Lineare Algebra Beispiele

[\begin{matrix} 2 & 6 & a 1 & 2 & b \end{matrix}]

$[\begin{array}{c} 2 & 6 & a \\ 1 & 2 & b \end{array}]$

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{matrix} \frac{2}{2} & \frac{6}{2} & \frac{a}{2} 1 & 2 & b \end{matrix}]

$[\begin{array}{c} \frac{2}{2} & \frac{6}{2} & \frac{a}{2} \\ 1 & 2 & b \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & 3 & \frac{a}{2} 1 & 2 & b \end{matrix}]

$[\begin{array}{c} 1 & 3 & \frac{a}{2} \\ 1 & 2 & b \end{array}]$

[\begin{matrix} 1 & 3 & \frac{a}{2} 1 & 2 & b \end{matrix}]

$[\begin{array}{c} 1 & 3 & \frac{a}{2} \\ 1 & 2 & b \end{array}]$

Schritt 1.2

Perform the row operation

R_{2} = R_{2} - R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

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Schritt 1.2.1

Perform the row operation

R_{2} = R_{2} - R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

[\begin{matrix} 1 & 3 & \frac{a}{2} 1 - 1 & 2 - 3 & b - \frac{a}{2} \end{matrix}]

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

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & 3 & \frac{a}{2} 0 & - 1 & b - \frac{a}{2} \end{matrix}]

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

[\begin{matrix} 1 & 3 & \frac{a}{2} 0 & - 1 & b - \frac{a}{2} \end{matrix}]

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

Schritt 1.3

Multiply each element of

R_{2}

$R_{2}$ by

- 1

$- 1$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Schritt 1.3.1

Multiply each element of

R_{2}

$R_{2}$ by

- 1

$- 1$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{matrix} 1 & 3 & \frac{a}{2} - 0 & - - 1 & - (b - \frac{a}{2}) \end{matrix}]

$[\begin{array}{c} 1 & 3 & \frac{a}{2} \\ - 0 & - - 1 & - (b - \frac{a}{2}) \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{2}$ .

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

Schritt 1.4

Perform the row operation

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

$1, 2$ a

$0$ .

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Schritt 1.4.1

Perform the row operation

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

$1, 2$ a

$0$ .

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

Schritt 1.4.2

Vereinfache

$R_{1}$ .

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

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

$2$