Lineare Algebra Beispiele

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

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

Schritt 1

Ermittele die normierte Zeilenstufenform.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{5}

$\frac{1}{5}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{5}

$\frac{1}{5}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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

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

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

Schritt 1.2

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

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.2.1

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

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

Schritt 1.2.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{4}{5} 0 & 1 & 3 0 & 2 & - \frac{9}{5} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - \frac{4}{5} \\ 0 & 1 & 3 \\ 0 & 2 & - \frac{9}{5} \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{4}{5} 0 & 1 & 3 0 & 2 & - \frac{9}{5} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - \frac{4}{5} \\ 0 & 1 & 3 \\ 0 & 2 & - \frac{9}{5} \end{array}]$

Schritt 1.3

Perform the row operation

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

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

3, 2

$3, 2$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.3.1

Perform the row operation

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

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

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{4}{5} 0 & 1 & 3 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - \frac{9}{5} - 2 \cdot 3 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - \frac{4}{5} \\ 0 & 1 & 3 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - \frac{9}{5} - 2 \cdot 3 \end{array}]$

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{4}{5} 0 & 1 & 3 0 & 0 & - \frac{39}{5} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - \frac{4}{5} \\ 0 & 1 & 3 \\ 0 & 0 & - \frac{39}{5} \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{4}{5} 0 & 1 & 3 0 & 0 & - \frac{39}{5} \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - \frac{4}{5} \\ 0 & 1 & 3 \\ 0 & 0 & - \frac{39}{5} \end{array}]$

Schritt 1.4

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{5}{39}

$- \frac{5}{39}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.4.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{5}{39}

$- \frac{5}{39}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{4}{5} 0 & 1 & 3 - \frac{5}{39} \cdot 0 & - \frac{5}{39} \cdot 0 & - \frac{5}{39} (- \frac{39}{5}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - \frac{4}{5} \\ 0 & 1 & 3 \\ - \frac{5}{39} \cdot 0 & - \frac{5}{39} \cdot 0 & - \frac{5}{39} (- \frac{39}{5}) \end{array}]$

Schritt 1.4.2

Vereinfache

R_{3}

$R_{3}$ .

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

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

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

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

Schritt 1.5

Perform the row operation

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

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

2, 3

$2, 3$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.5.1

Perform the row operation

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

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

2, 3

$2, 3$ a

0

$0$ .

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

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

Schritt 1.5.2

Vereinfache

$R_{2}$ .

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

Schritt 1.6

Perform the row operation

$R_{1} = R_{1} + \frac{4}{5} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.6.1

Perform the row operation

$R_{1} = R_{1} + \frac{4}{5} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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

Schritt 1.6.2

Vereinfache

$R_{1}$ .

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

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

$3$