Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 7 & 5 & 4 & 7 - 1 & 5 & 2 & 4 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 7 & 5 & 4 & 7 \\ - 1 & 5 & 2 & 4 \\ 7 & 0 & 2 & 2 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{7}

$\frac{1}{7}$ 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}{7}

$\frac{1}{7}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{7}{7} & \frac{5}{7} & \frac{4}{7} & \frac{7}{7} - 1 & 5 & 2 & 4 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{7}{7} & \frac{5}{7} & \frac{4}{7} & \frac{7}{7} \\ - 1 & 5 & 2 & 4 \\ 7 & 0 & 2 & 2 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 - 1 & 5 & 2 & 4 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ - 1 & 5 & 2 & 4 \\ 7 & 0 & 2 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 - 1 & 5 & 2 & 4 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ - 1 & 5 & 2 & 4 \\ 7 & 0 & 2 & 2 \end{array}]$

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{5}{7} & \frac{4}{7} & 1 - 1 + 1 \cdot 1 & 5 + \frac{5}{7} & 2 + \frac{4}{7} & 4 + 1 \cdot 1 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ - 1 + 1 \cdot 1 & 5 + \frac{5}{7} & 2 + \frac{4}{7} & 4 + 1 \cdot 1 \\ 7 & 0 & 2 & 2 \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & \frac{40}{7} & \frac{18}{7} & 5 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & \frac{40}{7} & \frac{18}{7} & 5 \\ 7 & 0 & 2 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & \frac{40}{7} & \frac{18}{7} & 5 7 & 0 & 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & \frac{40}{7} & \frac{18}{7} & 5 \\ 7 & 0 & 2 & 2 \end{array}]$

Schritt 1.3

Perform the row operation

R_{3} = R_{3} - 7 R_{1}

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Schritt 1.3.1

Perform the row operation

R_{3} = R_{3} - 7 R_{1}

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & \frac{40}{7} & \frac{18}{7} & 5 7 - 7 \cdot 1 & 0 - 7 (\frac{5}{7}) & 2 - 7 (\frac{4}{7}) & 2 - 7 \cdot 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & \frac{40}{7} & \frac{18}{7} & 5 \\ 7 - 7 \cdot 1 & 0 - 7 (\frac{5}{7}) & 2 - 7 (\frac{4}{7}) & 2 - 7 \cdot 1 \end{array}]$

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & \frac{40}{7} & \frac{18}{7} & 5 0 & - 5 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & \frac{40}{7} & \frac{18}{7} & 5 \\ 0 & - 5 & - 2 & - 5 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & \frac{40}{7} & \frac{18}{7} & 5 0 & - 5 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & \frac{40}{7} & \frac{18}{7} & 5 \\ 0 & - 5 & - 2 & - 5 \end{array}]$

Schritt 1.4

Multiply each element of

R_{2}

$R_{2}$ by

\frac{7}{40}

$\frac{7}{40}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Schritt 1.4.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{7}{40}

$\frac{7}{40}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 \frac{7}{40} \cdot 0 & \frac{7}{40} \cdot \frac{40}{7} & \frac{7}{40} \cdot \frac{18}{7} & \frac{7}{40} \cdot 5 0 & - 5 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ \frac{7}{40} \cdot 0 & \frac{7}{40} \cdot \frac{40}{7} & \frac{7}{40} \cdot \frac{18}{7} & \frac{7}{40} \cdot 5 \\ 0 & - 5 & - 2 & - 5 \end{array}]$

Schritt 1.4.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 & - 5 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 & - 5 & - 2 & - 5 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 & - 5 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 & - 5 & - 2 & - 5 \end{array}]$

Schritt 1.5

Perform the row operation

R_{3} = R_{3} + 5 R_{2}

$R_{3} = R_{3} + 5 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Schritt 1.5.1

Perform the row operation

R_{3} = R_{3} + 5 R_{2}

$R_{3} = R_{3} + 5 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 2 + 5 (\frac{9}{20}) & - 5 + 5 (\frac{7}{8}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 2 + 5 (\frac{9}{20}) & - 5 + 5 (\frac{7}{8}) \end{array}]$

