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\ranewthm{defn}{Definition}{\enndeOnNeutralSign}[X] \ranewthm{conv}{Konvention}{\enndeOnNeutralSign}[X] + \ranewthm{obs}{Beobachtung}{\enndeOnNeutralSign}[X] \ranewthm{e.g.}{Beipsiel}{\enndeOnNeutralSign}[X] \ranewthm{fact}{Fakt}{\enndeOnNeutralSign}[X] \ranewthm{rem}{Bemerkung}{\enndeOnNeutralSign}[X] @@ -7154,6 +7157,768 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve nähert als ${a\longrightarrow 0}$ in $\reell$, was zu erwarten ist. \end{enumerate} +%% ******************************************************************************** +%% FILE: body/uebung/ueb11.tex +%% ******************************************************************************** + +\setcounternach{chapter}{11} +\chapter[Woche 11]{Woche 11} + \label{ueb:11} + +\textbf{ACHTUNG.} +Diese Lösungen dienen \emph{nicht} als Musterlösungen sondern eher als Referenz. +Hier wird eingehender gearbeitet, als generell verlangt wird. +Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche vergleichen kann. + +%% AUFGABE 11-1 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 1]{} + \label{ueb:11:ex:1} +\let\sectionname\altsectionname + + \begin{enumerate}{\bfseries (a)} + %% AUFGABE 11-1(a) + \item + + Sei $A:=\begin{smatrix} + 1 &2 &3 &4\\ + 2 &5 &4 &11\\ + -1 &-2 &-2 &1\\ + 1 &3 &2 &13\\ + \end{smatrix}$, + eine ${m\times n}$-Matrix über dem Körper $\rtnl$, + mit $m=n=4$. + Um zu bestimmen, ob $A$ invertierbar ist, + bestimmen wir die Zeilenstufenform. + Dabei augmentieren wir $A$ mit der ${4\times 4}$-Identitätsmatrix, + um im Falle der Invertierbarkeit das Inverse von $A$ mit zu rechnen. + + \begin{longmathe}[mc]{RCL} + &\begin{matrix}{rrrr|rrrr} + 1 &2 &3 &4 &1 &0 &0 &0\\ + 2 &5 &4 &11 &0 &1 &0 &0\\ + -1 &-2 &-2 &1 &0 &0 &1 &0\\ + 1 &3 &2 &13 &0 &0 &0 &1\\ + \end{matrix}\\ + \xrightarrow{ + \substack{ + Z_{2}\mapsfrom Z_{2} - 2\cdot Z_{1}\\ + Z_{3}\mapsfrom Z_{3} + Z_{1}\\ + Z_{4}\mapsfrom Z_{4} - Z_{1}\\ + } + } + & + \begin{matrix}{rrrr|rrrr} + 1 &2 &3 &4 &1 &0 &0 &0\\ + 0 &1 &-2 &3 &-2 &1 &0 &0\\ + 0 &0 &1 &5 &1 &0 &1 &0\\ + 0 &1 &-1 &9 &-1 &0 &0 &1\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{1}\mapsfrom Z_{1} - 2\cdot Z_{2}\\ + Z_{4}\mapsfrom Z_{4} - Z_{2}\\ + } + } + & + \begin{matrix}{rrrr|rrrr} + 1 &0 &7 &-2 &5 &-2 &0 &0\\ + 0 &1 &-2 &3 &-2 &1 &0 &0\\ + 0 &0 &1 &5 &1 &0 &1 &0\\ + 0 &0 &1 &6 &1 &-1 &0 &1\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{1}\mapsfrom Z_{1} - 7\cdot Z_{3}\\ + Z_{2}\mapsfrom Z_{2} + 2\cdot Z_{3}\\ + Z_{4}\mapsfrom Z_{4} - Z_{3}\\ + } + } + & + \begin{matrix}{rrrr|rrrr} + 