Lineare Algebra Beispiele

$[\begin{array}{c} 6 & 8 & 9 \\ 3 & 5 & 7 \\ 9 & 4 & 2 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Schritt 1.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{8}{6} & \frac{9}{6} \\ 3 & 5 & 7 \\ 9 & 4 & 2 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{3}{2} \\ 3 & 5 & 7 \\ 9 & 4 & 2 \end{array}]$

Schritt 1.2

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 1.2.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{3}{2} \\ 3 - 3 \cdot 1 & 5 - 3 (\frac{4}{3}) & 7 - 3 (\frac{3}{2}) \\ 9 & 4 & 2 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{2}$ .

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

Schritt 1.3

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 1.3.1

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

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

Schritt 1.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{3}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & - 8 & - \frac{23}{2} \end{array}]$

Schritt 1.4

Perform the row operation $R_{3} = R_{3} + 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 1.4.1

Perform the row operation $R_{3} = R_{3} + 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{3}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 + 8 \cdot 0 & - 8 + 8 \cdot 1 & - \frac{23}{2} + 8 (\frac{5}{2}) \end{array}]$

Schritt 1.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{3}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 0 & \frac{17}{2} \end{array}]$

Schritt 1.5

Multiply each element of $R_{3}$ by $\frac{2}{17}$ to make the entry at $3, 3$ a $1$ .

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Schritt 1.5.1

Multiply each element of $R_{3}$ by $\frac{2}{17}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{3}{2} \\ 0 & 1 & \frac{5}{2} \\ \frac{2}{17} \cdot 0 & \frac{2}{17} \cdot 0 & \frac{2}{17} \cdot \frac{17}{2} \end{array}]$

Schritt 1.5.2

Vereinfache $R_{3}$ .

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

Schritt 1.6

Perform the row operation $R_{2} = R_{2} - \frac{5}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 1.6.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

Schritt 1.6.2

Vereinfache $R_{2}$ .

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

Schritt 1.7

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 1.7.1

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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

Schritt 1.7.2

Vereinfache $R_{1}$ .

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

Schritt 1.8

Perform the row operation $R_{1} = R_{1} - \frac{4}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 1.8.1

Perform the row operation $R_{1} = R_{1} - \frac{4}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Schritt 1.8.2

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 $3$

Schritt 3

The rank is the number of pivot columns.

$3$