Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 2 & 7 & - 1 & 8 & 6 \\ - 1 & - 3 & 2 & - 4 & 3 \\ 1 & 0 & - 6 & 5 & - 17 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

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 1.1.1

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

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 2 - 2 \cdot 1 & 7 - 2 \cdot 3 & - 1 - 2 \cdot - 1 & 8 - 2 \cdot 4 & 6 - 2 \cdot 1 \\ - 1 & - 3 & 2 & - 4 & 3 \\ 1 & 0 & - 6 & 5 & - 17 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 0 & 1 & 1 & 0 & 4 \\ - 1 & - 3 & 2 & - 4 & 3 \\ 1 & 0 & - 6 & 5 & - 17 \end{array}]$

Schritt 1.2

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

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Schritt 1.2.1

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

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 0 & 1 & 1 & 0 & 4 \\ - 1 + 1 \cdot 1 & - 3 + 1 \cdot 3 & 2 - 1 & - 4 + 1 \cdot 4 & 3 + 1 \cdot 1 \\ 1 & 0 & - 6 & 5 & - 17 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 0 & 1 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 & 4 \\ 1 & 0 & - 6 & 5 & - 17 \end{array}]$

Schritt 1.3

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

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Schritt 1.3.1

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

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 0 & 1 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 & 4 \\ 1 - 1 & 0 - 3 & - 6 + 1 & 5 - 4 & - 17 - 1 \end{array}]$

Schritt 1.3.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 0 & 1 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 & 4 \\ 0 & - 3 & - 5 & 1 & - 18 \end{array}]$

Schritt 1.4

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 1.4.1

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

$[\begin{array}{c} 1 & 3 & - 1 & 4 & 1 \\ 0 & 1 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 & 4 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 5 + 3 \cdot 1 & 1 + 3 \cdot 0 & - 18 + 3 \cdot 4 \end{array}]$

Schritt 1.4.2

Vereinfache $R_{4}$ .

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

Schritt 1.5

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

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Schritt 1.5.1

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

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

Schritt 1.5.2

Vereinfache $R_{4}$ .

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

Schritt 1.6

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

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Schritt 1.6.1

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

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

Schritt 1.6.2

Vereinfache $R_{1}$ .

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

Schritt 1.7

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

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Schritt 1.7.1

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

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

Schritt 1.7.2

Vereinfache $R_{2}$ .

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

Schritt 1.8

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

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Schritt 1.8.1

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

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

Schritt 1.8.2

Vereinfache $R_{1}$ .

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

Schritt 1.9

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

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Schritt 1.9.1

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

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

Schritt 1.9.2

Vereinfache $R_{1}$ .

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

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 $4$