1 &0 &0 &-37 &-2 &-2 &-7 &0\\ + 0 &1 &0 &13 &0 &1 &2 &0\\ + 0 &0 &1 &5 &1 &0 &1 &0\\ + 0 &0 &0 &1 &0 &-1 &-1 &1\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{1}\mapsfrom Z_{1} + 37\cdot Z_{4}\\ + Z_{2}\mapsfrom Z_{2} - 13\cdot Z_{4}\\ + Z_{3}\mapsfrom Z_{3} - 5\cdot Z_{4}\\ + } + } + & + \begin{matrix}{rrrr|rrrr} + 1 &0 &0 &0 &-2 &-39 &-44 &37\\ + 0 &1 &0 &0 &0 &14 &15 &-13\\ + 0 &0 &1 &0 &1 &5 &6 &-5\\ + 0 &0 &0 &1 &0 &-1 &-1 &1\\ + \end{matrix} + \\ + \end{longmathe} + + Also gilt ${\rank(A)=4=m=n}$. + Darum ist $A$ invertierbar.\\ + Der o.\,s. Berechnung zufolge gilt insbesondere + \fbox{$A^{-1}=\begin{smatrix} + -2 &-39 &-44 &37\\ + 0 &14 &15 &-13\\ + 1 &5 &6 &-5\\ + 0 &-1 &-1 &1\\ + \end{smatrix}$}. + + %% AUFGABE 11-1(b) + \item + + Sei $A:=\begin{smatrix} + 0 &0 &1\\ + 1 &0 &0\\ + 0 &1 &0\\ + \end{smatrix}$. + \textbf{Zu bestimmen:} Eine Darstellung von $A$ als Produkt aus Elementarmatrizen des III. Typs. + \textbf{Ansatz:} + Wir bestimmen mittels Zeilenoperationen + Elementarmatrizen, $E_{1},E_{2},\ldots,E_{N}$, vom Typ III + so dass + ${E_{n}\cdot\ldots \cdot E_{2}\cdot E_{1}\cdot A=\onematrix}$, + wobei {$\onematrix = $ die ${3\times 3}$-Identitätsmatrix}, + und daraus erhalten wir die Darstellung, + ${A=E_{1}^{-1}\cdot E_{2}^{-1}\cdot\ldots\cdot E_{N}^{-1}}$. + Man beachte dabei, dass das Inverse einer Elementarmatrix III. Typs + wiederum eine Elementarmatrix III. Typs bleibt. + + Zeilenoperationen: + + \begin{mathe}[mc]{rclclc} + \begin{matrix}{rrr} + 0 &0 &1\\ + 1 &0 &0\\ + 0 &1 &0\\ + \end{matrix} + &\xrightarrow{E_{1}:=S_{1,2}(+1)} + & + \begin{matrix}{rrr} + 1 &0 &1\\ + 1 &0 &0\\ + 0 &1 &0\\ + \end{matrix} + &\xrightarrow{E_{2}:=S_{2,1}(-1)} + & + \begin{matrix}{rrr} + 1 &0 &1\\ + 0 &0 &-1\\ + 0 &1 &0\\ + \end{matrix} + &\cdots + \\ + \cdots + &\xrightarrow{E_{3}:=S_{1,2}(+1)} + & + \begin{matrix}{rrr} + 1 &0 &0\\ + 0 &0 &-1\\ + 0 &1 &0\\ + \end{matrix} + &\xrightarrow{E_{4}:=S_{2,3}(+1)} + & + \begin{matrix}{rrr} + 1 &0 &0\\ + 0 &1 &-1\\ + 0 &1 &0\\ + \end{matrix} + &\cdots + \\ + \cdots + &\xrightarrow{E_{5}:=S_{3,2}(-1)} + & + \begin{matrix}{rrr} + 1 &0 &0\\ + 0 &1 &-1\\ + 0 &0 &1\\ + \end{matrix} + &\xrightarrow{E_{6}:=S_{2,3}(+1)} + & + \begin{matrix}{rrr} + 1 &0 &0\\ + 0 &1 &0\\ + 0 &0 &1\\ + \end{matrix} + .\\ + \end{mathe} + + Also gilt ${E_{6}\cdot\ldots\cdot E_{2}\cdot E_{1}\cdot A=\onematrix}$. + Daraus erhalten wir die Darstellung: + + \begin{mathe}[mc]{rcl} + A &= &(E_{6}\cdot\ldots\cdot E_{2}\cdot E_{1})^{-1}\\ + &= &E_{1}^{-1}\cdot E_{2}^{-1}\cdot\ldots\cdot E_{6}^{-1} + \quad\text{(siehe \cite[Lemma~4.3.1(7)~od.