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zmR5k3+;uj>p_825YUrz*g57LrLCShnW@t(#yMA`DrP%z>yfTIBs*DdHE~9Doov^$< zS%8(4-J+Orlgjn446*=)>!QrorzP8@_DOhX(P7p79&R}#?S`ioFtI6FW0f^9c|+s% z8jXseY4S#KO!<(&z}aI0xWgX0^k3Nj7jjb+L*>d>K462C$!&XBn?7LvKS&``8-oz6 z)A=po!tetaT<jV2Jb^omlKnc#^sp<=afcVgapLIZ$cc>?!LcY0(ofx=_Fzh~>*RS+ ztyV&izgYctcg#Uu2PQyeLU`fJu<VJHey$fJ3s}Pcsc2??7QA4J4TCeYH+6P#GBdLK i59VNO4d?7)<mBSvWM&S>&ceyc$puG2A+9I^_kRE*EEXaF diff --git a/docs/loesungen.tex b/docs/loesungen.tex index 8e44f40..b1f6b2f 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -57,6 +57,8 @@ %% | %% — body/uebung/ueb6.tex; %% | +%% — body/uebung/ueb7.tex; +%% | %% — body/ska/ska4.tex; %% | %% — body/ska/ska5.tex; @@ -72,6 +74,8 @@ %% — body/quizzes/quiz4.tex; %% | %% — body/quizzes/quiz5.tex; +%% | +%% — body/quizzes/quiz6.tex; %% | %% — back/index.tex; %% | @@ -1334,6 +1338,7 @@ \def\vectorspacespan{\mathop{\text{\upshape Lin}}} \def\dim{\mathop{\text{\upshape dim}}} +\def\rank{\mathop{\text{\upshape Rank}}} \def\onematrix{\text{\upshape\bfseries I}} \def\zeromatrix{\text{\upshape\bfseries 0}} \def\zerovector{\text{\upshape\bfseries 0}} @@ -1341,6 +1346,7 @@ \def\graph{\mathop{\text{\textup Gph}}} \def\domain{\mathop{\text{\textup dom}}} \def\range{\mathop{\text{\textup ran}}} +\def\functionspace{\mathop{\text{\textup Abb}}} \def\id{\text{\textup id}} \def\modfn{\mathop{\text{\textup mod}}} \def\divides{\mathbin{\mid}} @@ -4508,8 +4514,12 @@ Um \Cref{satz:1:ueb:6:ex:3} zu beweisen, brauchen wir zunächst folgendes Zwisch für alle $(a,b),(a',b'),(a'',b'')\in F$, ist dies erfüllt. - Anhand der o.\,s. Erkenntnisse darüber, welchen Axiomen $(F,+,\cdot)$ bereits genügt, - erhalten wir, dass $(F,+,\cdot)$ genau dann ein Körper, wenn in $(F,\cdot)$ jedes Element ein Inverses hat. + Darum erfüllt $(F,+,\cdot)$ jedes Axiom eines Körpers, + evtl. bis auf das Axiom für multiplikative Inverse. + Darum gilt: + $(F,+,\cdot)$ bildet einen Körper + $\Leftrightarrow$ + jedes Element in $F\ohne\{0_{F}\}$ hat ein multiplikatives Inverses. \end{proof} \end{einzug} @@ -4534,7 +4544,7 @@ Jetzt können wir uns dem Beweis von \Cref{satz:1:ueb:6:ex:3} widmen } \herRichtung - Angenommen, $a^{2}+b^{2}\neq 0$ für alle $(a,b)\in F\ohne\{0_{F}\}$. + Angenommen, $a^{2}+b^{2}\neq 0$ für alle $(a,b)\in F\ohne\{0_{F}\}$.\\ Sei $(a,b)\in F\ohne\{0_{F}\}$ beliebig. \textbf{Zu zeigen:} $(a,b)$ sei innerhalb $F$ invertierbar.