_{1}+U_{2}\subseteq V$ + gilt, + gilt $U_{1}+U_{2}=V=\reell^{4}$ + (Siehe \cite[Satz~5.4.3~(1)]{sinn2020}). + Darum können wir bspw. die \uline{kanonische Basis} für $\reell^{4}$ + + \begin{mathe}[mc]{c} + \left\{ + \begin{svector}1\\0\\0\\0\\\end{svector}, + \begin{svector}0\\1\\0\\0\\\end{svector}, + \begin{svector}0\\0\\1\\0\\\end{svector}, + \begin{svector}0\\0\\0\\1\\\end{svector} + \right\}\\ + \end{mathe} + + als Basis für $U_{1}+U_{2}$ verwenden. + + \textbf{Bemerkung.} In diesem letzten Teil hatten wir Glück. + Wenn sich $\dim(U_{1}+U_{2})0$, dann + + \begin{mathe}[mc]{rcl} + x^{k} + &= + &x^{k}-c_{k}^{-1}\cdot\zerovector\\ + &\eqcrefoverset{eq:1:\beweislabel}{=} + &x^{k}-c_{k}^{-1}\sum_{i=0}^{d}c_{i}(x-1)^{i}\\ + &= + &x^{k}-\sum_{i=0}^{k}c_{k}^{-1}c_{i}(x-1)^{i} + \quad\text{wegen Maximalität von $k$}\\ + &= + &\sum_{i=0}^{k-1} + -c_{k}^{-1}c_{i}(x-1)^{i} + +(x^{k}-(x-1)^{k}).\\ + \end{mathe} + + Mithilfe der binomischen Formel ist es leicht zu sehen, + dass der letzte Term ein Polynom vom Grade $