~Lemma~6.3.2]{sinn2020})}\\ + &= &S_{1,2}(+1)^{-1}\cdot S_{2,1}(-1)^{-1}\cdot S_{1,2}(+1)^{-1}\cdot S_{2,3}(+1)^{-1}\cdot S_{3,2}(-1)^{-1}\cdot S_{2,3}(+1)^{-1}\\ + &= &\boxed{S_{1,2}(-1)\cdot S_{2,1}(+1)\cdot S_{1,2}(-1)\cdot S_{2,3}(-1)\cdot S_{3,2}(+1)\cdot S_{2,3}(-1)}.\\ + \end{mathe} + + \end{enumerate} + +%% AUFGABE 11-2 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\setcounternach{section}{2} +\section[Aufgabe 2]{} + \label{ueb:11:ex:2} +\let\sectionname\altsectionname + + Seien $m,n\in\ntrl$ und $K$ ein Körper und $A$ eine ${m\times n}$-Matrix. + Wir betrachten die kanonisch definierte lineare Abbildung, + ${\phi_{A}:K^{n}\to K^{m},\,x\mapsto Ax}$, + und wollen \cite[Korollar~6.3.15]{sinn2020} hier beweisen. + Zunächst aber machen wir ein paar allgemeine Beobachtungen. + + \begin{obs} + \makelabel{obs:1:ueb:11:ex:2} + Per Definition und laut \cite[Lemma~5.4.7]{sinn2020} + gilt $\rank(A):=\text{Zeilenrang}\textoverset{Lemm}{=}\text{Spaltenrang}$. + Somit automatisch $\rank(A)\leq m$ und $\rank(A)\leq n$. + Darum gilt stets $\rank(A)\leq\min\{m,n\}$. + \end{obs} + + \begin{obs} + \makelabel{obs:2:ueb:11:ex:2} + Seien $w_{1},w_{2},\ldots,w_{n}\in K^{m}$ die Spalten von $A$. + Dann gilt + + \begin{mathe}[mc]{rcl} + \eqtag[eq:1:beob:ueb:11:ex:2] + \range(\phi_{A}) + &\textoverset{Defn}{=} + &\{\phi_{A}(x)\mid x\in K^{n}\}\\ + &= &\{Ax\mid x\in K^{n}\}\\ + &= &\{\sum_{i=1}^{n}x_{i}w_{1}\mid x_{1},x_{2},\ldots,x_{n}\in K\}\\ + &= &\underbrace{ + \vectorspacespan\{w_{1},w_{2},\ldots,w_{n}\} + }_{\text{Spaltenraum von $A$}}.\\ + \end{mathe} + + Insbesondere gilt per des Ranges und \cite[Lemma 5.4.7]{sinn2020} + + \begin{mathe}[mc]{rcl} + \eqtag[eq:2:beob:ueb:11:ex:2] + \rank(A) + &\textoverset{Lemm}{=} + &\text{Spaltenrang}(A)\\ + &\textoverset{Defn}{=} + &\dim(\vectorspacespan\{w_{1},w_{2},\ldots,w_{n}\})\\ + &\eqcrefoverset{eq:1:beob:ueb:11:ex:2}{=} + &\dim(\range(\phi_{A})).\\ + \end{mathe} + + Darum gilt die Behauptung. + \end{obs} + + \begin{enumerate}{\bfseries (a)} + %% AUFGABE 11-2(a) + \item\voritemise + \begin{schattierteboxdunn} + \begin{claim*} + $\phi_{A}$ ist injektiv $\Leftrightarrow$ $\rank(A)\geq n$. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + Für diesen Beweis machen wir von den Berechnungen in \cref{obs:2:ueb:11:ex:2} Gebrauch. + Wir inspizieren den linearen Unterraum + + \begin{mathe}[mc]{rcccl} + W &:= &\range(\phi_{A}) &= &\vectorspacespan\{w_{1},w_{2},\ldots,w_{n}\}\subseteq K^{m}\\ + \end{mathe} + + Offensichtlich ist $\cal{W}:=(w_{1},w_{2},\ldots,w_{n})$ ein Erzeugendensystem für $W$, sodass $\dim(W)\leq n$ gilt. + Des Weiteren gelten die Implikationen: + + \begin{mathe}[mc]{rcl} + \eqtag[eq:1:ueb:11:ex:2:a] + \dim(W)\geq n + &\Longrightarrow + &\dim(W)=n\quad\text{(da $\dim(W)\leq n$ sowieso)}\\ + &\Longrightarrow + &(w_{1},w_{2},\ldots,w_{n})\,\text{linear unabhängig}\\ + &&\quad\text{laut \cite[Korollar~5.4.4]{sinn2020}}\\ + &&\quad\text{und da $\cal{W}$ ein Erzsys. aus $n=\dim(W)$ Vektoren}\\ + &\Longrightarrow + &(w_{1},w_{2},\ldots,w_{n})\,\text{eine Basis für $W$}\\ + &&\quad\text{da $\cal{W}$ sowieso ein Erzeugendensystem für $W$ ist}\\ + &\Longrightarrow + &\dim(W)=n\\ + &\Longrightarrow + &\dim(W)\geq n\quad\text{(offensichtlich)}.\\ + \end{mathe} + + Da nun + $\rank(A)\eqcrefoverset{eq:2:beob:ueb:11:ex:2}{=}\dim(\range(\phi_{A}))=\dim(W)$, + erhalten wir + + \begin{longmathe}[mc]{RCL} + \rank(A)\geq n + &\Longleftrightarrow + &\dim(W)\geq n\\ + &\eqcrefoverset{eq:1:ueb:11:ex:2:a}{\Longleftrightarrow} + &(w_{1},w_{2},\ldots,w_{n})\,\text{linear unabhängig}\\ + &\textoverset{Defn}{\Longleftrightarrow} + &\forall{c_{1},c_{2},\ldots,c_{n}\in K:~} + \big( + \sum_{i=1}^{n}c_{i}w_{i}=\zerovector + \Rightarrow + c_{1},c_{2},\ldots,c_{n}=0_{K} + \big)\\ + &\Longleftrightarrow + &\forall{x\in K^{n}:~} + \big( + \underbrace{ + \sum_{i=1}^{n}x_{i}w_{i} + }_{ + =Ax=\phi_{A}(x) + } + =\zerovector + \Rightarrow + x=\zerovector + \big)\\ + &\Longleftrightarrow + &\forall{x\in K^{n}:~} + \big( + x\in\ker(\phi_{A})\Rightarrow x\in\{\zerovector\} + \big)\\ + &\Longleftrightarrow + &\ker(\phi_{A})\subseteq\{\zerovector\}\\ + &\textoverset{($\ast$)}{\Longleftrightarrow} + &\ker(\phi_{A})=\{\zerovector\}\\ + &\Longleftrightarrow + &\phi_{A}\,\text{injektiv} + \quad\text{(siehe \cite[Lemma~6.1.4(1)]{sinn2020})}.\\ + \end{longmathe} + + Hierbei gilt ($\ast$), + weil $\{\zerovector\}\subseteq\ker(\phi_{A})$ immer gilt, + weil wiederum $\phi_{A}(\zerovector)=\zerovector$ stets gilt (siehe \cite[Lemma~6.1.2]{sinn2020}). + Den o.\,s. doppelten Implikationen zufolge gilt die Behauptung. + \end{proof} + + %% AUFGABE 11-2(b) + \item\voritemise + \begin{schattierteboxdunn} + \begin{claim*} + $\phi_{A}$ ist surjektiv $\Leftrightarrow$ $\rank(A)\geq m$. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + Da $\phi_{A}$ linear ist, + ist $U:=\range(\phi_{A})$ ein linearer Unterraum von $K^{m}$ + (siehe \cite[Lemma~6.1.3]{sinn2020}). + Laut \cite[Satz~5.4.3(1)]{sinn2020}, + da $U\subseteq K^{m}$ ein linearer Unterraum eines endlichdimensionalen Vektorraumes ist, + gilt $U=K^{m}$ $\Leftrightarrow$ $\dim(U)=\dim(K^{m})(=m)$. + Darum gilt + + \begin{mathe}[mc]{rcl} + \phi_{A}~\text{surjektiv} + &\textoverset{Defn}{\Longleftrightarrow} + &\range(\phi_{A})=K^{m}\\ + &\Longleftrightarrow + &U=K^{m}\\ + &\Longleftrightarrow + &\dim(U)=m\\ + &\Longleftrightarrow + &\dim(\range(\phi_{A}))=m\\ + &\eqcrefoverset{eq:2:beob:ueb:11:ex:2}{\Longleftrightarrow} + &\rank(A)=m.