Lineare Algebra Beispiele

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

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

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

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Schritt 1.1.1

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

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

Schritt 1.1.2

Vereinfache $R_{2}$ .

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

Schritt 1.2

Perform the row operation $R_{1} = R_{1} - 5 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 1.2.1

Perform the row operation $R_{1} = R_{1} - 5 R_{3}$ to make the entry at $1, 3$ a $0$ .

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

Schritt 1.2.2

Vereinfache $R_{1}$ .

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

Schritt 1.3

Perform the row operation $R_{1} = R_{1} - 6 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 1.3.1

Perform the row operation $R_{1} = R_{1} - 6 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Schritt 1.3.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 13 \\ 0 & 1 & 0 & - 5 \\ 0 & 0 & 1 & 2 \end{array}]$

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

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