Lineare Algebra Beispiele

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

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

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} 1 & 0 - 1 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

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

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

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

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

Schritt 1.3

Perform the row operation

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

$R_{3} = R_{3} + R_{1}$ 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} + R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

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

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

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

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

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

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

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

Schritt 1.4

Multiply each element of

R_{2}

$R_{2}$ by

\frac{3}{2}

$\frac{3}{2}$ 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{3}{2}

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

2, 2

$2, 2$ a

1

$1$ .

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

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

Schritt 1.4.2

Vereinfache

R_{2}

$R_{2}$ .

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

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

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

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

Schritt 1.5

Perform the row operation

R_{3} = R_{3} - \frac{10}{3} R_{2}

$R_{3} = R_{3} - \frac{10}{3} R_{2}$ 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} - \frac{10}{3} R_{2}

$R_{3} = R_{3} - \frac{10}{3} R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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

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

Schritt 1.5.2

Vereinfache

R_{3}

$R_{3}$ .

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

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

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

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

Schritt 1.6

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 1.6.1

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

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

Schritt 1.6.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

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

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

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

$2$