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$V$. \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 $a=b$. + Dann ist $U_{2}:=\{f\in V\mid f(a)=f(b)=1\}$ \fbox{kein Untervektorraum}. \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}.) + Offensichtlich ist der Nullvektor, $0(\cdot)$, nicht in $U_{2}$, + da $0(a)=0\neq 1$. + 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: @@ -4700,6 +4693,7 @@ Dies ist zufälligerweise auch das Nullelement von $V$. \end{enumerate} %% AUFGABE 7-2 +\clearpage \let\altsectionname\sectionname \def\sectionname{Aufgabe} \section[Aufgabe 2]{} @@ -4752,9 +4746,9 @@ Dies ist zufälligerweise auch das Nullelement von $V$. \begin{mathe}[mc]{c} \begin{matrix}{ccc} -1 &2 &2\\ -0 &4 &5\\ -0 &0 &5\\ +\boxed{1} &2 &2\\ +0 &\boxed{4} &5\\ +0 &0 &\boxed{5}\\ \end{matrix}\\ \end{mathe} \end{algorithm} @@ -4786,40 +4780,21 @@ Dies ist zufälligerweise auch das Nullelement von $V$. \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: + zu untersuchen. Um dies zu bestimmen, können wir das Gaußverfahren anwenden. + Da wir in $\mathbf{F}_{5}$ arbeiten, genügt es, die Matrix über $\intgr$ zu behandeln, + und lediglich in den Zeilenoperationen Vielfache von $5$ zu vermeiden. + Da die Matrix dieselbe ist wie in \textbf{Aufgabe 7.2(a)} + und in dem Gaußverfahren dort Vielfache von $5$ vermieden wurden, + ist das Resultat \begin{mathe}[mc]{c} \begin{matrix}{ccl} -1 &2 &2\\ -0 &4 &5(=0)\\ -0 &3 &0\\ +\boxed{1} &2 &2\\ +0 &\boxed{4} &5(=0)\\ +0 &0 &5(=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 $\{\mathbf{v}_{1},\mathbf{v}_{2}\}$, @@ -4870,8 +4845,8 @@ Dies ist zufälligerweise auch das Nullelement von $V$. \begin{mathe}[mc]{c} \begin{matrix}{ccc} -1 &1+\imageinh &\imageinh\\ -0 &2+\imageinh &1+2\imageinh\\ +\boxed{1} &1+\imageinh &\imageinh\\ +0 &\boxed{2+\imageinh} &1+2\imageinh\\ 0 &0 &0\\ \end{matrix}\\ \end{mathe} @@ -4988,6 +4963,7 @@ Dies ist zufälligerweise auch das Nullelement von $V$. \end{enumerate} %% AUFGABE 7-3 +\clearpage \let\altsectionname\sectionname \def\sectionname{Aufgabe} \section[Aufgabe 3]{}