\\ + \end{mathe} + + Da laut \Cref{obs:1:ueb:11:ex:2} $\rank(A)\leq m$ stets gilt, + wissen wir außerdem, dass $\rank(A)=m$ $\Leftrightarrow$ $\rank(A)\geq m$. + Laut der o.\,s. Implikationen gilt also + $\phi_{A}$ ist surjektiv + $\Leftrightarrow$ $\rank(A)=m$ + $\Leftrightarrow$ $\rank(A)\geq m$. + Also gilt die Behauptung. + \end{proof} + + %% AUFGABE 11-2(c) + \item\voritemise + \begin{schattierteboxdunn} + \begin{claim*} + $\phi_{A}$ ist bijektiv $\Leftrightarrow$ $\rank(A)=m=n$. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + \hinRichtung + Angenommen, $\phi_{A}$ sei bijektiv. + Das heißt, $\phi_{A}$ ist injektiv und surjektiv. + Aus den anderen Teilaufgaben folgt + $\rank(A)\geq\max\{m,n\}$. + Darum gilt $\max\{m,n\}\rank(A)\leq\min\{m,n\}$ + (siehe \Cref{obs:1:ueb:11:ex:2}). + Folglich gelten $m=n$ und $\rank(A)=m=n$. + + \herRichtung + Aus $\rank(A)=m=n$ + folgen $\rank(A)\geq m$ und $\rank(A)\geq n$. + Laut der anderen Teilaufgaben ist $\phi_{A}$ + somit injektiv und surjektiv, + also bijektiv. + \end{proof} + \end{enumerate} + +%% AUFGABE 11-3 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\setcounternach{section}{3} +\section[Aufgabe 3]{} + \label{ueb:11:ex:3} +\let\sectionname\altsectionname + + \begin{enumerate}{\bfseries (a)} + %% AUFGABE 11-3(a) + \item\voritemise + \begin{schattierteboxdunn} + \begin{claim*} + Seien $n\in\ntrl$ und $K$ ein Körper. + Seien $A,B$ ${n\times n}$-Matrizen über $K$. + Falls $AB=\onematrix$, so gilt $BA=\onematrix$. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + Aus $AB=\onematrix$, folgt + + \begin{mathe}[mc]{rcccl} + \phi_{B}(x)=\phi_{B}(x') + &\Longrightarrow + &Bx=Bx' + &\Longrightarrow + &x=Ix=ABx=ABx'=Ix'=x'\\ + \end{mathe} + + für alle $x,x'\in K^{n}$. + Darum ist $\phi_{B}$ injektiv. + Da ${\phi_{B}:K^{n}\to K^{n}}$ eine injektive lineare Abbildung + zwischen zwei endlich dimensionalen Räumen der gleichen Dimension, + gilt laut \cite[Korollar~6.1.11]{sinn2020}, + dass $\phi_{B}$ auch surjektiv und somit bijektiv ist. + Darum existiert eine eindeutige (lineare) Abbildung, + ${\rho:K^{n}\to K^{n}}$, + so dass ($\dagger$) $\rho\circ\phi_{B}=\phi_{B}\circ\rho=\id$. + Daraus folgt, dass $\phi_{A}=\rho$, da + + \begin{mathe}[mc]{rclclclclcl} + \phi_{A} + &= &\phi_{A}\circ\id + &\overset{(\dagger)}{=} + &\phi_{A}\circ(\phi_{B}\circ\rho) + &= &(\underbrace{ + \phi_{A}\circ\phi_{B} + }_{=\phi_{AB}=\phi_{\onematrix}=\id} + )\circ\rho + &= &\id\circ\rho + &= &\rho. + \end{mathe} + + Darum gilt $\phi_{BA}=\phi_{B}\circ\phi_{A}=\phi_{B}\circ\rho\overset{(\dagger)}{=}\id=\phi_{\onematrix}$, + woraus sich $BA=\onematrix$ ergibt. + \end{proof} + + %% AUFGABE 11-3(b) + \item\voritemise + \begin{schattierteboxdunn} + \begin{claim*} + Seien $r,n\in\ntrl$ und $K$ ein Körper. + Seien $A$ eine ${n\times n}$-Matrix über $K$ mit $A^{r}=\zeromatrix$. + Dann ist $\onematrix-A$ invertierbar. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + Laut Teil (a) reicht es aus, eine ${n\times n}$-Matrix, $B$, zu finden, + so dass $(\onematrix-A)B=\onematrix$. + Wir untersuchen $B:=(\onematrix-A)\sum_{i=0}^{r-1}A^{i}$, + wobei wir hier $A^{0}:=\onematrix$ festlegen.\footnote{ + Diese Definition ist nicht willkürlich, + sondern zielführend, um algebraische Identitäten + wie etwa das Additionsgesetz für Exponenten + zu garantieren. + } + Es gilt + + \begin{longmathe}[mc]{RCL} + AB &= &(\onematrix-A)\sum_{i=0}^{r-1}A^{i}\\ + &= &\sum_{i=0}^{r-1}A^{i}-A\sum_{i=0}^{r-1}A^{i}\\ + &= &\sum_{i=0}^{r-1}A^{i}-\sum_{i=0}^{r-1}AA^{i}\\ + &= &\sum_{i=0}^{r-1}A^{i}-\sum_{i=0}^{r-1}A^{i+1}\\ + &= &\sum_{i=0}^{r-1}A^{i}-\sum_{i=1}^{r}A^{i}\\ + &= &A^{0}-A^{r} + \quad\text{(Teleskopsumme)}\\ + &= &\onematrix-\zeromatrix + \quad\text{(per Voraussetzung)}.\\ + \end{longmathe} + + Darum gilt $(\onematrix-A)B=\onematrix$ + und damit laut Teil (a) $B(\onematrix-A)=\onematrix$. + Insbesondere ist $\onematrix-A$ invertierbar, und zwar durch $B$. + \end{proof} + \end{enumerate} + +%% BONUSAUFGABE 11-4 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Bonusaufgabe} +\setcounternach{section}{4} +\section[Bonusaufgabe 4]{} + \label{ueb:11:ex:bonus:4} +\let\sectionname\altsectionname + + \begin{enumerate}{\bfseries (a)} + %% BONUSAUFGABE 11-4(a) + \item + $f:\reell^{4}\to\reell^{2},\,(x_{1},x_{2},x_{3},x_{4})\mapsto(x_{2},x_{4})$ + über dem Körper $\reell$ + ist \fbox{linear}, + da $f=\phi_{A}$ mit $A=\begin{smatrix} + 0 &1 &0 &0\\ + 0 &0 &0 &1\\ + \end{smatrix}$. + %% BONUSAUFGABE 11-4(b) + \item + $f:\kmplx^{3}\to\kmplx^{3},\,(x_{1},x_{2},x_{3})\mapsto(x_{2},x_{4})$ + über dem Körper $\kmplx$ + ist \fbox{linear}, + da $f=\phi_{A}$ mit $A=\begin{smatrix} + 2 &0 &0 &\imageinh\\ + -1 &0 &0 &0\\ + 0 &0 &3-2\imageinh &1\\ + \end{smatrix}$. + %% BONUSAUFGABE 11-4(c) + \item + $f:\rtnl^{2}\to\rtnl^{3},\,(x_{1},x_{2})\mapsto(x_{1}+x_{2},1,x_{1})$ + über dem Körper $\rtnl$ + ist \fbox{nicht