=for timestamp
Mo Mär 20 17:57:58 CET 2006

=head2 64. Hausaufgabe

=head3 Geometrie-Buch Seite 22, Aufgabe 1e

Löse das Gleichungssystem:

M<\begin{array}{rrr|l}
{} 2 & -3 & -1 & 4 \\
{} 3 & -1 & 2 & 5 \\
{} 3 & -8 & -5 & 5 \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 3 & -1 & 2 & 5 \\
{} 3 & -8 & -5 & 5 \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 0 & \frac{7}{2} & \frac{7}{2} & -1 \\
{} 3 & -8 & -5 & 5 \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 0 & \frac{7}{2} & \frac{7}{2} & -1 \\
{} 0 & -\frac{7}{2} & -\frac{7}{2} & -\frac{7}{2} \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 0 & \frac{7}{2} & \frac{7}{2} & -1 \\
{} 0 & 0 & 0 & -\frac{7}{2} \\
\end{array}>

Widerspruch; also keine Lösungen

=head3 Geometrie-Buch Seite 33, Aufgabe 1

Löse die Gleichungssysteme mit dem Gauß-Verfahren:

=over

=item b)

M<\begin{array}{rrr|l}
{} -1 & -1 & 1 & 0 \\
{} 3 & 1 & 2 & 11 \\
{} -1 & -1 & 4 & 9 \\
\\
{} 1 & 1 & -1 & 0 \\
{} 0 & -2 & 5 & 11 \\
{} 0 & 0 & 3 & 9 \\
\\
{} 1 & 1 & -1 & 0 \\
{} 0 & 1 & -\frac{5}{2} & -\frac{11}{2} \\
{} 0 & 0 & 3 & 9 \\
\end{array}>

M<x_3 = 3;>

M<x_2 = -\frac{11}{2} + \frac{5}{2} x_3 = 2;>

M<x_1 = x_3 - x_2 = 1;>

=item d)

M<\begin{array}{rrr|l}
{} -1 & 1 & 1 & 0 \\
{} -1 & 4 & 2 & 0 \\
{} 2 & 2 & 3 & 0 \\
\\
{} 1 & -1 & -1 & 0 \\
{} 0 & 3 & 1 & 0 \\
{} 0 & 10 & 7 & 0 \\
\\
{} 1 & -1 & -1 & 0 \\
{} 0 & 1 & \frac{1}{3} & 0 \\
{} 0 & 0 & \frac{11}{3} & 0 \\
\end{array}>

M<x_3 = x_2 = x_1 = 0;>

=item e)

M<\begin{array}{rrr|l}
{} 2 & -3 & -1 & 4 \\
{} 3 & -1 & 2 & 5 \\
{} 3 & -8 & -5 & 5 \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 0 & 7 & 7 & 0 \\
{} 0 & -\frac{7}{2} & -\frac{7}{2} & -1 \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 0 & 1 & 1 & 0 \\
{} 0 & -\frac{7}{2} & -\frac{7}{2} & -1 \\
\\
{} 1 & -\frac{3}{2} & -\frac{1}{2} & 2 \\
{} 0 & 1 & 1 & 0 \\
{} 0 & 1 & 1 & \frac{2}{7} \\
\end{array}>

Widerspruch; also keine Lösungen

=item f)

M<\begin{array}{rrr|l}
{} -1 & 1 & 2 & 0 \\
{} 1 & -3 & 4 & 0 \\
{} 2 & -4 & 2 & 0 \\
\\
{} 1 & -1 & -2 & 0 \\
{} 0 & -2 & 6 & 0 \\
{} 0 & -2 & 6 & 0 \\
\\
{} 1 & -1 & -2 & 0 \\
{} 0 & 1 & -3 & 0 \\
\end{array}>

M<x_2 = 3 x_3;>

M<x_1 = 2 x_3 + x_2 = 2 x_3 + 3 x_3 = 5 x_3;>

=for timestamp
Di Mär 21 16:45:32 CET 2006

M<L = \left\{ (x_1, x_2, x_3) \middle| x_1 = 5k \wedge x_2 = 3k \wedge x_3 = k;
k \in \mathds{R} \right\}\!;>

M<\vec X = k \left(\!\begin{smallmatrix}5\\3\\1\end{smallmatrix}\!\right)\!;
\quad k \in \mathds{R};>

=back