Schritt 1.5.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 & 0 & \frac{1}{4} & - \frac{5}{8} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 & 0 & \frac{1}{4} & - \frac{5}{8} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 & 0 & \frac{1}{4} & - \frac{5}{8} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 & 0 & \frac{1}{4} & - \frac{5}{8} \end{array}]$

Schritt 1.6

Multiply each element of

R_{3}

$R_{3}$ by

4

$4$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Schritt 1.6.1

Multiply each element of

R_{3}

$R_{3}$ by

4

$4$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 4 \cdot 0 & 4 \cdot 0 & 4 (\frac{1}{4}) & 4 (- \frac{5}{8}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 4 \cdot 0 & 4 \cdot 0 & 4 (\frac{1}{4}) & 4 (- \frac{5}{8}) \end{array}]$

Schritt 1.6.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & \frac{9}{20} & \frac{7}{8} 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & \frac{9}{20} & \frac{7}{8} \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

Schritt 1.7

Perform the row operation

R_{2} = R_{2} - \frac{9}{20} R_{3}

$R_{2} = R_{2} - \frac{9}{20} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Schritt 1.7.1

Perform the row operation

R_{2} = R_{2} - \frac{9}{20} R_{3}

$R_{2} = R_{2} - \frac{9}{20} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 - \frac{9}{20} \cdot 0 & 1 - \frac{9}{20} \cdot 0 & \frac{9}{20} - \frac{9}{20} \cdot 1 & \frac{7}{8} - \frac{9}{20} (- \frac{5}{2}) 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 - \frac{9}{20} \cdot 0 & 1 - \frac{9}{20} \cdot 0 & \frac{9}{20} - \frac{9}{20} \cdot 1 & \frac{7}{8} - \frac{9}{20} (- \frac{5}{2}) \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

Schritt 1.7.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & \frac{4}{7} & 1 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{4}{7} & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

Schritt 1.8

Perform the row operation

R_{1} = R_{1} - \frac{4}{7} R_{3}

$R_{1} = R_{1} - \frac{4}{7} R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Schritt 1.8.1

Perform the row operation

R_{1} = R_{1} - \frac{4}{7} R_{3}

$R_{1} = R_{1} - \frac{4}{7} R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 - \frac{4}{7} \cdot 0 & \frac{5}{7} - \frac{4}{7} \cdot 0 & \frac{4}{7} - \frac{4}{7} \cdot 1 & 1 - \frac{4}{7} (- \frac{5}{2}) 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - \frac{4}{7} \cdot 0 & \frac{5}{7} - \frac{4}{7} \cdot 0 & \frac{4}{7} - \frac{4}{7} \cdot 1 & 1 - \frac{4}{7} (- \frac{5}{2}) \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

Schritt 1.8.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & 0 & \frac{17}{7} 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & 0 & \frac{17}{7} \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{7} & 0 & \frac{17}{7} 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{7} & 0 & \frac{17}{7} \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

Schritt 1.9

Perform the row operation

R_{1} = R_{1} - \frac{5}{7} R_{2}

$R_{1} = R_{1} - \frac{5}{7} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Schritt 1.9.1

Perform the row operation

R_{1} = R_{1} - \frac{5}{7} R_{2}

$R_{1} = R_{1} - \frac{5}{7} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & 0 - \frac{5}{7} \cdot 0 & \frac{17}{7} - \frac{5}{7} \cdot 2 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & 0 - \frac{5}{7} \cdot 0 & \frac{17}{7} - \frac{5}{7} \cdot 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - \frac{5}{2} \end{array}]$

Schritt 1.9.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 1 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 1 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 1 0 & 1 & 0 & 2 0 & 0 & 1 & - \frac{5}{2} \end{matrix} ⎤ ⎥ ⎥ ⎦

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

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_{22},

$a_{11}, a_{22},$ and

a_{33}

$a_{33}$

Pivot Columns:

1, 2,

$1, 2,$ and

3

$3$