Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 9 & 12 & 8 6 & 4 & 5 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 9 & 12 & 8 \\ 6 & 4 & 5 \\ 3 & 8 & 2 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{9}

$\frac{1}{9}$ 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}{9}

$\frac{1}{9}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{9}{9} & \frac{12}{9} & \frac{8}{9} 6 & 4 & 5 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{9}{9} & \frac{12}{9} & \frac{8}{9} \\ 6 & 4 & 5 \\ 3 & 8 & 2 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 6 & 4 & 5 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 6 & 4 & 5 \\ 3 & 8 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 6 & 4 & 5 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 6 & 4 & 5 \\ 3 & 8 & 2 \end{array}]$

Schritt 1.2

Perform the row operation

R_{2} = R_{2} - 6 R_{1}

$R_{2} = R_{2} - 6 R_{1}$ 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} - 6 R_{1}

$R_{2} = R_{2} - 6 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 6 - 6 \cdot 1 & 4 - 6 (\frac{4}{3}) & 5 - 6 (\frac{8}{9}) 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 6 - 6 \cdot 1 & 4 - 6 (\frac{4}{3}) & 5 - 6 (\frac{8}{9}) \\ 3 & 8 & 2 \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & - 4 & - \frac{1}{3} 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & - 4 & - \frac{1}{3} \\ 3 & 8 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & - 4 & - \frac{1}{3} 3 & 8 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & - 4 & - \frac{1}{3} \\ 3 & 8 & 2 \end{array}]$

Schritt 1.3

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

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Schritt 1.3.1

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & - 4 & - \frac{1}{3} 3 - 3 \cdot 1 & 8 - 3 (\frac{4}{3}) & 2 - 3 (\frac{8}{9}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & - 4 & - \frac{1}{3} \\ 3 - 3 \cdot 1 & 8 - 3 (\frac{4}{3}) & 2 - 3 (\frac{8}{9}) \end{array}]$

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & - 4 & - \frac{1}{3} 0 & 4 & - \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & - 4 & - \frac{1}{3} \\ 0 & 4 & - \frac{2}{3} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & - 4 & - \frac{1}{3} 0 & 4 & - \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & - 4 & - \frac{1}{3} \\ 0 & 4 & - \frac{2}{3} \end{array}]$

Schritt 1.4

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{4}

$- \frac{1}{4}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Schritt 1.4.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{4}

$- \frac{1}{4}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} (- \frac{1}{3}) 0 & 4 & - \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} (- \frac{1}{3}) \\ 0 & 4 & - \frac{2}{3} \end{array}]$

Schritt 1.4.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 & 4 & - \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 & 4 & - \frac{2}{3} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 & 4 & - \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 & 4 & - \frac{2}{3} \end{array}]$

Schritt 1.5

Perform the row operation

R_{3} = R_{3} - 4 R_{2}

$R_{3} = R_{3} - 4 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Schritt 1.5.1

Perform the row operation

R_{3} = R_{3} - 4 R_{2}

$R_{3} = R_{3} - 4 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & - \frac{2}{3} - 4 (\frac{1}{12}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & - \frac{2}{3} - 4 (\frac{1}{12}) \end{array}]$

Schritt 1.5.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 & 0 & - 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 & 0 & - 1 \end{array}]$

Schritt 1.6

Multiply each element of

R_{3}

$R_{3}$ by

- 1

$- 1$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Schritt 1.6.1

Multiply each element of

R_{3}

$R_{3}$ by

- 1

$- 1$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} - 0 & - 0 & - - 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ - 0 & - 0 & - - 1 \end{array}]$

Schritt 1.6.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & \frac{1}{12} 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & \frac{1}{12} \\ 0 & 0 & 1 \end{array}]$

Schritt 1.7

Perform the row operation

R_{2} = R_{2} - \frac{1}{12} R_{3}

$R_{2} = R_{2} - \frac{1}{12} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Schritt 1.7.1

Perform the row operation

R_{2} = R_{2} - \frac{1}{12} R_{3}

$R_{2} = R_{2} - \frac{1}{12} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 - \frac{1}{12} \cdot 0 & 1 - \frac{1}{12} \cdot 0 & \frac{1}{12} - \frac{1}{12} \cdot 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 - \frac{1}{12} \cdot 0 & 1 - \frac{1}{12} \cdot 0 & \frac{1}{12} - \frac{1}{12} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.7.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & \frac{8}{9} 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{8}{9} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.8

Perform the row operation

R_{1} = R_{1} - \frac{8}{9} R_{3}

$R_{1} = R_{1} - \frac{8}{9} R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Schritt 1.8.1

Perform the row operation

R_{1} = R_{1} - \frac{8}{9} R_{3}

$R_{1} = R_{1} - \frac{8}{9} R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 - \frac{8}{9} \cdot 0 & \frac{4}{3} - \frac{8}{9} \cdot 0 & \frac{8}{9} - \frac{8}{9} \cdot 1 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - \frac{8}{9} \cdot 0 & \frac{4}{3} - \frac{8}{9} \cdot 0 & \frac{8}{9} - \frac{8}{9} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.8.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{4}{3} & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{4}{3} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.9

Perform the row operation

R_{1} = R_{1} - \frac{4}{3} R_{2}

$R_{1} = R_{1} - \frac{4}{3} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Schritt 1.9.1

Perform the row operation

R_{1} = R_{1} - \frac{4}{3} R_{2}

$R_{1} = R_{1} - \frac{4}{3} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & 0 - \frac{4}{3} \cdot 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & 0 - \frac{4}{3} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.9.2

Vereinfache

R_{1}

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

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:

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

$a_{33}$

Pivot Columns:

$1, 2,$ and

$3$