Lineare Algebra Beispiele

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

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

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

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Schritt 1.1.1

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

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

Schritt 1.1.2

Vereinfache $R_{1}$ .

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

Schritt 1.2

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

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Schritt 1.2.1

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

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

Schritt 1.2.2

Vereinfache $R_{2}$ .

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

Schritt 1.3

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

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Schritt 1.3.1

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

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

Schritt 1.3.2

Vereinfache $R_{3}$ .

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

Schritt 1.4

Multiply each element of $R_{2}$ by $- \frac{2}{5}$ to make the entry at $2, 2$ a $1$ .

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Schritt 1.4.1

Multiply each element of $R_{2}$ by $- \frac{2}{5}$ to make the entry at $2, 2$ a $1$ .

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

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 $3$