linear}, + weil $f(\zerovector)=(0,1,0)\neq\zerovector$, + während $f(\zerovector)=\zerovector$ + für lineare Abbildungen + gelten muss (siehe \cite[Lemma~6.1.2]{sinn2020}). + %% BONUSAUFGABE 11-4(d) + \item + $f:\reell\to\reell,\,x\mapsto\sqrt{2}\cdot x$ + über dem Körper $\reell$ + ist \fbox{linear}, + da $f$ hier nichts anders als Skalarmultiplikation ist, + und Skalarmultiplikation ist linear. + %% BONUSAUFGABE 11-4(e) + \item + $f:\reell\to\reell,\,x\mapsto x^{2}$ + über dem Körper $\reell$ + ist \fbox{nicht linear}, + da bspw. $f(3)=9\neq 3\cdot 1=3\cdot f(1)$, + d.\,h., das Homogenitätsaxiom ist offensichtlich verletzt. + %% BONUSAUFGABE 11-4(f) + \item + $f:\mathbb{F}_{2}\to\mathbb{F}_{2},\,x\mapsto x^{2}$ + über dem Körper $\mathbb{F}_{2}$ + ist gleich der Identitätsabbbildung, + da $x^{2}=x$ für alle $x\in\mathbb{F}_{2}$. + Damit ist $f$ trivialerweise \fbox{linear}. + %% BONUSAUFGABE 11-4(g) + \item + $f:\rtnl^{2}\to\rtnl^{2},\,(x_{1},x_{2})\mapsto(x_{1}x_{2},x_{1}+x_{2})$ + über dem Körper $\reell$ + ist \fbox{nicht linear}, + da bspw. $f(3,3)=(9,6)\neq 3\cdot(1,2)=3\cdot f(1,1)$, + d.\,h., das Homogenitätsaxiom ist offensichtlich verletzt. + \end{enumerate} + +%% BONUSAUFGABE 11-5 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Bonusaufgabe} +\setcounternach{section}{5} +\section[Bonusaufgabe 5]{} + \label{ueb:11:ex:bonus:5} +\let\sectionname\altsectionname + + Sei $A:=\begin{smatrix} + 11 &-5 &4 &6\\ + -15 &7 &-6 &-8\\ + 2 &-1 &1 &1\\ + \end{smatrix}$, + eine ${m\times n}$-Matrix über dem Körper $\rtnl$, + mit $m=n=4$. + Unser \textbf{Ziel} besteht darin, + invertierbare Matrizen, + $P\in\rtnl^{m\times m}$ + und + $Q\in\rtnl^{n\times n}$, + zu bestimmen, + so dass $P\cdot A\cdot Q=\begin{smatrix} + \einser_{r} &0\\ + 0 &0\\ + \end{smatrix}$ wie in \cite[Satz~6.3.10]{sinn2020}. + Um eine Möglichkeit für $P$ zu bestimmen, + berechnen wir die Zeilenstufenform + der augmentieren Matrix $\left(A \vert \onematrix\right)$, + und entnehmen aus der rechten Hälfte $P$. + + \begin{longmathe}[mc]{RCL} + &\begin{matrix}{rrrr|rrr} + 11 &-5 &4 &6 &1 &0 &0\\ + -15 &7 &-6 &-8 &0 &1 &0\\ + 2 &-1 &1 &1 &0 &0 &1\\ + \end{matrix}\\ + \xrightarrow{ + \substack{ + Z_{2}\mapsfrom 15\cdot Z_{1} + 11\cdot Z_{2}\\ + Z_{3}\mapsfrom 2\cdot Z_{1} - 11\cdot Z_{3}\\ + } + } + & + \begin{matrix}{rrrr|rrr} + 11 &-5 &4 &6 &1 &0 &0\\ + 0 &2 &-6 &2 &15 &11 &0\\ + 0 &1 &-3 &1 &2 &0 &-11\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{1}\mapsfrom 5\cdot Z_{2} + 2\cdot Z_{1}\\ + Z_{4}\mapsfrom Z_{2} - 2\cdot Z_{3}\\ + } + } + & + \begin{matrix}{rrrr|rrr} + 