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Aber man kann auch in di \subsection*{Aufgabe 7} Dieser Plan geht leider nicht auf. Es ist kein Fehler, aber es bringt uns nicht näher an die Zeilenstufenform. Eine gute Strategie ist es Spalten um Spalten in die Zeilenstufenform zu bringen. -%\subsection*{Aufgabe 8} +\subsection*{Aufgabe 8} +I zu III add +$\left(\begin{matrix} +1&3&-2&1\\ +3&-1&5&0\\ +0&4&0&3 +\end{matrix}\right)$ +II zu III add +$\left(\begin{matrix} +1&3&-2&1\\ +3&-1&5&0\\ +3&3&5&3 +\end{matrix}\right)$\\ +Die Idee geht leider nicht auf, da III nach der ersten Operation ganz neue Koeffizienten hat. + +Nach der ersten Null würde ich die Spalte in die Zeilenform bringen. + +\subsection*{Aufgabe 9} +\begin{enumerate}[label=\alph*)] +\item In der Treppenform ist der erste Keoffizient jeder Zeile, der nicht null ist ein Pivot. + +$\left(\begin{matrix} +2&4&-2&-5\\ +0&-1&1&7\\ +0&0&5&4 +\end{matrix}\right)$\\Die Pivots aus diesem Beispiel sind 2, -1, 5. + +\item +Die \textit{normierte Zeilenstufenform} ist die Treppenform wobei alle Pivot-Elemente den Wert 1 haben. + +\item +Die \textit{Reduzierte Zeilenstufenform (rref)} ist eine normierte Zeilenstufenform, +wobei alle Koeffizienten (ohne die Konstanten) und denn Pivots Null sind. +\end{enumerate} + +\subsection*{Aufgabe 10} +\begin{enumerate}[label=\alph*)] +\item II: $0=1\Rightarrow\mathbb{L}=\{\}$ +\item II: $0=0$. Es muss also nur noch I erfüllt werden. Dazu gibt es unendlich viele Lösungen. +\item Da II und III identisch sind, können wir eine streichen. Das lineare Gleichungssystem ist unterbestimmt und es gibt unendlich viele Lösungen. +\item Das lineare Gleichungssystem ist unterbestimmt und es gibt unendlich viele Lösungen. +\item IV: $0=1\Rightarrow\mathbb{L}=\{\}$ +\end{enumerate} + + \section*{Wahr oder falsch 4.4} + \section*{Üben und Anwenden 4.4} \subsection*{Aufgabe 3} \begin{enumerate}[label=\alph*)]