Lineare Algebra Beispiele

$[\begin{array}{c} 4 & 3 & 26 \\ 3 & 11 & - 7 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{4}$ to make the entry at

$1, 1$ a

$1$ .

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Schritt 1.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{4}$ to make the entry at

$1, 1$ a

$1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{3}{4} & \frac{26}{4} \\ 3 & 11 & - 7 \end{array}]$

Schritt 1.1.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{4} & \frac{13}{2} \\ 3 & 11 & - 7 \end{array}]$

Schritt 1.2

Perform the row operation

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Schritt 1.2.1

Perform the row operation

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{3}{4} & \frac{13}{2} \\ 3 - 3 \cdot 1 & 11 - 3 (\frac{3}{4}) & - 7 - 3 (\frac{13}{2}) \end{array}]$

Schritt 1.2.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{4} & \frac{13}{2} \\ 0 & \frac{35}{4} & - \frac{53}{2} \end{array}]$

Schritt 1.3

Multiply each element of

$R_{2}$ by

$\frac{4}{35}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.3.1

Multiply each element of

$R_{2}$ by

$\frac{4}{35}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & \frac{3}{4} & \frac{13}{2} \\ \frac{4}{35} \cdot 0 & \frac{4}{35} \cdot \frac{35}{4} & \frac{4}{35} (- \frac{53}{2}) \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{4} & \frac{13}{2} \\ 0 & 1 & - \frac{106}{35} \end{array}]$

Schritt 1.4

Perform the row operation

$R_{1} = R_{1} - \frac{3}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.4.1

Perform the row operation

$R_{1} = R_{1} - \frac{3}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{3}{4} \cdot 0 & \frac{3}{4} - \frac{3}{4} \cdot 1 & \frac{13}{2} - \frac{3}{4} (- \frac{106}{35}) \\ 0 & 1 & - \frac{106}{35} \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{307}{35} \\ 0 & 1 & - \frac{106}{35} \end{array}]$

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

$2$