Lineare Algebra Beispiele

$[\begin{array}{c} - 3 & 6 & - 1 & 1 & - 7 \\ 1 & - 2 & 2 & 3 & - 1 \\ 2 & - 4 & 5 & 8 & - 4 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of $R_{1}$ by $- \frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

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Schritt 1.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 6 & - \frac{1}{3} \cdot - 1 & - \frac{1}{3} \cdot 1 & - \frac{1}{3} \cdot - 7 \\ 1 & - 2 & 2 & 3 & - 1 \\ 2 & - 4 & 5 & 8 & - 4 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 1 & - 2 & 2 & 3 & - 1 \\ 2 & - 4 & 5 & 8 & - 4 \end{array}]$

Schritt 1.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 1.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 1 - 1 & - 2 + 2 & 2 - \frac{1}{3} & 3 + \frac{1}{3} & - 1 - \frac{7}{3} \\ 2 & - 4 & 5 & 8 & - 4 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 0 & 0 & \frac{5}{3} & \frac{10}{3} & - \frac{10}{3} \\ 2 & - 4 & 5 & 8 & - 4 \end{array}]$

Schritt 1.3

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 1.3.1

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 0 & 0 & \frac{5}{3} & \frac{10}{3} & - \frac{10}{3} \\ 2 - 2 \cdot 1 & - 4 - 2 \cdot - 2 & 5 - 2 (\frac{1}{3}) & 8 - 2 (- \frac{1}{3}) & - 4 - 2 (\frac{7}{3}) \end{array}]$

Schritt 1.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 0 & 0 & \frac{5}{3} & \frac{10}{3} & - \frac{10}{3} \\ 0 & 0 & \frac{13}{3} & \frac{26}{3} & - \frac{26}{3} \end{array}]$

Schritt 1.4

Multiply each element of $R_{2}$ by $\frac{3}{5}$ to make the entry at $2, 3$ a $1$ .

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Schritt 1.4.1

Multiply each element of $R_{2}$ by $\frac{3}{5}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ \frac{3}{5} \cdot 0 & \frac{3}{5} \cdot 0 & \frac{3}{5} \cdot \frac{5}{3} & \frac{3}{5} \cdot \frac{10}{3} & \frac{3}{5} (- \frac{10}{3}) \\ 0 & 0 & \frac{13}{3} & \frac{26}{3} & - \frac{26}{3} \end{array}]$

Schritt 1.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 0 & 0 & 1 & 2 & - 2 \\ 0 & 0 & \frac{13}{3} & \frac{26}{3} & - \frac{26}{3} \end{array}]$

Schritt 1.5

Perform the row operation $R_{3} = R_{3} - \frac{13}{3} R_{2}$ to make the entry at $3, 3$ a $0$ .

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Schritt 1.5.1

Perform the row operation $R_{3} = R_{3} - \frac{13}{3} R_{2}$ to make the entry at $3, 3$ a $0$ .

$[\begin{array}{c} 1 & - 2 & \frac{1}{3} & - \frac{1}{3} & \frac{7}{3} \\ 0 & 0 & 1 & 2 & - 2 \\ 0 - \frac{13}{3} \cdot 0 & 0 - \frac{13}{3} \cdot 0 & \frac{13}{3} - \frac{13}{3} \cdot 1 & \frac{26}{3} - \frac{13}{3} \cdot 2 & - \frac{26}{3} - \frac{13}{3} \cdot - 2 \end{array}]$

Schritt 1.5.2

Vereinfache $R_{3}$ .

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

Schritt 1.6

Perform the row operation $R_{1} = R_{1} - \frac{1}{3} R_{2}$ to make the entry at $1, 3$ a $0$ .

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Schritt 1.6.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{3} R_{2}$ to make the entry at $1, 3$ a $0$ .

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

Schritt 1.6.2

Vereinfache $R_{1}$ .

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

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: $a_{11}$ and $a_{23}$

Pivot Columns: $1$ and $3$

Schritt 3

The rank is the number of pivot columns.

$2$