Lineare Algebra Beispiele

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

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{3}$ to make the entry at

$1, 1$ a

$1$ .

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Schritt 1.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{3}$ to make the entry at

$1, 1$ a

$1$ .

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

Schritt 1.1.2

Vereinfache

$R_{1}$ .

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

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

$0$ .

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

Schritt 1.2.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{10}{3} & \frac{4}{3} \\ - 1 & - 1 & 1 \end{array}]$

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

$0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{10}{3} & \frac{4}{3} \\ - 1 + 1 \cdot 1 & - 1 + \frac{1}{3} & 1 + \frac{1}{3} \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{10}{3} & \frac{4}{3} \\ 0 & - \frac{2}{3} & \frac{4}{3} \end{array}]$

Schritt 1.4

Multiply each element of

$R_{2}$ by

$\frac{3}{10}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.4.1

Multiply each element of

$R_{2}$ by

$\frac{3}{10}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{3} \\ \frac{3}{10} \cdot 0 & \frac{3}{10} \cdot \frac{10}{3} & \frac{3}{10} \cdot \frac{4}{3} \\ 0 & - \frac{2}{3} & \frac{4}{3} \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{2}$ .

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

Schritt 1.5

Perform the row operation

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

$3, 2$ a

$0$ .

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Schritt 1.5.1

Perform the row operation

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

$3, 2$ a

$0$ .

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

Schritt 1.5.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{3} \\ 0 & 1 & \frac{2}{5} \\ 0 & 0 & \frac{8}{5} \end{array}]$

Schritt 1.6

Multiply each element of

$R_{3}$ by

$\frac{5}{8}$ to make the entry at

$3, 3$ a

$1$ .

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Schritt 1.6.1

Multiply each element of

$R_{3}$ by

$\frac{5}{8}$ to make the entry at

$3, 3$ a

$1$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{3} \\ 0 & 1 & \frac{2}{5} \\ \frac{5}{8} \cdot 0 & \frac{5}{8} \cdot 0 & \frac{5}{8} \cdot \frac{8}{5} \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{3}$ .

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

Schritt 1.7

Perform the row operation

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

$2, 3$ a

$0$ .

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Schritt 1.7.1

Perform the row operation

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

$2, 3$ a

$0$ .

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

Schritt 1.7.2

Vereinfache

$R_{2}$ .

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

Schritt 1.8

Perform the row operation

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

$1, 3$ a

$0$ .

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Schritt 1.8.1

Perform the row operation

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

$1, 3$ a

$0$ .

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

Schritt 1.8.2

Vereinfache

$R_{1}$ .

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

Schritt 1.9

Perform the row operation

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

$1, 2$ a

$0$ .

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Schritt 1.9.1

Perform the row operation

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

$1, 2$ a

$0$ .

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

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

$3$