\\ Per Annahme gilt nun $r:=a^{2}+b^{2}\neq 0$ und somit ist $r$ innerhalb $K$ invertierbar. @@ -4579,10 +4589,601 @@ Jetzt können wir uns dem Beweis von \Cref{satz:1:ueb:6:ex:3} widmen woraus sich $a^{2}+b^{2}\neq 0$ ergibt. \end{proof} -\setcounternach{part}{2} -\part{Selbstkontrollenaufgaben} +%% ******************************************************************************** +%% FILE: body/uebung/ueb7.tex +%% ******************************************************************************** - \def\chaptername{SKA Blatt} +\setcounternach{chapter}{7} +\chapter[Woche 7]{Woche 7} + \label{ueb:7} + +\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 7-1 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 1]{} + \label{ueb:7:ex:1} +\let\sectionname\altsectionname + +Betrachte den Vektorraum $V:=\functionspace(\reell,\reell)$ über dem Körper $\reell$. +Wir bezeichnen mit $0(\cdot)$ die konstante Funktion, die überall gleich $0$ ist. +Dies ist zufälligerweise auch das Nullelement von $V$. + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 7-1a + \item + \begin{claim*} + Seien $a,b\in\reell$ beliebig. + Dann \fbox{ist} $U_{1}:=\{f\in V\mid f(a)=f(b)\}$ ein Untervektorraum (»linearer Unterraum«). + \end{claim*} + + \begin{proof} + Wir gehen die Axiome aus der Definition durch: + + \begin{kompaktitem}[\rtab][\rtab] + \item[{\bfseries (NL)}] + Offensichtlich gilt $0(\cdot)\in U_{1}$, da diese Funktion überall und damit insbesondere auf $\{a,b\}$ gleich ist. + Also, $U_{1}\neq\leer$. + + \item[{\bfseries (LK)}] + Seien $\alpha,\beta\in\reell$ und $f,g\in U_{1}$. + \textbf{Zu zeigen:} $\alpha\cdot f+\beta\cdot g\in U_{1}$\\ + Es gilt + + \begin{mathe}[mc]{rcl} + (\alpha\cdot f+\beta\cdot g)(a) + &= &\alpha\cdot f(a)+\beta\cdot g(a)\\ + &\overset{(\ast)}{=} + &\alpha\cdot f(b)+\beta\cdot g(b)\\ + &= &(\alpha\cdot f+\beta\cdot g)(b)\\ + \end{mathe} + + Hier gilt $(\ast)$, weil $f,g\in U_{1}$.\\ + Darum gilt per Konstruktion, dass $\alpha\cdot f+\beta\cdot g\in U_{1}$. + \end{kompaktitem} + + Darum bildet $U_{1}$ einen Untervektorraum. + \end{proof} + %% AUFGABE 7-1b + \item + \begin{claim*} + Seien $a,b\in\reell$ beliebig. + Dann ist $U_{2}:=\{f\in V\mid f(a)=f(b)=1\}$ genau dann ein Untervektorraum, wenn $a=b$. + \end{claim*} + + \begin{proof} + \herRichtung + Falls $a=b$, + dann gilt offensichtlich $f(a)=f(b)$ für alle $f\in V$, + sodass $U_{2}=V$, + woraus sich trivialerweise ergibt, + dass $U_{2}$ ein Untervektorraum ist. + + \hinRichtung + Offensichtlich ist der Nullvektor, $0(\cdot)$, nicht in $U_{2}$. + Darum kann $U_{2}$ kein Untervektorraum sein. + (Siehe \cite[Lemma~5.1.3]{sinn2020}.) + \end{proof} + + Alternativ für die $\Rightarrow$-Richtung kann man folgendermaßen argumentieren: + die konstante Funktion $1(\cdot)$ liegt in $U_{2}$, + aber $0\cdot 1(\cdot)\notin U_{2}$, + sodass $U_{2}$ nicht unter Skalarmultiplikation stabil ist. + %% AUFGABE 7-1c + \item + \begin{claim*} + Die Teilmenge $U_{3}:=\{f\in V\mid f\,\text{injektiv}\}$ \fbox{ist kein Untervektorraum} von $V$. + \end{claim*} + + \begin{proof} + Offensichtlich ist der Nullvektor, $0(\cdot)$, nicht in $U_{3}$, + weil diese konstante Funktion nicht injektiv ist. + Darum kann $U_{3}$ kein Untervektorraum sein. + (Siehe \cite[Lemma~5.1.3]{sinn2020}.) + \end{proof} + %% AUFGABE 7-1d + \item + \begin{claim*} + Die Teilmenge $U_{4}:=\{f\in V\mid f\,\text{surjektiv}\}$ \fbox{ist kein Untervektorraum} von $V$. + \end{claim*} + + \begin{proof} + Offensichtlich ist der Nullvektor, $0(\cdot)$, nicht in $U_{4}$, + weil die konstante Funktion nicht surjektiv ist. + Darum kann $U_{4}$ kein Untervektorraum sein. + (Siehe \cite[Lemma~5.1.3]{sinn2020}.) + \end{proof} +\end{enumerate} + +%% AUFGABE 7-2 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 2]{} + \label{ueb:7:ex:2} +\let\sectionname\altsectionname + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 7-2a + \item + \begin{claim*} + Sei $K=\rtnl$. Dann sind + ${\mathbf{v}_{1}=\begin{svector}1\\2\\2\\\end{svector}}$, + ${\mathbf{v}_{2}=\begin{svector}3\\2\\1\\\end{svector}}$, und + ${\mathbf{v}_{3}=\begin{svector}2\\1\\-1\\\end{svector}}$ + über $K$ \fbox{linear unabhängig}. + \end{claim*} + + \begin{proof} + Es reicht aus, den (Zeilen)rang der Matrix + + \begin{mathe}[mc]{rcl} + A &:= &\begin{matrix}{ccc} +1 &3 &2\\ +2 &2 &1\\ +2 &1 &-1\\ +\end{matrix} + \end{mathe} + + zu untersuchen. Wir berechnen + + \begin{algorithm}[\rtab][\rtab] + Reduktion der Matrix, $A$, mittels des Gaußverfahrens:\\ + Zeilentransformationen + ${Z_{2} \leftsquigarrow 3\cdot Z_{1}-Z_{2}}$ + und + ${Z_{3} \leftsquigarrow 2\cdot Z_{1}-Z_{3}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{matrix}{ccc} +1 &2 &2\\ +0 &4 &5\\ +0 &3 &5\\ +\end{matrix}\\ + \end{mathe} + + Zeilentransformation + ${Z_{3} \leftsquigarrow 4\cdot Z_{3}-3\cdot Z_{2}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{matrix}{ccc} +1 &2 &2\\ +0 &4 &5\\ +0 &0 &5\\ +\end{matrix}\\ + \end{mathe} + \end{algorithm} + + Der Zeilenstufenform entnimmt man, $\rank(A)=3$. + Darum sind alle $3$ Vektoren, + $\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\}$, + linear unabhängig. + \end{proof} + + %% AUFGABE 7-2b + \item + \begin{claim*} + Sei $K=\mathbb{F}_{5}$. Dann sind + ${\mathbf{v}_{1}=\begin{svector}1\\2\\2\\\end{svector}}$, + ${\mathbf{v}_{2}=\begin{svector}3\\2\\1\\\end{svector}}$, und + ${\mathbf{v}_{3}=\begin{svector}2\\1\\4\\\end{svector}}$ + über $K$ \fbox{nicht linear unabhängig}. + \end{claim*} + + \begin{proof} + Es reicht aus, den (Zeilen)rang der Matrix + + \begin{mathe}[mc]{rcl} + A &:= &\begin{matrix}{ccc} +1 &3 &2\\ +2 &2 &1\\ +2 &1 &4\\ +\end{matrix} + \end{mathe} + + zu untersuchen. Wir berechnen + + \begin{algorithm}[\rtab][\rtab] + Reduktion der Matrix, $A$, mittels des Gaußverfahrens:\footnote{ + Wir achten hier besonders darauf, + niemals mit einem Vielfach von $5$ zu multiplizieren! + }\\ + Zeilentransformationen + ${Z_{2} \leftsquigarrow 3\cdot Z_{1}-Z_{2}}$ + und + ${Z_{3} \leftsquigarrow 2\cdot Z_{1}-Z_{3}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{matrix}{ccl} +1 &2 &2\\ +0 &4 &5(=0)\\ +0 &3 &0\\ +\end{matrix}\\ + \end{mathe} + + Zeilentransformation + ${Z_{3} \leftsquigarrow 4\cdot Z_{3}-3\cdot Z_{2}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{matrix}{ccc} +1 &2 &2\\ +0 &4 &0\\ +0 &0 &0\\ +\end{matrix}\\ + \end{mathe} + \end{algorithm} + + Der Zeilenstufenform entnimmt man, $\rank(A)=2$. + Darum sind nur $2$ der Vektoren, und zwar + $\{\mathbf{v}_{1},\mathbf{v}_{2}\}$, + linear unabhängig. + Der Vektor, $\mathbf{v}_{3}$, hingegen, hängt linear von diesen ab. + \end{proof} + + %% AUFGABE 7-2c + \item + \begin{claim*} + Sei $K=\kmplx$. Dann sind + ${\mathbf{v}_{1}=\begin{svector}1\\\imageinh\\0\\\end{svector}}$, + ${\mathbf{v}_{2}=\begin{svector}1+\imageinh\\-\imageinh\\1-2\imageinh\\\end{svector}}$, und + ${\mathbf{v}_{3}=\begin{svector}\imageinh\\1-\imageinh\\2-\imageinh\\\end{svector}}$ + über $K$ \fbox{nicht linear unabhängig}. + \end{claim*} + + \begin{proof} + Es reicht aus, den (Zeilen)rang der Matrix + + \begin{mathe}[mc]{rcl} + A &:= &\begin{matrix}{ccc} +1 &1+\imageinh &\imageinh\\ +\imageinh &-\imageinh &1-\imageinh\\ +0 &1-2\imageinh &2-\imageinh\\ +\end{matrix} + \end{mathe} + + zu untersuchen. Wir berechnen + + \begin{algorithm}[\rtab][\rtab] + Reduktion der Matrix, $A$, mittels des Gaußverfahrens:\\ + Zeilentransformation + ${Z_{2} \leftsquigarrow \imageinh\cdot Z_{2}+Z_{1}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{matrix}{ccc} +1 &1+\imageinh &\imageinh\\ +0 &2+\imageinh &1+2\imageinh\\ +0 &1-2\imageinh &2-\imageinh\\ +\end{matrix}\\ + \end{mathe} + + Zeilentransformation + ${Z_{3} \leftsquigarrow \imageinh\cdot Z_{3}-Z_{2}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{matrix}{ccc} +1 &1+\imageinh &\imageinh\\ +0 &2+\imageinh &1+2\imageinh\\ +0 &0 &0\\ +\end{matrix}\\ + \end{mathe} + \end{algorithm} + + Der Zeilenstufenform entnimmt man, $\rank(A)=2$. + Darum sind nur $2$ der Vektoren, und zwar + $\{\mathbf{v}_{1},\mathbf{v}_{2}\}$, + linear unabhängig. + Der Vektor, $\mathbf{v}_{3}$, hingegen, hängt linear von diesen ab. + \end{proof} + + Wir betrachten nun dieselbe Aufgabe, nur über $\reell$ statt $\kmplx$: + + \begin{claim*} + Sei $K=\reell$. Dann sind + ${\mathbf{v}_{1}=\begin{svector}1\\0\\0\\1\\0\\0\\\end{svector}}$, + ${\mathbf{v}_{2}=\begin{svector}1\\1\\0\\-1\\1\\-2\\\end{svector}}$, und + ${\mathbf{v}_{3}=\begin{svector}0\\1\\1\\-1\\2\\-1\\\end{svector}}$ + über $K$ \fbox{linear