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\def\onematrix{\text{\upshape\bfseries I}} \def\zeromatrix{\text{\upshape\bfseries 0}} @@ -1343,6 +1343,8 @@ \def\graph{\mathop{\text{\textup Gph}}} \def\id{\text{\textup id}} \def\modfn{\mathop{\text{\textup mod}}} +\def\divides{\mathbin{\mid}} +\def\ggT{\mathop{\text{\upshape ggT}}} \makeatother \begin{document} @@ -1578,7 +1580,7 @@ Dies führt zu einem Fallunterschied: \right\} }$, oder etwas kompakter formuliert, - ${L_{\alpha,\beta}=\begin{svector}32\\-4\\0\\0\\\end{svector} + \span\left\{\begin{svector}26\\-4\\1\\0\\\end{svector}, \begin{svector}-13\\2\\1\\1\\\end{svector}\right\}}$. + ${L_{\alpha,\beta}=\begin{svector}32\\-4\\0\\0\\\end{svector} + \vectorspacespan\left\{\begin{svector}26\\-4\\1\\0\\\end{svector}, \begin{svector}-13\\2\\1\\1\\\end{svector}\right\}}$. \end{enumerate} %% FALL 2 @@ -1618,7 +1620,7 @@ Dies führt zu einem Fallunterschied: \right\} }$, oder etwas kompakter formuliert, - ${L_{\alpha,\beta}=\begin{svector}32\\-4\\0\\0\\\end{svector} + \frac{\beta-8}{\alpha-4}\cdot\begin{svector}26\\-4\\1\\0\\\end{svector} + \span\left\{\begin{svector}-13\\2\\1\\1\\\end{svector}\right\}}$. + ${L_{\alpha,\beta}=\begin{svector}32\\-4\\0\\0\\\end{svector} + \frac{\beta-8}{\alpha-4}\cdot\begin{svector}26\\-4\\1\\0\\\end{svector} + \vectorspacespan\left\{\begin{svector}-13\\2\\1\\1\\\end{svector}\right\}}$. \end{enumerate} Wir fassen die Lösung für alle Fälle zusammen: @@ -1626,8 +1628,8 @@ Wir fassen die Lösung für alle Fälle zusammen: \begin{mathe}[mc]{rcl} L_{\alpha,\beta} &= &\begin{cases}[m]{lcl} \leer &: &\alpha=4,\,\beta\neq 8\\ - \mathbf{u} + \span\{\mathbf{v},\mathbf{w}\} &: &\alpha=4,\,\beta=8\\ - \mathbf{u} + \frac{\alpha-4}{\beta-8}\mathbf{v} + \span\{\mathbf{w}\} &: &\alpha\neq 4\\ + \mathbf{u} + \vectorspacespan\{\mathbf{v},\mathbf{w}\} &: &\alpha=4,\,\beta=8\\ + \mathbf{u} + \frac{\alpha-4}{\beta-8}\mathbf{v} + \vectorspacespan\{\mathbf{w}\} &: &\alpha\neq 4\\ \end{cases} \end{mathe} @@ -2283,7 +2285,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve Da $\mathbf{w},\mathbf{w}^{\prime}\neq\zerovector$ bedeutet dies, dass $\mathbf{w},\mathbf{w}^{\prime}$ \emph{linear unabhängig} sind. ($\to$ Warum??)\\ Also gilt für den Untervektorraum - $U:=\span\{\mathbf{w},\mathbf{w}^{\prime}\}$, + $U:=\vectorspacespan\{\mathbf{w},\mathbf{w}^{\prime}\}$, dass $\dim(U)=2$.\\ Da $U\subseteq\reell^{2}$ Vektorräume sind und $\dim(U)=2=\dim(\reell^{2})$, folgt hieraus, dass $U=\reell^{2}$. ($\to$ Warum??)\\ @@ -2294,7 +2296,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve \mathbf{\xi} &:= &\mathbf{v}^{\prime}-\mathbf{v}\in\reell^{2}.