Lineare Algebra Beispiele

[\begin{array}{c} 3 & 2 & 8 \\ 2 & - 4 & - 7 \end{array}]

$[\begin{array}{c} 3 & 2 & 8 \\ 2 & - 4 & - 7 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{3}

$\frac{1}{3}$ 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}{3}

$\frac{1}{3}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{8}{3} \\ 2 & - 4 & - 7 \end{array}]

$[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{8}{3} \\ 2 & - 4 & - 7 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 2 & - 4 & - 7 \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 2 & - 4 & - 7 \end{array}]$

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 2 & - 4 & - 7 \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 2 & - 4 & - 7 \end{array}]$

Schritt 1.2

Perform the row operation

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

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

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

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 2 - 2 \cdot 1 & - 4 - 2 (\frac{2}{3}) & - 7 - 2 (\frac{8}{3}) \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 2 - 2 \cdot 1 & - 4 - 2 (\frac{2}{3}) & - 7 - 2 (\frac{8}{3}) \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & - \frac{16}{3} & - \frac{37}{3} \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & - \frac{16}{3} & - \frac{37}{3} \end{array}]$

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & - \frac{16}{3} & - \frac{37}{3} \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & - \frac{16}{3} & - \frac{37}{3} \end{array}]$

Schritt 1.3

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{3}{16}

$- \frac{3}{16}$ 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{3}{16}

$- \frac{3}{16}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ - \frac{3}{16} \cdot 0 & - \frac{3}{16} (- \frac{16}{3}) & - \frac{3}{16} (- \frac{37}{3}) \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ - \frac{3}{16} \cdot 0 & - \frac{3}{16} (- \frac{16}{3}) & - \frac{3}{16} (- \frac{37}{3}) \end{array}]$

Schritt 1.3.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & 1 & \frac{37}{16} \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & 1 & \frac{37}{16} \end{array}]$

[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & 1 & \frac{37}{16} \end{array}]

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{8}{3} \\ 0 & 1 & \frac{37}{16} \end{array}]$

Schritt 1.4

Perform the row operation

R_{1} = R_{1} - \frac{2}{3} R_{2}

$R_{1} = R_{1} - \frac{2}{3} R_{2}$ 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{2}{3} R_{2}

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

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & \frac{8}{3} - \frac{2}{3} \cdot \frac{37}{16} \\ 0 & 1 & \frac{37}{16} \end{array}]

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & \frac{8}{3} - \frac{2}{3} \cdot \frac{37}{16} \\ 0 & 1 & \frac{37}{16} \end{array}]$

Schritt 1.4.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 & \frac{9}{8} \\ 0 & 1 & \frac{37}{16} \end{array}]

$[\begin{array}{c} 1 & 0 & \frac{9}{8} \\ 0 & 1 & \frac{37}{16} \end{array}]$

[\begin{array}{c} 1 & 0 & \frac{9}{8} \\ 0 & 1 & \frac{37}{16} \end{array}]

$[\begin{array}{c} 1 & 0 & \frac{9}{8} \\ 0 & 1 & \frac{37}{16} \end{array}]$

[\begin{array}{c} 1 & 0 & \frac{9}{8} \\ 0 & 1 & \frac{37}{16} \end{array}]

$[\begin{array}{c} 1 & 0 & \frac{9}{8} \\ 0 & 1 & \frac{37}{16} \end{array}]$

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}

$a_{11}$ and

a_{22}

$a_{22}$

Pivot Columns:

1

$1$ and

2

$2$