Lineare Algebra Beispiele

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

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 $1$ .

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

Schritt 1.1.2

Vereinfache $R_{1}$ .

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

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 $0$ .

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

Schritt 1.2.2

Vereinfache $R_{2}$ .

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

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 $0$ .

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

Schritt 1.3.2

Vereinfache $R_{3}$ .

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

Schritt 1.4

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

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Schritt 1.4.1

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

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

Schritt 1.4.2

Vereinfache $R_{4}$ .

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

Schritt 1.5

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

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Schritt 1.5.1

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

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

Schritt 1.5.2

Vereinfache $R_{2}$ .

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

Schritt 1.6

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

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Schritt 1.6.1

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

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

Schritt 1.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & - 34 & - 22 \\ 0 & - \frac{1}{2} & - \frac{7}{2} & 0 \end{array}]$

Schritt 1.7

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

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Schritt 1.7.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & - 34 & - 22 \\ 0 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & - \frac{7}{2} + \frac{1}{2} \cdot 9 & 0 + \frac{1}{2} \cdot 6 \end{array}]$

Schritt 1.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & - 34 & - 22 \\ 0 & 0 & 1 & 3 \end{array}]$

Schritt 1.8

Multiply each element of $R_{3}$ by $- \frac{1}{34}$ to make the entry at $3, 3$ a $1$ .

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Schritt 1.8.1

Multiply each element of $R_{3}$ by $- \frac{1}{34}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ - \frac{1}{34} \cdot 0 & - \frac{1}{34} \cdot 0 & - \frac{1}{34} \cdot - 34 & - \frac{1}{34} \cdot - 22 \\ 0 & 0 & 1 & 3 \end{array}]$

Schritt 1.8.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & 1 & \frac{11}{17} \\ 0 & 0 & 1 & 3 \end{array}]$

Schritt 1.9

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

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Schritt 1.9.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & 1 & \frac{11}{17} \\ 0 - 0 & 0 - 0 & 1 - 1 & 3 - \frac{11}{17} \end{array}]$

Schritt 1.9.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & 1 & \frac{11}{17} \\ 0 & 0 & 0 & \frac{40}{17} \end{array}]$

Schritt 1.10

Multiply each element of $R_{4}$ by $\frac{17}{40}$ to make the entry at $4, 4$ a $1$ .

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Schritt 1.10.1

Multiply each element of $R_{4}$ by $\frac{17}{40}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & 1 & \frac{11}{17} \\ \frac{17}{40} \cdot 0 & \frac{17}{40} \cdot 0 & \frac{17}{40} \cdot 0 & \frac{17}{40} \cdot \frac{40}{17} \end{array}]$

Schritt 1.10.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 & 0 & 1 & \frac{11}{17} \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 1.11

Perform the row operation $R_{3} = R_{3} - \frac{11}{17} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 1.11.1

Perform the row operation $R_{3} = R_{3} - \frac{11}{17} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & 9 & 6 \\ 0 - \frac{11}{17} \cdot 0 & 0 - \frac{11}{17} \cdot 0 & 1 - \frac{11}{17} \cdot 0 & \frac{11}{17} - \frac{11}{17} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 1.11.2

Vereinfache $R_{3}$ .

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

Schritt 1.12

Perform the row operation $R_{2} = R_{2} - 6 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Schritt 1.12.1

Perform the row operation $R_{2} = R_{2} - 6 R_{4}$ to make the entry at $2, 4$ a $0$ .

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

Schritt 1.12.2

Vereinfache $R_{2}$ .

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

Schritt 1.13

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

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Schritt 1.13.1

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

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

Schritt 1.13.2

Vereinfache $R_{2}$ .

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

Schritt 1.14

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

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Schritt 1.14.1

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

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

Schritt 1.14.2

Vereinfache $R_{1}$ .

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

Schritt 1.15

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

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Schritt 1.15.1

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

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

Schritt 1.15.2

Vereinfache $R_{1}$ .

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

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: $a_{11}, a_{22}, a_{33},$ and $a_{44}$

Pivot Columns: $1, 2, 3,$ and $4$