\\ \end{mathe} - Dann $\mathbf{\xi}\in U=\span\{\mathbf{w},\mathbf{w}^{\prime}\}$. + Dann $\mathbf{\xi}\in U=\vectorspacespan\{\mathbf{w},\mathbf{w}^{\prime}\}$. Folglich existieren Skalare $\alpha,\beta\in\reell$, so dass $\alpha\mathbf{w}+\beta\mathbf{w}^{\prime}=\mathbf{\xi}$ gilt.\\ @@ -2623,8 +2625,8 @@ Wir arbeiten im Vektorraum $\reell^{3}$ und betrachten die Vektoren \end{mathe} \textbf{Zu berechnen:} - $U:=\span\{\mathbf{v}_{1},\mathbf{v}_{2}\} - \cap\span\{\mathbf{w}_{1},\mathbf{w}_{2}\}$ + $U:=\vectorspacespan\{\mathbf{v}_{1},\mathbf{v}_{2}\} + \cap\vectorspacespan\{\mathbf{w}_{1},\mathbf{w}_{2}\}$ als Untervektorraum von $\reell^{3}$.\\ Zu diesem Zwecke betrachte einen beliebigen Vektor, $\mathbf{\xi}\in\reell^{3}$. Es gilt @@ -2755,7 +2757,7 @@ Wir können nun \eqcref{eq:0:ueb:3:ex:1} fortsetzen und erhalten 28\mathbf{v}_{1}+-8\mathbf{v}_{2} }_{=:\mathbf{u}} )\\ - &\Longleftrightarrow &\mathbf{\xi}\in\span\{\mathbf{u}\}\\ + &\Longleftrightarrow &\mathbf{\xi}\in\vectorspacespan\{\mathbf{u}\}\\ \end{mathe} für alle $\mathbf{\xi}\in\reell^{3}$.\\ @@ -2773,12 +2775,12 @@ Aus \eqcref{eq:1:ueb:3:ex:1} ergibt sich der zu berechnende Untervektorraum als \begin{mathe}[mc]{rcccccccl} - \span\{\mathbf{v}_{1},\mathbf{v}_{2}\} - \cap\span\{\mathbf{w}_{1},\mathbf{w}_{2}\} + \vectorspacespan\{\mathbf{v}_{1},\mathbf{v}_{2}\} + \cap\vectorspacespan\{\mathbf{w}_{1},\mathbf{w}_{2}\} &= &U - &= &\span\{\mathbf{u}\} - &= &\span\{44\begin{svector}1\\1\\1\\\end{svector}\} - &= &\span\{\begin{svector}1\\1\\1\\\end{svector}\}.\\ + &= &\vectorspacespan\{\mathbf{u}\} + &= &\vectorspacespan\{44\begin{svector}1\\1\\1\\\end{svector}\} + &= &\vectorspacespan\{\begin{svector}1\\1\\1\\\end{svector}\}.\\ \end{mathe} %% AUFGABE 3-2 @@ -3473,7 +3475,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \end{mathe} Also gilt \eqcref{eq:1:\beweislabel}. - Also gilt $\Phi(1)$ + Also gilt $\Phi(1)$. \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] Sei $n>1$. @@ -3708,7 +3710,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. Angenommen, $\Phi(k)$ gilt für alle $k1$. - Angenommen, $\Phi(k)$ gilt für alle $k1$ vorausgesetzt! + Wir haben etwas ausführlicher gezeigt, dass die Menge $X'$ mindestens $n-1$ Goldfische enthält. + Wenn wir $\tilde{x}:=x_{0}$ wählen entspricht dies der Größe des Schnitts $X_{0}\cap X_{1}$. + Das \uline{Diagramm} mag andeuten, dass dieser Schnitt nicht leer ist, aber das Diagramm täuscht. + Im Induktionsschritt setzen wir nur voraus, dass $n\geq 1$. + Darum ist $n-1>0$ nur garantiert, wenn stattdessen $n'\geq 2$ vorausgesetzt wird. Das heißt das Induktionsargument ist faul, weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird.