Lineare Algebra Beispiele

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2.5 & 5 & 1 & 0 & 0 & 8 4.5 & 3 & 0 & 1 & 0 & 9 5 & 10 & 0 & 0 & 1 & 10 - 54 & - 60 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 2.5 & 5 & 1 & 0 & 0 & 8 \\ 4.5 & 3 & 0 & 1 & 0 & 9 \\ 5 & 10 & 0 & 0 & 1 & 10 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2.5}

$\frac{1}{2.5}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Schritt 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2.5}

$\frac{1}{2.5}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2.5}{2.5} & \frac{5}{2.5} & \frac{1}{2.5} & \frac{0}{2.5} & \frac{0}{2.5} & \frac{8}{2.5} 4.5 & 3 & 0 & 1 & 0 & 9 5 & 10 & 0 & 0 & 1 & 10 - 54 & - 60 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{2.5}{2.5} & \frac{5}{2.5} & \frac{1}{2.5} & \frac{0}{2.5} & \frac{0}{2.5} & \frac{8}{2.5} \\ 4.5 & 3 & 0 & 1 & 0 & 9 \\ 5 & 10 & 0 & 0 & 1 & 10 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 0.4 & 0 & 0 & 3.2 4.5 & 3 & 0 & 1 & 0 & 9 5 & 10 & 0 & 0 & 1 & 10 - 54 & - 60 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 4.5 & 3 & 0 & 1 & 0 & 9 \\ 5 & 10 & 0 & 0 & 1 & 10 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 0.4 & 0 & 0 & 3.2 4.5 & 3 & 0 & 1 & 0 & 9 5 & 10 & 0 & 0 & 1 & 10 - 54 & - 60 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 4.5 & 3 & 0 & 1 & 0 & 9 \\ 5 & 10 & 0 & 0 & 1 & 10 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.2

Perform the row operation

R_{2} = R_{2} - 4.5 R_{1}

$R_{2} = R_{2} - 4.5 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Schritt 1.2.1

Perform the row operation

R_{2} = R_{2} - 4.5 R_{1}

$R_{2} = R_{2} - 4.5 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 0.4 & 0 & 0 & 3.2 4.5 - 4.5 \cdot 1 & 3 - 4.5 \cdot 2 & 0 - 4.5 \cdot 0.4 & 1 - 4.5 \cdot 0 & 0 - 4.5 \cdot 0 & 9 - 4.5 \cdot 3.2 5 & 10 & 0 & 0 & 1 & 10 - 54 & - 60 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 4.5 - 4.5 \cdot 1 & 3 - 4.5 \cdot 2 & 0 - 4.5 \cdot 0.4 & 1 - 4.5 \cdot 0 & 0 - 4.5 \cdot 0 & 9 - 4.5 \cdot 3.2 \\ 5 & 10 & 0 & 0 & 1 & 10 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 0.4 & 0 & 0 & 3.2 0 & - 6 & - 1.8 & 1 & 0 & - 5.4 5 & 10 & 0 & 0 & 1 & 10 - 54 & - 60 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & - 6 & - 1.8 & 1 & 0 & - 5.4 \\ 5 & 10 & 0 & 0 & 1 & 10 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.3

Perform the row operation

$R_{3} = R_{3} - 5 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Schritt 1.3.1

Perform the row operation

$R_{3} = R_{3} - 5 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & - 6 & - 1.8 & 1 & 0 & - 5.4 \\ 5 - 5 \cdot 1 & 10 - 5 \cdot 2 & 0 - 5 \cdot 0.4 & 0 - 5 \cdot 0 & 1 - 5 \cdot 0 & 10 - 5 \cdot 3.2 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & - 6 & - 1.8 & 1 & 0 & - 5.4 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ - 54 & - 60 & 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.4

Perform the row operation

$R_{4} = R_{4} + 54 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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Schritt 1.4.1

Perform the row operation

$R_{4} = R_{4} + 54 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & - 6 & - 1.8 & 1 & 0 & - 5.4 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ - 54 + 54 \cdot 1 & - 60 + 54 \cdot 2 & 0 + 54 \cdot 0.4 & 0 + 54 \cdot 0 & 0 + 54 \cdot 0 & 0 + 54 \cdot 3.2 \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & - 6 & - 1.8 & 1 & 0 & - 5.4 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ 0 & 48 & 21.6 & 0 & 0 & 172.8 \end{array}]$

Schritt 1.5

Multiply each element of

$R_{2}$ by

$- \frac{1}{6}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.5.1

Multiply each element of

$R_{2}$ by

$- \frac{1}{6}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 1.8 & - \frac{1}{6} \cdot 1 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 5.4 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ 0 & 48 & 21.6 & 0 & 0 & 172.8 \end{array}]$

Schritt 1.5.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ 0 & 48 & 21.6 & 0 & 0 & 172.8 \end{array}]$

Schritt 1.6

Perform the row operation

$R_{4} = R_{4} - 48 R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Schritt 1.6.1

Perform the row operation

$R_{4} = R_{4} - 48 R_{2}$ to make the entry at

$4, 2$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ 0 - 48 \cdot 0 & 48 - 48 \cdot 1 & 21.6 - 48 \cdot 0.3 & 0 - 48 \cdot - 0.1\overline{6} & 0 - 48 \cdot 0 & 172.8 - 48 \cdot 0.9 \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & - 2 & 0 & 1 & - 6 \\ 0 & 0 & 7.1\overline{9} & 8 & 0 & 129.5\overline{9} \end{array}]$

Schritt 1.7

Multiply each element of

$R_{3}$ by

$- \frac{1}{2}$ to make the entry at

$3, 3$ a

$1$ .

