Lineare Algebra Beispiele

$[\begin{array}{c} 2 & 2 & 4 & 0 \\ - 4 & - 4 & - 8 & 0 \\ 0 & - 7 & 21 & 0 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{2}{2} & \frac{4}{2} & \frac{0}{2} \\ - 4 & - 4 & - 8 & 0 \\ 0 & - 7 & 21 & 0 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ - 4 & - 4 & - 8 & 0 \\ 0 & - 7 & 21 & 0 \end{array}]$

Schritt 1.2

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

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Schritt 1.2.1

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

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ - 4 + 4 \cdot 1 & - 4 + 4 \cdot 1 & - 8 + 4 \cdot 2 & 0 + 4 \cdot 0 \\ 0 & - 7 & 21 & 0 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 7 & 21 & 0 \end{array}]$

Schritt 1.3

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ 0 & - 7 & 21 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.4

Multiply each element of $R_{2}$ by $- \frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

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Schritt 1.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot - 7 & - \frac{1}{7} \cdot 21 & - \frac{1}{7} \cdot 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.4.2

Vereinfache $R_{2}$ .

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

Schritt 1.5

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

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Schritt 1.5.1

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

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

Schritt 1.5.2

Vereinfache $R_{1}$ .

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

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

Pivot Columns: $1$ and $2$