Lineare Algebra Beispiele

$[\begin{array}{c} 2 & - 1 & 1 & 6 & 8 \\ 1 & - 2 & - 4 & 3 & - 2 \\ - 7 & 8 & 10 & 3 & - 10 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{- 1}{2} & \frac{1}{2} & \frac{6}{2} & \frac{8}{2} \\ 1 & - 2 & - 4 & 3 & - 2 \\ - 7 & 8 & 10 & 3 & - 10 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 1 & - 2 & - 4 & 3 & - 2 \\ - 7 & 8 & 10 & 3 & - 10 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 1 - 1 & - 2 + \frac{1}{2} & - 4 - \frac{1}{2} & 3 - 3 & - 2 - 4 \\ - 7 & 8 & 10 & 3 & - 10 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & - \frac{3}{2} & - \frac{9}{2} & 0 & - 6 \\ - 7 & 8 & 10 & 3 & - 10 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

Schritt 1.3

Perform the row operation $R_{3} = R_{3} + 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 1.3.1

Perform the row operation $R_{3} = R_{3} + 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & - \frac{3}{2} & - \frac{9}{2} & 0 & - 6 \\ - 7 + 7 \cdot 1 & 8 + 7 (- \frac{1}{2}) & 10 + 7 (\frac{1}{2}) & 3 + 7 \cdot 3 & - 10 + 7 \cdot 4 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

Schritt 1.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & - \frac{3}{2} & - \frac{9}{2} & 0 & - 6 \\ 0 & \frac{9}{2} & \frac{27}{2} & 24 & 18 \\ 4 & - 5 & - 7 & 0 & 4 \end{array}]$

Schritt 1.4

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

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Schritt 1.4.1

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

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & - \frac{3}{2} & - \frac{9}{2} & 0 & - 6 \\ 0 & \frac{9}{2} & \frac{27}{2} & 24 & 18 \\ 4 - 4 \cdot 1 & - 5 - 4 (- \frac{1}{2}) & - 7 - 4 (\frac{1}{2}) & 0 - 4 \cdot 3 & 4 - 4 \cdot 4 \end{array}]$

Schritt 1.4.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & - \frac{3}{2} & - \frac{9}{2} & 0 & - 6 \\ 0 & \frac{9}{2} & \frac{27}{2} & 24 & 18 \\ 0 & - 3 & - 9 & - 12 & - 12 \end{array}]$

Schritt 1.5

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

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Schritt 1.5.1

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

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} (- \frac{9}{2}) & - \frac{2}{3} \cdot 0 & - \frac{2}{3} \cdot - 6 \\ 0 & \frac{9}{2} & \frac{27}{2} & 24 & 18 \\ 0 & - 3 & - 9 & - 12 & - 12 \end{array}]$

Schritt 1.5.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & \frac{9}{2} & \frac{27}{2} & 24 & 18 \\ 0 & - 3 & - 9 & - 12 & - 12 \end{array}]$

Schritt 1.6

Perform the row operation $R_{3} = R_{3} - \frac{9}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 1.6.1

Perform the row operation $R_{3} = R_{3} - \frac{9}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 - \frac{9}{2} \cdot 0 & \frac{9}{2} - \frac{9}{2} \cdot 1 & \frac{27}{2} - \frac{9}{2} \cdot 3 & 24 - \frac{9}{2} \cdot 0 & 18 - \frac{9}{2} \cdot 4 \\ 0 & - 3 & - 9 & - 12 & - 12 \end{array}]$

Schritt 1.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & 0 & 0 & 24 & 0 \\ 0 & - 3 & - 9 & - 12 & - 12 \end{array}]$

Schritt 1.7

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

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Schritt 1.7.1

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

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & 0 & 0 & 24 & 0 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 9 + 3 \cdot 3 & - 12 + 3 \cdot 0 & - 12 + 3 \cdot 4 \end{array}]$

Schritt 1.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & 0 & 0 & 24 & 0 \\ 0 & 0 & 0 & - 12 & 0 \end{array}]$

Schritt 1.8

Multiply each element of $R_{3}$ by $\frac{1}{24}$ to make the entry at $3, 4$ a $1$ .

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Schritt 1.8.1

Multiply each element of $R_{3}$ by $\frac{1}{24}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ \frac{0}{24} & \frac{0}{24} & \frac{0}{24} & \frac{24}{24} & \frac{0}{24} \\ 0 & 0 & 0 & - 12 & 0 \end{array}]$

Schritt 1.8.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 12 & 0 \end{array}]$

Schritt 1.9

Perform the row operation $R_{4} = R_{4} + 12 R_{3}$ to make the entry at $4, 4$ a $0$ .

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Schritt 1.9.1

Perform the row operation $R_{4} = R_{4} + 12 R_{3}$ to make the entry at $4, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & \frac{1}{2} & 3 & 4 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 \\ 0 + 12 \cdot 0 & 0 + 12 \cdot 0 & 0 + 12 \cdot 0 & - 12 + 12 \cdot 1 & 0 + 12 \cdot 0 \end{array}]$

Schritt 1.9.2

Vereinfache $R_{4}$ .

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

Schritt 1.10

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

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Schritt 1.10.1

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

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

Schritt 1.10.2

Vereinfache $R_{1}$ .

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

Schritt 1.11

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

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Schritt 1.11.1

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

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

Schritt 1.11.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 2 & 0 & 6 \\ 0 & 1 & 3 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

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 $a_{34}$

Pivot Columns: $1, 2,$ and $4$