Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 2 & 5 & 0 & 5 1 & 0 & - 3 & - 2 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 2 & 5 & 0 & 5 \\ 1 & 0 & - 3 & - 2 \\ 2 & - 1 & 2 & 4 \end{array}]$

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}

$\frac{1}{2}$ 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}

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

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{2} & \frac{5}{2} & \frac{0}{2} & \frac{5}{2} 1 & 0 & - 3 & - 2 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{2}{2} & \frac{5}{2} & \frac{0}{2} & \frac{5}{2} \\ 1 & 0 & - 3 & - 2 \\ 2 & - 1 & 2 & 4 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 1 & 0 & - 3 & - 2 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 1 & 0 & - 3 & - 2 \\ 2 & - 1 & 2 & 4 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 1 & 0 & - 3 & - 2 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 1 & 0 & - 3 & - 2 \\ 2 & - 1 & 2 & 4 \end{array}]$

Schritt 1.2

Perform the row operation

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

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

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

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 1 - 1 & 0 - \frac{5}{2} & - 3 - 0 & - 2 - \frac{5}{2} 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 1 - 1 & 0 - \frac{5}{2} & - 3 - 0 & - 2 - \frac{5}{2} \\ 2 & - 1 & 2 & 4 \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} \\ 2 & - 1 & 2 & 4 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} 2 & - 1 & 2 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} \\ 2 & - 1 & 2 & 4 \end{array}]$

Schritt 1.3

Perform the row operation

R_{3} = R_{3} - 2 R_{1}

$R_{3} = R_{3} - 2 R_{1}$ 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} - 2 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} 2 - 2 \cdot 1 & - 1 - 2 (\frac{5}{2}) & 2 - 2 \cdot 0 & 4 - 2 (\frac{5}{2}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} \\ 2 - 2 \cdot 1 & - 1 - 2 (\frac{5}{2}) & 2 - 2 \cdot 0 & 4 - 2 (\frac{5}{2}) \end{array}]$

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} 0 & - 6 & 2 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} \\ 0 & - 6 & 2 & - 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} 0 & - 6 & 2 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & - \frac{5}{2} & - 3 & - \frac{9}{2} \\ 0 & - 6 & 2 & - 1 \end{array}]$

Schritt 1.4

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{2}{5}

$- \frac{2}{5}$ 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{2}{5}

$- \frac{2}{5}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{2} & 0 & \frac{5}{2} - \frac{2}{5} \cdot 0 & - \frac{2}{5} (- \frac{5}{2}) & - \frac{2}{5} \cdot - 3 & - \frac{2}{5} (- \frac{9}{2}) 0 & - 6 & 2 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} (- \frac{5}{2}) & - \frac{2}{5} \cdot - 3 & - \frac{2}{5} (- \frac{9}{2}) \\ 0 & - 6 & 2 & - 1 \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & 1 & \frac{6}{5} & \frac{9}{5} \\ 0 & - 6 & 2 & - 1 \end{array}]$

Schritt 1.5

Perform the row operation

$R_{3} = R_{3} + 6 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Schritt 1.5.1

Perform the row operation

$R_{3} = R_{3} + 6 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & 1 & \frac{6}{5} & \frac{9}{5} \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & 2 + 6 (\frac{6}{5}) & - 1 + 6 (\frac{9}{5}) \end{array}]$

Schritt 1.5.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & 1 & \frac{6}{5} & \frac{9}{5} \\ 0 & 0 & \frac{46}{5} & \frac{49}{5} \end{array}]$

Schritt 1.6

Multiply each element of

$R_{3}$ by

$\frac{5}{46}$ to make the entry at

$3, 3$ a

$1$ .

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Schritt 1.6.1

Multiply each element of

$R_{3}$ by

$\frac{5}{46}$ to make the entry at

$3, 3$ a

$1$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & 1 & \frac{6}{5} & \frac{9}{5} \\ \frac{5}{46} \cdot 0 & \frac{5}{46} \cdot 0 & \frac{5}{46} \cdot \frac{46}{5} & \frac{5}{46} \cdot \frac{49}{5} \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & 1 & \frac{6}{5} & \frac{9}{5} \\ 0 & 0 & 1 & \frac{49}{46} \end{array}]$

Schritt 1.7

Perform the row operation

$R_{2} = R_{2} - \frac{6}{5} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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Schritt 1.7.1

Perform the row operation

$R_{2} = R_{2} - \frac{6}{5} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 - \frac{6}{5} \cdot 0 & 1 - \frac{6}{5} \cdot 0 & \frac{6}{5} - \frac{6}{5} \cdot 1 & \frac{9}{5} - \frac{6}{5} \cdot \frac{49}{46} \\ 0 & 0 & 1 & \frac{49}{46} \end{array}]$

Schritt 1.7.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & \frac{5}{2} \\ 0 & 1 & 0 & \frac{12}{23} \\ 0 & 0 & 1 & \frac{49}{46} \end{array}]$

Schritt 1.8

Perform the row operation

$R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.8.1

Perform the row operation

$R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot 1 & 0 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot \frac{12}{23} \\ 0 & 1 & 0 & \frac{12}{23} \\ 0 & 0 & 1 & \frac{49}{46} \end{array}]$

Schritt 1.8.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{55}{46} \\ 0 & 1 & 0 & \frac{12}{23} \\ 0 & 0 & 1 & \frac{49}{46} \end{array}]$

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

$3$