Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 3 & 7 & 5.65 \\ 1 & 4 & 10 & 7.4 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

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

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Schritt 1.1.1

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

$[\begin{array}{c} 1 & 3 & 7 & 5.65 \\ 1 - 1 & 4 - 3 & 10 - 7 & 7.4 - 5.65 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 7 & 5.65 \\ 0 & 1 & 3 & 1.75 \end{array}]$

Schritt 1.2

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 1.2.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 7 - 3 \cdot 3 & 5.65 - 3 \cdot 1.75 \\ 0 & 1 & 3 & 1.75 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0.4 \\ 0 & 1 & 3 & 1.75 \end{array}]$

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

Pivot Columns: $1$ and $2$