unabhängig}. + \end{claim*} + + \begin{proof} + Es reicht aus, den (Zeilen)rang der Matrix + + \begin{mathe}[mc]{rcl} + A &:= &\begin{smatrix} +1&1&0\\ +0&1&1\\ +0&0&1\\ +1&-1&-1\\ +0&1&2\\ +0&-2&-1\\ +\end{smatrix} + \end{mathe} + + zu untersuchen. Wir berechnen + + \begin{algorithm}[\rtab][\rtab] + Reduktion der Matrix, $A$, mittels des Gaußverfahrens:\\ + Zeilentransformation + ${Z_{4} \leftsquigarrow \imageinh\cdot Z_{1}-Z_{4}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{smatrix} +1&1&0\\ +0&1&1\\ +0&0&1\\ +0&2&1\\ +0&1&2\\ +0&-2&-1\\ +\end{smatrix}\\ + \end{mathe} + + Zeilentransformation + ${Z_{3} \leftrightsquigarrow Z_{6}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{smatrix} +1&1&0\\ +0&1&1\\ +0&-2&-1\\ +0&2&1\\ +0&1&2\\ +0&0&1\\ +\end{smatrix}\\ + \end{mathe} + + Zeilentransformationen + ${Z_{3} \leftrightsquigarrow 2\cdot Z_{2}+Z_{3}}$, + ${Z_{4} \leftrightsquigarrow 2\cdot Z_{2}-Z_{4}}$, + und + ${Z_{5} \leftrightsquigarrow -1\cdot Z_{2}+Z_{5}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{smatrix} +1&1&0\\ +0&1&1\\ +0&0&1\\ +0&0&1\\ +0&0&1\\ +0&0&1\\ +\end{smatrix}\\ + \end{mathe} + + Zeilentransformationen + ${Z_{4} \leftrightsquigarrow Z_{4}-Z_{3}}$, + ${Z_{5} \leftrightsquigarrow Z_{5}-Z_{3}}$, + und + ${Z_{6} \leftrightsquigarrow Z_{6}-Z_{3}}$ + anwenden: + + \begin{mathe}[mc]{c} + \begin{smatrix} +1&1&0\\ +0&1&1\\ +0&0&1\\ +0&0&0\\ +0&0&0\\ +0&0&0\\ +\end{smatrix}\\ + \end{mathe} + \end{algorithm} + + Der Zeilenstufenform entnimmt man, $\rank(A)=3$. + Darum sind alle $3$ Vektoren, + $\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\}$, + linear unabhängig. + \end{proof} + +\end{enumerate} + +%% AUFGABE 7-3 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 3]{} + \label{ueb:7:ex:3} +\let\sectionname\altsectionname + +Seien $K$ ein Körper und $V$ ein Vektorraum über $K$. +Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$. + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 7-3a + \item + \begin{claim*} + Die folgende Aussage ist \fbox{gültig}:\\ + Angenommen es existieren linear unabhängige Vektoren, + $\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{n}\in V$ + und Skalare $c_{i}\in K\ohne\{0\}$, + so dass $\mathbf{v}_{i}=c_{i}\mathbf{w}_{i}$ + für alle $i\in\{1,2,\ldots,n\}$. + Dann bilden + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + ein linear unabhängiges System. + \end{claim*} + + \begin{proof} + Wir zeigen dies direkt. + Sei $\alpha_{i}\in K$ für $i\in\{1,2,\ldots,n\}$ + und so dass + mit $\sum_{i=1}^{n}\alpha_{i}\cdot\mathbf{v}_{i}=\zerovector$.\\ + \textbf{Zu zeigen:} $\forall{i\in\{1,2,\ldots,n\}:~}\alpha_{i}=0$.