Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 5 & - 4 1 & 4 & 3 & - 2 2 & 7 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 0 & 1 & 5 & - 4 \\ 1 & 4 & 3 & - 2 \\ 2 & 7 & 1 & - 2 \end{array}]$

Schritt 1

Swap

R_{2}

$R_{2}$ with

R_{1}

$R_{1}$ to put a nonzero entry at

1, 1

$1, 1$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 4 & 3 & - 2 0 & 1 & 5 & - 4 2 & 7 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 4 & 3 & - 2 \\ 0 & 1 & 5 & - 4 \\ 2 & 7 & 1 & - 2 \end{array}]$

Schritt 2

Perform the row operation

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

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

3, 1

$3, 1$ a

0

$0$ .

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Schritt 2.1

Perform the row operation

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

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

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 4 & 3 & - 2 0 & 1 & 5 & - 4 2 - 2 \cdot 1 & 7 - 2 \cdot 4 & 1 - 2 \cdot 3 & - 2 - 2 \cdot - 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 4 & 3 & - 2 \\ 0 & 1 & 5 & - 4 \\ 2 - 2 \cdot 1 & 7 - 2 \cdot 4 & 1 - 2 \cdot 3 & - 2 - 2 \cdot - 2 \end{array}]$

Schritt 2.2

Vereinfache

R_{3}

$R_{3}$ .

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

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

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

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

Schritt 3

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 3.1

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

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

Schritt 3.2

Vereinfache

R_{3}

$R_{3}$ .

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

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

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

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

Schritt 4

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{2}

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

3, 4

$3, 4$ a

1

$1$ .

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Schritt 4.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{2}

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

3, 4

$3, 4$ a

1

$1$ .

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

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

Schritt 4.2

Vereinfache

R_{3}

$R_{3}$ .

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

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

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

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

Schritt 5

Perform the row operation

R_{2} = R_{2} + 4 R_{3}

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

2, 4

$2, 4$ a

0

$0$ .

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Schritt 5.1

Perform the row operation

R_{2} = R_{2} + 4 R_{3}

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

2, 4

$2, 4$ a

0

$0$ .

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

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

Schritt 5.2

Vereinfache

R_{2}

$R_{2}$ .

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

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

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

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

Schritt 6

Perform the row operation

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

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

1, 4

$1, 4$ a

0

$0$ .

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Schritt 6.1

Perform the row operation

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

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

1, 4

$1, 4$ a

0

$0$ .

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

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

Schritt 6.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

Schritt 7

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 7.1

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

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

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

Schritt 7.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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