«K12/K13» 64. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 64. Hausaufgabe

0.0.1.1 ↑ Geometrie-Buch Seite 22, Aufgabe 1e

Löse das Gleichungssystem:

\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}$\begin{array}{cccc}\hfill 2& \hfill -3& \hfill -1& 4\hfill \\ \hfill 3& \hfill -1& \hfill 2& 5\hfill \\ \hfill 3& \hfill -8& \hfill -5& 5\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 3& \hfill -1& \hfill 2& 5\hfill \\ \hfill 3& \hfill -8& \hfill -5& 5\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 0& \hfill \frac{7}{2}& \hfill \frac{7}{2}& -1\hfill \\ \hfill 3& \hfill -8& \hfill -5& 5\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 0& \hfill \frac{7}{2}& \hfill \frac{7}{2}& -1\hfill \\ \hfill 0& \hfill -\frac{7}{2}& \hfill -\frac{7}{2}& -\frac{7}{2}\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 0& \hfill \frac{7}{2}& \hfill \frac{7}{2}& -1\hfill \\ \hfill 0& \hfill 0& \hfill 0& -\frac{7}{2}\hfill \\ \hfill \end{array}$

Widerspruch; also keine Lösungen

0.0.1.2 ↑ Geometrie-Buch Seite 33, Aufgabe 1

Löse die Gleichungssysteme mit dem Gauß-Verfahren:

b)

\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}$\begin{array}{cccc}\hfill -1& \hfill -1& \hfill 1& 0\hfill \\ \hfill 3& \hfill 1& \hfill 2& 11\hfill \\ \hfill -1& \hfill -1& \hfill 4& 9\hfill \\ \hfill \\ \hfill 1& \hfill 1& \hfill -1& 0\hfill \\ \hfill 0& \hfill -2& \hfill 5& 11\hfill \\ \hfill 0& \hfill 0& \hfill 3& 9\hfill \\ \hfill \\ \hfill 1& \hfill 1& \hfill -1& 0\hfill \\ \hfill 0& \hfill 1& \hfill -\frac{5}{2}& -\frac{11}{2}\hfill \\ \hfill 0& \hfill 0& \hfill 3& 9\hfill \\ \hfill \end{array}$

x_3 = 3;$x_3=3;$

x_2 = -\frac{11}{2} + \frac{5}{2} x_3 = 2;$x_2=-\frac{11}{2}+\frac{5}{2}{x}_3=2;$

x_1 = x_3 - x_2 = 1;$x_1=x_3-x_2=1;$

d)

\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}$\begin{array}{cccc}\hfill -1& \hfill 1& \hfill 1& 0\hfill \\ \hfill -1& \hfill 4& \hfill 2& 0\hfill \\ \hfill 2& \hfill 2& \hfill 3& 0\hfill \\ \hfill \\ \hfill 1& \hfill -1& \hfill -1& 0\hfill \\ \hfill 0& \hfill 3& \hfill 1& 0\hfill \\ \hfill 0& \hfill 10& \hfill 7& 0\hfill \\ \hfill \\ \hfill 1& \hfill -1& \hfill -1& 0\hfill \\ \hfill 0& \hfill 1& \hfill \frac{1}{3}& 0\hfill \\ \hfill 0& \hfill 0& \hfill \frac{11}{3}& 0\hfill \\ \hfill \end{array}$

x_3 = x_2 = x_1 = 0;$x_3=x_2=x_1=0;$

e)

\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}$\begin{array}{cccc}\hfill 2& \hfill -3& \hfill -1& 4\hfill \\ \hfill 3& \hfill -1& \hfill 2& 5\hfill \\ \hfill 3& \hfill -8& \hfill -5& 5\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 0& \hfill 7& \hfill 7& 0\hfill \\ \hfill 0& \hfill -\frac{7}{2}& \hfill -\frac{7}{2}& -1\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 0& \hfill 1& \hfill 1& 0\hfill \\ \hfill 0& \hfill -\frac{7}{2}& \hfill -\frac{7}{2}& -1\hfill \\ \hfill \\ \hfill 1& \hfill -\frac{3}{2}& \hfill -\frac{1}{2}& 2\hfill \\ \hfill 0& \hfill 1& \hfill 1& 0\hfill \\ \hfill 0& \hfill 1& \hfill 1& \frac{2}{7}\hfill \\ \hfill \end{array}$

Widerspruch; also keine Lösungen

f)

\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}$\begin{array}{cccc}\hfill -1& \hfill 1& \hfill 2& 0\hfill \\ \hfill 1& \hfill -3& \hfill 4& 0\hfill \\ \hfill 2& \hfill -4& \hfill 2& 0\hfill \\ \hfill \\ \hfill 1& \hfill -1& \hfill -2& 0\hfill \\ \hfill 0& \hfill -2& \hfill 6& 0\hfill \\ \hfill 0& \hfill -2& \hfill 6& 0\hfill \\ \hfill \\ \hfill 1& \hfill -1& \hfill -2& 0\hfill \\ \hfill 0& \hfill 1& \hfill -3& 0\hfill \\ \hfill \end{array}$

x_2 = 3 x_3;$x_2=3x_3;$

x_1 = 2 x_3 + x_2 = 2 x_3 + 3 x_3 = 5 x_3;$x_1=2x_3+x_2=2x_3+3x_3=5x_3;$

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\}\!;$L=\left\{\left(x_1,x_2,x_3\right)\mid x_1=5k\wedge x_2=3k\wedge x_3=k;k\in \mathbb{ℝ}\right\};$

\vec X = k \left(\!\begin{smallmatrix}5\\3\\1\end{smallmatrix}\!\right)\!; \quad k \in \mathds{R};$\overrightarrow{X}=k\left(\begin{array}{c}5\\ 3\\ 1\end{array}\right);k\in \mathbb{ℝ};$