\\ + Es gilt + + \begin{longmathe}[mc]{RCL} + \sum_{i=1}^{n}\alpha_{i}\cdot\mathbf{v}_{i}=\zerovector + &\Longleftrightarrow + &\sum_{i=1}^{n}\alpha_{i}\cdot(c_{i}\mathbf{w}_{i})=\zerovector\\ + &\Longleftrightarrow + &\sum_{i=1}^{n}(\alpha_{i}c_{i})\cdot\mathbf{w}_{i}=\zerovector\\ + &\Longrightarrow + &\forall{i\in\{1,2,\ldots,n\}:~}\alpha_{i}c_{i}=0,\\ + &&\text{da $\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{n}$ linear unabhängig}\\ + &\Longleftrightarrow + &\forall{i\in\{1,2,\ldots,n\}:~}\alpha_{i}=0\,\text{oder}\,c_{i}=0, + \quad\text{da $K$ ein Körper ist}\\ + &\Longleftrightarrow + &\forall{i\in\{1,2,\ldots,n\}:~}\alpha_{i}=0, + \quad\text{da $c_{i}\neq 0$ für alle $i$}.\\ + \end{longmathe} + + Darum ist gilt $\alpha_{i}=0$ für alle $i\in\{1,2,\ldots,n\}$. + Folglich sind + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + linear unabhängig. + \end{proof} + + %% AUFGABE 7-3b + \item + \begin{claim*} + Die folgende Aussage ist \fbox{ungültig}:\\ + Angenommen, + $\mathbf{v}_{n}\notin\vectorspacespan(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n-1})$. + Dann ist + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + linear unabhängig. + \end{claim*} + + \begin{proof} + Folgendes ist ein Gegenbeispiel. + Sei $n\geq 3$ beliebig. + Sei $V=K^{2}$ + und betrachte die Vektoren + + \begin{mathe}[mc]{rclqrcl} + \mathbf{v}_{1}=\mathbf{v}_{2}=\ldots=\mathbf{v}_{n-1} &:= &\begin{svector}1\\0\\\end{svector} + &\mathbf{v}_{n} &:= &\begin{svector}0\\1\\\end{svector}\\ + \end{mathe} + + Dann gilt $\mathbf{v}_{n}\notin\vectorspacespan(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n-1})$, + weil $\vectorspacespan(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n-1}) + =\vectorspacespan(\mathbf{v}_{1}) + =\{\begin{svector}t\\0\\\end{svector}\mid t\in K\}\notni \begin{svector}0\\1\\\end{svector}$. + Andererseits sind die $n-1\geq 2$ Vektoren, + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + per Wahl nicht linear unabhängig (weil die alle gleich sind). + Also sind die Vektoren, + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + ebenfalls nicht linear unabhängig. + Darum gilt die behauptete Implikation nicht im Allgemeinen. + \end{proof} + + %% AUFGABE 7-3c + \item + \begin{claim*} + Die folgende Aussage ist \fbox{ungültig}:\\ + Das System + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + ist genau dann linear unabhängig, + wenn jedes echte Teilsystem linear unabhängig ist. + \end{claim*} + + \begin{proof} + Der $\Rightarrow$-Teil ist offensichtlich wahr. + Es kann also nur die $\Leftarrow$-Richtung schiefgehen. + Wir betrachten ein Gegenbeispiel mit $n=3$ Vektoren. + Sei $V=K^{2}$ und betrachte + + \begin{mathe}[mc]{rclqrclqrcl} + \mathbf{v}_{1} &:= &\begin{svector}0\\1\\\end{svector}, + &\mathbf{v}_{2} &:= &\begin{svector}1\\0\\\end{svector}, + &\mathbf{v}_{3} &:= &\begin{svector}1\\1\\\end{svector}.