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Schritt 1.7.1

Multiply each element of

$R_{3}$ by

$- \frac{1}{2}$ to make the entry at

$3, 3$ a

$1$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 1 & - \frac{1}{2} \cdot - 6 \\ 0 & 0 & 7.1\overline{9} & 8 & 0 & 129.5\overline{9} \end{array}]$

Schritt 1.7.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 7.1\overline{9} & 8 & 0 & 129.5\overline{9} \end{array}]$

Schritt 1.8

Perform the row operation

$R_{4} = R_{4} - 7.1\overline{9} R_{3}$ to make the entry at

$4, 3$ a

$0$ .

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Schritt 1.8.1

Perform the row operation

$R_{4} = R_{4} - 7.1\overline{9} R_{3}$ to make the entry at

$4, 3$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 - 7.1\overline{9} \cdot 0 & 0 - 7.1\overline{9} \cdot 0 & 7.1\overline{9} - 7.1\overline{9} \cdot 1 & 8 - 7.1\overline{9} \cdot 0 & 0 - 7.1\overline{9} \cdot - 0.5 & 129.5\overline{9} - 7.1\overline{9} \cdot 3 \end{array}]$

Schritt 1.8.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 8 & 3.6 & 108 \end{array}]$

Schritt 1.9

Multiply each element of

$R_{4}$ by

$\frac{1}{8}$ to make the entry at

$4, 4$ a

$1$ .

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Schritt 1.9.1

Multiply each element of

$R_{4}$ by

$\frac{1}{8}$ to make the entry at

$4, 4$ a

$1$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ \frac{0}{8} & \frac{0}{8} & \frac{0}{8} & \frac{8}{8} & \frac{3.6}{8} & \frac{108}{8} \end{array}]$

Schritt 1.9.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & - 0.1\overline{6} & 0 & 0.9 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.10

Perform the row operation

$R_{2} = R_{2} + 0.1\overline{6} R_{4}$ to make the entry at

$2, 4$ a

$0$ .

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Schritt 1.10.1

Perform the row operation

$R_{2} = R_{2} + 0.1\overline{6} R_{4}$ to make the entry at

$2, 4$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 + 0.1\overline{6} \cdot 0 & 1 + 0.1\overline{6} \cdot 0 & 0.3 + 0.1\overline{6} \cdot 0 & - 0.1\overline{6} + 0.1\overline{6} \cdot 1 & 0 + 0.1\overline{6} \cdot 0.44\overline{9} & 0.9 + 0.1\overline{6} \cdot 13.4\overline{9} \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.10.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0.3 & 0 & 0.074\overline{9} & 3.15 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.11

Perform the row operation

$R_{2} = R_{2} - 0.3 R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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Schritt 1.11.1

Perform the row operation

$R_{2} = R_{2} - 0.3 R_{3}$ to make the entry at

$2, 3$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 - 0.3 \cdot 0 & 1 - 0.3 \cdot 0 & 0.3 - 0.3 \cdot 1 & 0 - 0.3 \cdot 0 & 0.074\overline{9} - 0.3 \cdot - 0.5 & 3.15 - 0.3 \cdot 3 \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.11.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0.4 & 0 & 0 & 3.2 \\ 0 & 1 & 0 & 0 & 0.224\overline{9} & 2.24\overline{9} \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.12

Perform the row operation

$R_{1} = R_{1} - 0.4 R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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Schritt 1.12.1

Perform the row operation

$R_{1} = R_{1} - 0.4 R_{3}$ to make the entry at

$1, 3$ a

$0$ .

$[\begin{array}{c} 1 - 0.4 \cdot 0 & 2 - 0.4 \cdot 0 & 0.4 - 0.4 \cdot 1 & 0 - 0.4 \cdot 0 & 0 - 0.4 \cdot - 0.5 & 3.2 - 0.4 \cdot 3 \\ 0 & 1 & 0 & 0 & 0.224\overline{9} & 2.24\overline{9} \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.12.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & 0 & 0.2 & 2 \\ 0 & 1 & 0 & 0 & 0.224\overline{9} & 2.24\overline{9} \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.13

Perform the row operation

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.13.1

Perform the row operation

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0.2 - 2 \cdot 0.224\overline{9} & 2 - 2 \cdot 2.24\overline{9} \\ 0 & 1 & 0 & 0 & 0.224\overline{9} & 2.24\overline{9} \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 1.13.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & - 0.24\overline{9} & - 2.4\overline{9} \\ 0 & 1 & 0 & 0 & 0.224\overline{9} & 2.24\overline{9} \\ 0 & 0 & 1 & 0 & - 0.5 & 3 \\ 0 & 0 & 0 & 1 & 0.44\overline{9} & 13.4\overline{9} \end{array}]$

Schritt 2

The pivot positions are the locations with the leading

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

$a_{11}, a_{22}, a_{33},$ and

$a_{44}$

Pivot Columns:

$1, 2, 3,$ and

$4$