Lineare Algebra Beispiele

[\begin{matrix} 4 & 7 11 & 20 \end{matrix}]

$[\begin{array}{c} 4 & 7 \\ 11 & 20 \end{array}]$

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}{4}

$\frac{1}{4}$ 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}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{matrix} \frac{4}{4} & \frac{7}{4} 11 & 20 \end{matrix}]

$[\begin{array}{c} \frac{4}{4} & \frac{7}{4} \\ 11 & 20 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & \frac{7}{4} 11 & 20 \end{matrix}]

$[\begin{array}{c} 1 & \frac{7}{4} \\ 11 & 20 \end{array}]$

[\begin{matrix} 1 & \frac{7}{4} 11 & 20 \end{matrix}]

$[\begin{array}{c} 1 & \frac{7}{4} \\ 11 & 20 \end{array}]$

Schritt 1.2

Perform the row operation

R_{2} = R_{2} - 11 R_{1}

$R_{2} = R_{2} - 11 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.2.1

Perform the row operation

R_{2} = R_{2} - 11 R_{1}

$R_{2} = R_{2} - 11 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & \frac{7}{4} 11 - 11 \cdot 1 & 20 - 11 (\frac{7}{4}) \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{7}{4} \\ 11 - 11 \cdot 1 & 20 - 11 (\frac{7}{4}) \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & \frac{7}{4} 0 & \frac{3}{4} \end{matrix}]

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

[\begin{matrix} 1 & \frac{7}{4} 0 & \frac{3}{4} \end{matrix}]

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

Schritt 1.3

Multiply each element of

R_{2}

$R_{2}$ by

\frac{4}{3}

$\frac{4}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.3.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{4}{3}

$\frac{4}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{matrix} 1 & \frac{7}{4} \frac{4}{3} \cdot 0 & \frac{4}{3} \cdot \frac{3}{4} \end{matrix}]

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

Schritt 1.3.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & \frac{7}{4} 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & \frac{7}{4} \\ 0 & 1 \end{array}]$

[\begin{matrix} 1 & \frac{7}{4} 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & \frac{7}{4} \\ 0 & 1 \end{array}]$

Schritt 1.4

Perform the row operation

R_{1} = R_{1} - \frac{7}{4} R_{2}

$R_{1} = R_{1} - \frac{7}{4} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.4.1

Perform the row operation

R_{1} = R_{1} - \frac{7}{4} R_{2}

$R_{1} = R_{1} - \frac{7}{4} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{matrix} 1 - \frac{7}{4} \cdot 0 & \frac{7}{4} - \frac{7}{4} \cdot 1 0 & 1 \end{matrix}]

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

Schritt 1.4.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

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

$2$