\\ + \end{mathe} + + Es ist einfach zu sehen, dass die echten Teilsysteme + + \begin{mathe}[mc]{cccccc} + (\mathbf{v}_{1}), + &(\mathbf{v}_{2}), + &(\mathbf{v}_{3}), + &(\mathbf{v}_{1},\mathbf{v}_{2}) + &(\mathbf{v}_{1},\mathbf{v}_{3}) + &(\mathbf{v}_{2},\mathbf{v}_{3})\\ + \end{mathe} + + linear unabhängig sind. Aber (vor allem weil $V$ nur $2$-dimensional ist) + $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$ + ist nicht linear unabhängig, da + + \begin{mathe}[mc]{rcl} + 1\cdot\mathbf{v}_{1} + +1\cdot\mathbf{v}_{2} + +-1\cdot\mathbf{v}_{3} + &= &\zerovector.\\ + \end{mathe} + + Darum ist gilt die behauptete Implikation nicht im Allgemeinen. + \end{proof} + + %% AUFGABE 7-3d + \item + \begin{claim*} + Die folgende Aussage ist \fbox{gültig}:\\ + Angenommen, + $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + sei linear unabhängig. + Dann für alle $i\in\{1,2,\ldots,n\}\ohne\{1\}$ + und $c\in K$ + bilden + $\mathbf{u},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + ein linear unabhängiges System, + wobei ${\mathbf{u}:=\mathbf{v}_{1}+c\mathbf{v}_{i}}$. + \end{claim*} + + \begin{proof} + Wir zeigen dies direkt. + Sei $\alpha_{j}\in K$ für $j\in\{1,2,\ldots,n\}$ + und so dass + mit $\alpha_{1}\mathbf{u}+\sum_{j=2}^{n}\alpha_{j}\cdot\mathbf{v}_{j}=\zerovector$.\\ + \textbf{Zu zeigen:} $\forall{j\in\{1,2,\ldots,n\}:~}\alpha_{j}=0$.\\ + Es gilt + + \begin{longmathe}[mc]{RCL} + \alpha_{1}\mathbf{u}+\sum_{j=2}^{n}\alpha_{j}\cdot\mathbf{v}_{j}=\zerovector + &\Longleftrightarrow + &\alpha_{1}\mathbf{v}_{1} + +\alpha_{1}c\mathbf{v}_{i} + +\sum_{j=2}^{n}\alpha_{j}\cdot\mathbf{v}_{j}=\zerovector\\ + &\Longleftrightarrow + &c\alpha_{1}\mathbf{v}_{i} + +\sum_{j=1}^{n}\alpha_{j}\cdot\mathbf{v}_{j}=\zerovector\\ + &\Longleftrightarrow + &\sum_{j=1}^{n}\beta_{j}\mathbf{v}_{j}=\zerovector,\\ + &&\text{% + wobei $\beta_{j}=\alpha_{j}$ für $j\in\{1,2,\ldots,n\}\ohne\{i\}$ + und $\beta_{i}=c\alpha_{1}+\alpha_{i}$ + }\\ + &\Longrightarrow + &\forall{j\in\{1,2,\ldots,n\}:~}\beta_{j}=0,\\ + &&\text{da $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ linear unabhängig}\\ + &\Longleftrightarrow + &c\alpha_{1}+\alpha_{i}=0 + \,\text{und}\, + \forall{j\in\{1,2,\ldots,n\}\ohne\{i\}:~}\alpha_{j}=0\\ + &\Longrightarrow + &c\cdot 0+\alpha_{i}=0 + \,\text{und}\, + \forall{j\in\{1,2,\ldots,n\}\ohne\{i\}:~}\alpha_{j}=0\\ + &\Longleftrightarrow + &\forall{j\in\{1,2,\ldots,n\}:~}\alpha_{j}=0\\ + \end{longmathe} + + Darum ist gilt $\alpha_{i}=0$ für alle $i\in\{1,2,\ldots,n\}$. + Folglich sind + $\mathbf{u},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$ + linear unabhängig. + \end{proof} +\end{enumerate} %% ******************************************************************************** %% FILE: body/ska/ska4.tex @@ -6643,11 +7244,6 @@ Sind $n,m\in\intgr$ teilerfremd, dann ist $[m]$ innerhalb $\intgr/n\intgr$ inver Falls $n$ nicht prim ist, muss man sich allerdings bei der Injektivitätsargumentation mehr bemühen. Einfacher ist also natürlich die Anwendung von dem Lemma von B\'ezout. -\setcounternach{part}{3} -\part{Quizzes} - - \def\chaptername{Quiz} - %% ******************************************************************************** %% FILE: body/quizzes/quiz1.tex %% ******************************************************************************** @@ -6957,6 +7553,78 @@ Wir betrachten die Komposition ${g\circ f:X\to Z}$ Das heißt, $p\divides\begin{svector}2n\\n\\\end{svector}$. \end{proof} +%% ******************************************************************************** +%% FILE: body/quizzes/quiz6.tex +%% ******************************************************************************** + +\setcounternach{chapter}{6} +\chapter[Woche 6]{Woche 6} + \label{quiz:6} + +\begin{enumerate}{\bfseries 1.} + \item + Seien $n\in\ntrlpos$. + Man bezeichne mit $(\intgr/n\intgr)^{\times}$ + die Menge der bzgl. Multiplikation modulo $n$ + invertierbaren Elemente in $\intgr/n\intgr$. + + \begin{claim*} + Für $a\in\intgr$ + gilt + $[a]\in(\intgr/n\intgr)^{\times}$ + gdw. $\exists{u,v\in\intgr:~}ua+vn=1$ + gdw. $\ggT(a,n)=1$ + \end{claim*} + + \begin{proof} + Es gilt + + \begin{mathe}[mc]{rcl} + [a]\in(\intgr/n\intgr)^{\times} + &\Longleftrightarrow + &\exists{u\in\intgr:~}\overbrace{[u]\cdot [a]}^{[u\cdot a]=}=[1]\\ + &\Longleftrightarrow &\exists{u\in\intgr:~}u\cdot a\equiv 1\mod n\\ + &\Longleftrightarrow &\exists{u\in\intgr:~}\exists{v\in\intgr:~}ua+vn=1\\ + &\Longleftrightarrow &\ggT(a,n)=1 + \quad\text{(wegen des Lemmas von B\'ezout).} + \end{mathe} + + Also gilt + $[a]\in(\intgr/n\intgr)^{\times}$ + $\Leftrightarrow$ + $\exists{u,v\in\intgr:~}ua+vn=1$ + $\Leftrightarrow$ + $\ggT(a,n)=1$. + \end{proof} + \item + \begin{claim*} + Sei $n=9$. Dann gilt $(\intgr/n\intgr)^{\times}=\{[1],[2],[4],[5],[7],[8]\}$. + \end{claim*} + + \begin{proof} + Da + + \hraum + \begin{tabular}[mc]{|L|CCCCCCCCC|} + \hline + \hline + a &0 &1 &2 &3 &4 &5 &6 &7 &8\\ + \hline + \ggT(a,n=9) &9 &1 &1 &3 &1 &1 &3 &1 &1\\ + \hline + \hline + \end{tabular} + \hraum + + gilt der letzten Aufgabe zufolge + $% + (\intgr/n\intgr)^{\times} + =\{[a]\mid a\in\{0,1,2,\ldots,n-1\},\,\ggT(a,n)=1\} + =\{[1],[2],[4],[5],[7],[8]\}% + $. + \end{proof} +\end{enumerate} + %% ******************************************************************************** %% FILE: back/index.tex %% ********************************************************************************