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+\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} +\usepackage{enumitem} +\author{Beat Jäckle (Jb)} +\begin{document} +Lösungen für die Rohfassung: Mathematik fürs Gymnasium +%\setcounter{section}{4} +%\setcounter{subsection}{2} +\subsection*{Selber erklären 4.4} +\subsubsection*{Aufgabe 1} +Wir notieren die Koeffizienten und Konstanten der Normalform in einer Tabelle. Dadurch müssen wir viel weniger Zeichen aufschreiben. + +$\left(\begin{matrix} +2 & -1 & 7 & 11 & 1 \\ +1 & 0 & 3 & -21 & 19\\ +0 & -5 & 1 & 4 & -8\\ +-19.5 & 0 & 0 & 6 & 8.5 +\end{matrix}\right)$ + +\subsection*{Aufgabe 2} +Überprüfe, dass das LGS tatsächlich in der Normalform ist, bevor du beginnst. + +Vergiss die Vorzeichen nicht und schreib eine 0, falls die Variable, welche der Position entspricht nicht vorkommt. + +\subsection*{Aufgabe 3} +\begin{itemize} +\item zwei Zeilen vertauschen +\item eine Zeile mit einer Zahl ungleich Null multiplizieren +\item eine Zeile zu einer Zeile dazu addieren +\end{itemize} +Wenn wir zwei Zeilen vertauschen, dann werden weiterhin beide Gleichungen erfüllt. + +\subsection*{Aufgabe 4} +Diese Umformung ist falsch. Wenn man zu jedem Koeffizient und zur Konstante einer Zeile 1 dazu addiert, ist es wie mit der Gleichung $1x+1y+1z=1$ zu addieren. Diese Gleichung muss aber nicht wahr sein. + +\subsection*{Aufgabe 5} +In der zweiten Zeile ist der erste Koeffizient Null. In der dritten Zeile sind die ersten zwei Koeffizienten Null. In der vierten sind die ersten drei Koeffizienten Null. Und so weiter. + +Der Vorteil der Treppenform ist, dass die wir von unten nach oben die Gleichungen einfach auflösen können und die Lösungsmenge bestimmen können, da in der untersten Gleichung (normalerweise) nur noch eine Unbekannte enthalten ist. + +\subsection*{Aufgabe 6} +Das ist nicht die Treppenform die wir besprochen haben. Aber man kann auch in dieser speziellen Form die Lösung einfacher bestimmen, da die erste Gleichung aus nur noch einer Variable besteht. + +\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} + +\end{document} \ No newline at end of file