Lineare Algebra Beispiele

[\begin{matrix} 4 & 14 & h - 2 & - 7 & 4 \end{matrix}]

$[\begin{array}{c} 4 & 14 & h \\ - 2 & - 7 & 4 \end{array}]$

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

$1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{14}{4} & \frac{h}{4} \\ - 2 & - 7 & 4 \end{array}]$

Schritt 1.1.2

Vereinfache

$R_{1}$ .

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

Schritt 1.2

Perform the row operation

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

$2, 1$ a

$0$ .

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Schritt 1.2.1

Perform the row operation

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

$2, 1$ a

$0$ .

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

Schritt 1.2.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{7}{2} & \frac{h}{4} \\ 0 & 0 & 4 + \frac{h}{2} \end{array}]$

Schritt 1.3

Multiply each element of

$R_{2}$ by

$\frac{1}{4 + \frac{h}{2}}$ to make the entry at

$2, 3$ a

$1$ .

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Schritt 1.3.1

Multiply each element of

$R_{2}$ by

$\frac{1}{4 + \frac{h}{2}}$ to make the entry at

$2, 3$ a

$1$ .

$[\begin{array}{c} 1 & \frac{7}{2} & \frac{h}{4} \\ \frac{0}{4 + \frac{h}{2}} & \frac{0}{4 + \frac{h}{2}} & \frac{4 + \frac{h}{2}}{4 + \frac{h}{2}} \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{7}{2} & \frac{h}{4} \\ 0 & 0 & 1 \end{array}]$

Schritt 1.4

Perform the row operation

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

$1, 3$ a

$0$ .

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Schritt 1.4.1

Perform the row operation

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

$1, 3$ a

$0$ .

$[\begin{array}{c} 1 - \frac{h}{4} \cdot 0 & \frac{7}{2} - \frac{h}{4} \cdot 0 & \frac{h}{4} - \frac{h}{4} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{1}$ .

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

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

$a_{23}$

Pivot Columns:

$1$ and

$3$