Lineare Algebra Beispiele

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

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

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Schritt 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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

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

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 \\ 0 \\ 3 \end{array}]

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

[\begin{array}{c} 1 \\ 0 \\ 3 \end{array}]

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

Schritt 1.2

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

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Schritt 1.2.1

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 \\ 0 \\ 3 - 3 \cdot 1 \end{array}]

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

Schritt 1.2.2

Vereinfache

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}]

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

[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}]

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

[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}]

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

Schritt 2

The pivot positions are the locations with the leading

1

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

a_{11}

$a_{11}$

Pivot Columns:

1

$1$