22 &0 &-22 &22 &77 &55 &0\\ + 0 &2 &-6 &2 &15 &11 &0\\ + 0 &0 &0 &0 &11 &11 &22\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{1}\mapsfrom Z_{1}:11\\ + Z_{3}\mapsfrom Z_{3}:11\\ + } + } + & + \begin{matrix}{rrrr|rrr} + 2 &0 &-2 &2 &7 &5 &0\\ + 0 &2 &-6 &2 &15 &11 &0\\ + 0 &0 &0 &0 &1 &1 &2\\ + \end{matrix} + \\ + \end{longmathe} + + Setze also \fbox{$P:=\begin{smatrix} + 7 &5 &0\\ + 15 &11 &0\\ + 1 &1 &2\\ + \end{smatrix}$}. + Unter dieser Wahl gilt $P\cdot A=$ linke Hälfte der letzten Matrix in der o.\,s. Berechnung. + + Um nun $Q$ zu bestimmen, führen wir den gleichen Ansatz auf $(P\cdot A)^{T}$ aus, + bis die erwünschte (transponierte) Form erreicht ist. + D.\,h., wir starten jetzt mit $\left((P\cdot A)^{T}\vert \onematrix\right)$, + reduzieren, und zum Schluss entnehmen wir $Q$ als die transponierte rechte Hälfte + des Resultates: + + \begin{longmathe}[mc]{RCL} + &\begin{matrix}{rrr|rrrr} + 2 &0 &0 &1 &0 &0 &0\\ + 0 &2 &0 &0 &1 &0 &0\\ + -2 &-6 &0 &0 &0 &1 &0\\ + 2 &2 &0 &0 &0 &0 &1\\ + \end{matrix}\\ + \xrightarrow{ + \substack{ + Z_{3}\mapsfrom Z_{3} + Z_{1}\\ + Z_{4}\mapsfrom Z_{4} - Z_{1}\\ + } + } + & + \begin{matrix}{rrr|rrrr} + 2 &0 &0 &1 &0 &0 &0\\ + 0 &2 &0 &0 &1 &0 &0\\ + 0 &-6 &0 &1 &0 &1 &0\\ + 0 &2 &0 &-1 &0 &0 &1\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{3}\mapsfrom Z_{3} + 3\cdot Z_{2}\\ + Z_{4}\mapsfrom Z_{4} - Z_{2}\\ + } + } + & + \begin{matrix}{rrr|rrrr} + 2 &0 &0 &1 &0 &0 &0\\ + 0 &2 &0 &0 &1 &0 &0\\ + 0 &0 &0 &1 &3 &1 &0\\ + 0 &0 &0 &-1 &-1 &0 &1\\ + \end{matrix} + \\ + \xrightarrow{ + \substack{ + Z_{1}\mapsfrom Z_{1}:2\\ + Z_{2}\mapsfrom Z_{2}:2\\ + } + } + & + \begin{matrix}{rrr|rrrr} + 1 &0 &0 &1/2 &0 &0 &0\\ + 0 &1 &0 &0 &1/2 &0 &0\\ + 0 &0 &0 &1 &3 &1 &0\\ + 0 &0 &0 &-1 &-1 &0 &1\\ + \end{matrix} + \\ + \end{longmathe} + + Setze also \fbox{$Q:=\begin{smatrix} + 1/2 &0 &1 &-1\\ + 0 &1/2 &3 &-1\\ + 0 &0 &1 &0\\ + 0 &0 &0 &1\\ + \end{smatrix}$}. + Der o.\,s. Berechnung zur Folge gilt + $Q^{T}\cdot(P\cdot A)^{T}=\begin{smatrix} + \einser_{r} &0\\ + 0 &0\\ + \end{smatrix}$ + und damit sollte + $P\cdot A\cdot Q + =\begin{smatrix} + \einser_{r} &0\\ + 0 &0\\ + \end{smatrix}^{T} + =\begin{smatrix} + \einser_{r} &0\\ + 0 &0\\ + \end{smatrix}$ + gelten. + Wir prüfen dies und erhalten: + + \begin{mathe}[mc]{rcl} + P\cdot A\cdot Q + &= &\begin{matrix}{rrrr} + 1 &0 &0 &0\\ + 0 &1 &0 &0\\ + 0 &0 &0 &0\\ + \end{matrix}.\\ + \end{mathe} + + Darum sind $P,Q$ in der Tat passende invertierbare Matrizen, + so dass $P\cdot A\cdot Q$ der erwünschten Form ist. + \setcounternach{part}{2} \part{Selbstkontrollenaufgaben}