Lineare Algebra Beispiele

- 3 x + y - z = - 2

$- 3 x + y - z = - 2$ ,

- 3 x + z = 4

$- 3 x + z = 4$ ,

y - 5 z = 0

$y - 5 z = 0$

Schritt 1

Write the system as a matrix.

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

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

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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 2.1.1

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

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

Schritt 2.1.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

Schritt 2.2

Perform the row operation

R_{2} = R_{2} + 3 R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

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Schritt 2.2.1

Perform the row operation

R_{2} = R_{2} + 3 R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} - 3 + 3 \cdot 1 & 0 + 3 (- \frac{1}{3}) & 1 + 3 (\frac{1}{3}) & 4 + 3 (\frac{2}{3}) 0 & 1 & - 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Schritt 2.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & - 1 & 2 & 6 0 & 1 & - 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & - 1 & 2 & 6 \\ 0 & 1 & - 5 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & - 1 & 2 & 6 0 & 1 & - 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & - 1 & 2 & 6 \\ 0 & 1 & - 5 & 0 \end{array}]$

Schritt 2.3

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 2.3.1

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 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} - 0 & - - 1 & - 1 \cdot 2 & - 1 \cdot 6 0 & 1 & - 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ - 0 & - - 1 & - 1 \cdot 2 & - 1 \cdot 6 \\ 0 & 1 & - 5 & 0 \end{array}]$

Schritt 2.3.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 & 1 & - 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 & 1 & - 5 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 & 1 & - 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 & 1 & - 5 & 0 \end{array}]$

Schritt 2.4

Perform the row operation

R_{3} = R_{3} - R_{2}

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

3, 2

$3, 2$ a

0

$0$ .

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Schritt 2.4.1

Perform the row operation

R_{3} = R_{3} - R_{2}

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

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 - 0 & 1 - 1 & - 5 + 2 & 0 + 6 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 - 0 & 1 - 1 & - 5 + 2 & 0 + 6 \end{array}]$

Schritt 2.4.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 & 0 & - 3 & 6 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 & 0 & - 3 & 6 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 & 0 & - 3 & 6 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 & 0 & - 3 & 6 \end{array}]$

Schritt 2.5

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

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

3, 3

$3, 3$ a

1

$1$ .

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Schritt 2.5.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

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

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 6 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 6 \end{array}]$

Schritt 2.5.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 & 0 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 & 0 & 1 & - 2 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 & 1 & - 2 & - 6 0 & 0 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 & 1 & - 2 & - 6 \\ 0 & 0 & 1 & - 2 \end{array}]$

Schritt 2.6

Perform the row operation

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

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

2, 3

$2, 3$ a

0

$0$ .

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Schritt 2.6.1

Perform the row operation

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

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

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 6 + 2 \cdot - 2 0 & 0 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 6 + 2 \cdot - 2 \\ 0 & 0 & 1 & - 2 \end{array}]$

Schritt 2.6.2

Vereinfache

R_{2}

$R_{2}$ .

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

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

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

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

Schritt 2.7

Perform the row operation

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

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

1, 3

$1, 3$ a

0

$0$ .

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Schritt 2.7.1

Perform the row operation

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

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

1, 3

$1, 3$ a

0

$0$ .

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

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

Schritt 2.7.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

Schritt 2.8

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

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Schritt 2.8.1

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

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

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

Schritt 2.8.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Schritt 3

Use the result matrix to declare the final solution to the system of equations.

x = - 2

$x = - 2$

y = - 10

$y = - 10$

z = - 2

$z = - 2$

Schritt 4

The solution is the set of ordered pairs that make the system true.

(- 2, - 10, - 2)

$(- 2, - 10, - 2)$

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