Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 7 & - 3 & 5 - 2 & 5 & - 8 - 1 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 7 & - 3 & 5 \\ - 2 & 5 & - 8 \\ - 1 & 3 & 4 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{7}

$\frac{1}{7}$ 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}{7}

$\frac{1}{7}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{7}{7} & \frac{- 3}{7} & \frac{5}{7} - 2 & 5 & - 8 - 1 & 3 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{7}{7} & \frac{- 3}{7} & \frac{5}{7} \\ - 2 & 5 & - 8 \\ - 1 & 3 & 4 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{7} & \frac{5}{7} - 2 & 5 & - 8 - 1 & 3 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ - 2 & 5 & - 8 \\ - 1 & 3 & 4 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{7} & \frac{5}{7} - 2 & 5 & - 8 - 1 & 3 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ - 2 & 5 & - 8 \\ - 1 & 3 & 4 \end{array}]$

Schritt 1.2

Perform the row operation

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

$2, 1$ a

$0$ .

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Schritt 1.2.1

Perform the row operation

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

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ - 2 + 2 \cdot 1 & 5 + 2 (- \frac{3}{7}) & - 8 + 2 (\frac{5}{7}) \\ - 1 & 3 & 4 \end{array}]$

Schritt 1.2.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & \frac{29}{7} & - \frac{46}{7} \\ - 1 & 3 & 4 \end{array}]$

Schritt 1.3

Perform the row operation

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

$3, 1$ a

$0$ .

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Schritt 1.3.1

Perform the row operation

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

$3, 1$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & \frac{29}{7} & - \frac{46}{7} \\ - 1 + 1 \cdot 1 & 3 - \frac{3}{7} & 4 + \frac{5}{7} \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & \frac{29}{7} & - \frac{46}{7} \\ 0 & \frac{18}{7} & \frac{33}{7} \end{array}]$

Schritt 1.4

Multiply each element of

$R_{2}$ by

$\frac{7}{29}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.4.1

Multiply each element of

$R_{2}$ by

$\frac{7}{29}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ \frac{7}{29} \cdot 0 & \frac{7}{29} \cdot \frac{29}{7} & \frac{7}{29} (- \frac{46}{7}) \\ 0 & \frac{18}{7} & \frac{33}{7} \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & 1 & - \frac{46}{29} \\ 0 & \frac{18}{7} & \frac{33}{7} \end{array}]$

Schritt 1.5

Perform the row operation

$R_{3} = R_{3} - \frac{18}{7} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Schritt 1.5.1

Perform the row operation

$R_{3} = R_{3} - \frac{18}{7} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & 1 & - \frac{46}{29} \\ 0 - \frac{18}{7} \cdot 0 & \frac{18}{7} - \frac{18}{7} \cdot 1 & \frac{33}{7} - \frac{18}{7} (- \frac{46}{29}) \end{array}]$

Schritt 1.5.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & 1 & - \frac{46}{29} \\ 0 & 0 & \frac{255}{29} \end{array}]$

Schritt 1.6

Multiply each element of

$R_{3}$ by

$\frac{29}{255}$ to make the entry at

$3, 3$ a

$1$ .

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Schritt 1.6.1

Multiply each element of

$R_{3}$ by

$\frac{29}{255}$ to make the entry at

$3, 3$ a

$1$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & 1 & - \frac{46}{29} \\ \frac{29}{255} \cdot 0 & \frac{29}{255} \cdot 0 & \frac{29}{255} \cdot \frac{255}{29} \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & 1 & - \frac{46}{29} \\ 0 & 0 & 1 \end{array}]$

Schritt 1.7

Perform the row operation

$R_{2} = R_{2} + \frac{46}{29} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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Schritt 1.7.1

Perform the row operation

$R_{2} = R_{2} + \frac{46}{29} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 + \frac{46}{29} \cdot 0 & 1 + \frac{46}{29} \cdot 0 & - \frac{46}{29} + \frac{46}{29} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.7.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & \frac{5}{7} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.8

Perform the row operation

$R_{1} = R_{1} - \frac{5}{7} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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Schritt 1.8.1

Perform the row operation

$R_{1} = R_{1} - \frac{5}{7} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

$[\begin{array}{c} 1 - \frac{5}{7} \cdot 0 & - \frac{3}{7} - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.8.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{7} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.9

Perform the row operation

$R_{1} = R_{1} + \frac{3}{7} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.9.1

Perform the row operation

$R_{1} = R_{1} + \frac{3}{7} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 + \frac{3}{7} \cdot 0 & - \frac{3}{7} + \frac{3}{7} \cdot 1 & 0 + \frac{3}{7} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.9.2

Vereinfache

$R_{1}$ .

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

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

$3$