Lineare Algebra Beispiele

[\begin{matrix} 3 & - 2 & 14 1 & 3 & 1 \end{matrix}]

$[\begin{array}{c} 3 & - 2 & 14 \\ 1 & 3 & 1 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{3}$ to make the entry at

$1, 1$ a

$1$ .

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Schritt 1.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{3}$ to make the entry at

$1, 1$ a

$1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{14}{3} \\ 1 & 3 & 1 \end{array}]$

Schritt 1.1.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{14}{3} \\ 1 & 3 & 1 \end{array}]$

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

$0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{14}{3} \\ 1 - 1 & 3 + \frac{2}{3} & 1 - \frac{14}{3} \end{array}]$

Schritt 1.2.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{14}{3} \\ 0 & \frac{11}{3} & - \frac{11}{3} \end{array}]$

Schritt 1.3

Multiply each element of

$R_{2}$ by

$\frac{3}{11}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.3.1

Multiply each element of

$R_{2}$ by

$\frac{3}{11}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{14}{3} \\ \frac{3}{11} \cdot 0 & \frac{3}{11} \cdot \frac{11}{3} & \frac{3}{11} (- \frac{11}{3}) \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{14}{3} \\ 0 & 1 & - 1 \end{array}]$

Schritt 1.4

Perform the row operation

$R_{1} = R_{1} + \frac{2}{3} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.4.1

Perform the row operation

$R_{1} = R_{1} + \frac{2}{3} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & \frac{14}{3} + \frac{2}{3} \cdot - 1 \\ 0 & 1 & - 1 \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{1}$ .

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

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:

$a_{11}$ and

$a_{22}$

Pivot Columns:

$1$ and

$2$