Lineare Algebra Beispiele

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 & 2 1 & 6 & 2 & 1 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 3 & 2 & 4 & 2 \\ 1 & 6 & 2 & 1 \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{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{matrix} \frac{3}{3} & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 1 & 6 & 2 & 1 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 1 & 6 & 2 & 1 \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 1 & 6 & 2 & 1 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 1 & 6 & 2 & 1 \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 1 & 6 & 2 & 1 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 1 & 6 & 2 & 1 \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

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{2}{3} & \frac{4}{3} & \frac{2}{3} 1 - 1 & 6 - \frac{2}{3} & 2 - \frac{4}{3} & 1 - \frac{2}{3} 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 1 - 1 & 6 - \frac{2}{3} & 2 - \frac{4}{3} & 1 - \frac{2}{3} \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 4 & 2 & 2 & 4 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 4 & 2 & 2 & 4 \\ 2 & 1 & 1 & 2 \end{array}]$

Schritt 1.3

Perform the row operation

R_{3} = R_{3} - 4 R_{1}

$R_{3} = R_{3} - 4 R_{1}$ 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} - 4 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 4 - 4 \cdot 1 & 2 - 4 (\frac{2}{3}) & 2 - 4 (\frac{4}{3}) & 4 - 4 (\frac{2}{3}) 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 4 - 4 \cdot 1 & 2 - 4 (\frac{2}{3}) & 2 - 4 (\frac{4}{3}) & 4 - 4 (\frac{2}{3}) \\ 2 & 1 & 1 & 2 \end{array}]$

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} \\ 2 & 1 & 1 & 2 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} 2 & 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} \\ 2 & 1 & 1 & 2 \end{array}]$

Schritt 1.4

Perform the row operation

R_{4} = R_{4} - 2 R_{1}

$R_{4} = R_{4} - 2 R_{1}$ to make the entry at

4, 1

$4, 1$ a

0

$0$ .

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Schritt 1.4.1

Perform the row operation

R_{4} = R_{4} - 2 R_{1}

$R_{4} = R_{4} - 2 R_{1}$ to make the entry at

4, 1

$4, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} 2 - 2 \cdot 1 & 1 - 2 (\frac{2}{3}) & 1 - 2 (\frac{4}{3}) & 2 - 2 (\frac{2}{3}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} \\ 2 - 2 \cdot 1 & 1 - 2 (\frac{2}{3}) & 1 - 2 (\frac{4}{3}) & 2 - 2 (\frac{2}{3}) \end{array}]$

Schritt 1.4.2

Vereinfache

R_{4}

$R_{4}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} \\ 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} \\ 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} 0 & \frac{16}{3} & \frac{2}{3} & \frac{1}{3} 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Schritt 1.5

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 1.5.1

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{matrix} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \frac{3}{16} \cdot 0 & \frac{3}{16} \cdot \frac{16}{3} & \frac{3}{16} \cdot \frac{2}{3} & \frac{3}{16} \cdot \frac{1}{3} 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ \frac{3}{16} \cdot 0 & \frac{3}{16} \cdot \frac{16}{3} & \frac{3}{16} \cdot \frac{2}{3} & \frac{3}{16} \cdot \frac{1}{3} \\ 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} \\ 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{array}]$

Schritt 1.5.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & - \frac{2}{3} & - \frac{10}{3} & \frac{4}{3} \\ 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{array}]$

Schritt 1.6

Perform the row operation

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

$3, 2$ a

$0$ .

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Schritt 1.6.1

Perform the row operation

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

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & - \frac{10}{3} + \frac{2}{3} \cdot \frac{1}{8} & \frac{4}{3} + \frac{2}{3} \cdot \frac{1}{16} \\ 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & 0 & - \frac{13}{4} & \frac{11}{8} \\ 0 & - \frac{1}{3} & - \frac{5}{3} & \frac{2}{3} \end{array}]$

Schritt 1.7

Perform the row operation

$R_{4} = R_{4} + \frac{1}{3} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Schritt 1.7.1

Perform the row operation

$R_{4} = R_{4} + \frac{1}{3} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & 0 & - \frac{13}{4} & \frac{11}{8} \\ 0 + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & - \frac{5}{3} + \frac{1}{3} \cdot \frac{1}{8} & \frac{2}{3} + \frac{1}{3} \cdot \frac{1}{16} \end{array}]$

Schritt 1.7.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & 0 & - \frac{13}{4} & \frac{11}{8} \\ 0 & 0 & - \frac{13}{8} & \frac{11}{16} \end{array}]$

Schritt 1.8

Multiply each element of

$R_{3}$ by

$- \frac{4}{13}$ to make the entry at

$3, 3$ a

$1$ .

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Schritt 1.8.1

Multiply each element of

$R_{3}$ by

$- \frac{4}{13}$ to make the entry at

$3, 3$ a

$1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ - \frac{4}{13} \cdot 0 & - \frac{4}{13} \cdot 0 & - \frac{4}{13} (- \frac{13}{4}) & - \frac{4}{13} \cdot \frac{11}{8} \\ 0 & 0 & - \frac{13}{8} & \frac{11}{16} \end{array}]$

Schritt 1.8.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & - \frac{13}{8} & \frac{11}{16} \end{array}]$

Schritt 1.9

Perform the row operation

$R_{4} = R_{4} + \frac{13}{8} R_{3}$ to make the entry at

$4, 3$ a

$0$ .

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Schritt 1.9.1

Perform the row operation

$R_{4} = R_{4} + \frac{13}{8} R_{3}$ to make the entry at

$4, 3$ a

$0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 + \frac{13}{8} \cdot 0 & 0 + \frac{13}{8} \cdot 0 & - \frac{13}{8} + \frac{13}{8} \cdot 1 & \frac{11}{16} + \frac{13}{8} (- \frac{11}{26}) \end{array}]$

Schritt 1.9.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & \frac{1}{8} & \frac{1}{16} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.10

Perform the row operation

$R_{2} = R_{2} - \frac{1}{8} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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Schritt 1.10.1

Perform the row operation

$R_{2} = R_{2} - \frac{1}{8} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 - \frac{1}{8} \cdot 0 & 1 - \frac{1}{8} \cdot 0 & \frac{1}{8} - \frac{1}{8} \cdot 1 & \frac{1}{16} - \frac{1}{8} (- \frac{11}{26}) \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.10.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{2}{3} \\ 0 & 1 & 0 & \frac{3}{26} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.11

Perform the row operation

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

$1, 3$ a

$0$ .

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Schritt 1.11.1

Perform the row operation

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

$1, 3$ a

$0$ .

$[\begin{array}{c} 1 - \frac{4}{3} \cdot 0 & \frac{2}{3} - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & \frac{2}{3} - \frac{4}{3} (- \frac{11}{26}) \\ 0 & 1 & 0 & \frac{3}{26} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.11.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & 0 & \frac{16}{13} \\ 0 & 1 & 0 & \frac{3}{26} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.12

Perform the row operation

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

$1, 2$ a

$0$ .

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Schritt 1.12.1

Perform the row operation

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

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & 0 - \frac{2}{3} \cdot 0 & \frac{16}{13} - \frac{2}{3} \cdot \frac{3}{26} \\ 0 & 1 & 0 & \frac{3}{26} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 1.12.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{15}{13} \\ 0 & 1 & 0 & \frac{3}{26} \\ 0 & 0 & 1 & - \frac{11}{26} \\ 0 & 0 & 0 & 0 \end{array}]$

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:

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

$a_{33}$

Pivot Columns:

$1, 2,$ and

$3$