Lineare Algebra Beispiele

R = [\begin{matrix} 5 & 8 1 & 2 \end{matrix}]

$R = [\begin{array}{c} 5 & 8 \\ 1 & 2 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{5}

$\frac{1}{5}$ 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}{5}

$\frac{1}{5}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{matrix} \frac{5}{5} & \frac{8}{5} 1 & 2 \end{matrix}]

$[\begin{array}{c} \frac{5}{5} & \frac{8}{5} \\ 1 & 2 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & \frac{8}{5} 1 & 2 \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 1 & 2 \end{array}]$

[\begin{matrix} 1 & \frac{8}{5} 1 & 2 \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 1 & 2 \end{array}]$

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 & \frac{8}{5} 1 - 1 & 2 - \frac{8}{5} \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 1 - 1 & 2 - \frac{8}{5} \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & \frac{8}{5} 0 & \frac{2}{5} \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 0 & \frac{2}{5} \end{array}]$

[\begin{matrix} 1 & \frac{8}{5} 0 & \frac{2}{5} \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 0 & \frac{2}{5} \end{array}]$

Schritt 1.3

Multiply each element of

R_{2}

$R_{2}$ by

\frac{5}{2}

$\frac{5}{2}$ 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

\frac{5}{2}

$\frac{5}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{matrix} 1 & \frac{8}{5} \frac{5}{2} \cdot 0 & \frac{5}{2} \cdot \frac{2}{5} \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ \frac{5}{2} \cdot 0 & \frac{5}{2} \cdot \frac{2}{5} \end{array}]$

Schritt 1.3.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & \frac{8}{5} 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 0 & 1 \end{array}]$

[\begin{matrix} 1 & \frac{8}{5} 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & \frac{8}{5} \\ 0 & 1 \end{array}]$

Schritt 1.4

Perform the row operation

R_{1} = R_{1} - \frac{8}{5} R_{2}

$R_{1} = R_{1} - \frac{8}{5} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Schritt 1.4.1

Perform the row operation

R_{1} = R_{1} - \frac{8}{5} R_{2}

$R_{1} = R_{1} - \frac{8}{5} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{matrix} 1 - \frac{8}{5} \cdot 0 & \frac{8}{5} - \frac{8}{5} \cdot 1 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 - \frac{8}{5} \cdot 0 & \frac{8}{5} - \frac{8}{5} \cdot 1 \\ 0 & 1 \end{array}]$

Schritt 1.4.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

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:

r_{11}

$r_{11}$ and

r_{22}

$r_{22}$

Pivot Columns:

1

$1$ and

2

$2$