Lineare Algebra Beispiele

$[\begin{array}{c} 1 & - 7 \\ 2 & - 10 \\ 3 & - 13 \\ 4 & - 16 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

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

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Schritt 1.1.1

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

$[\begin{array}{c} 1 & - 7 \\ 2 - 2 \cdot 1 & - 10 - 2 \cdot - 7 \\ 3 & - 13 \\ 4 & - 16 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 4 \\ 3 & - 13 \\ 4 & - 16 \end{array}]$

Schritt 1.2

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

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Schritt 1.2.1

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

$[\begin{array}{c} 1 & - 7 \\ 0 & 4 \\ 3 - 3 \cdot 1 & - 13 - 3 \cdot - 7 \\ 4 & - 16 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 4 \\ 0 & 8 \\ 4 & - 16 \end{array}]$

Schritt 1.3

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Schritt 1.3.1

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 4 \\ 0 & 8 \\ 4 - 4 \cdot 1 & - 16 - 4 \cdot - 7 \end{array}]$

Schritt 1.3.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 4 \\ 0 & 8 \\ 0 & 12 \end{array}]$

Schritt 1.4

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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Schritt 1.4.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 7 \\ \frac{0}{4} & \frac{4}{4} \\ 0 & 8 \\ 0 & 12 \end{array}]$

Schritt 1.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 1 \\ 0 & 8 \\ 0 & 12 \end{array}]$

Schritt 1.5

Perform the row operation $R_{3} = R_{3} - 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 1.5.1

Perform the row operation $R_{3} = R_{3} - 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 1 \\ 0 - 8 \cdot 0 & 8 - 8 \cdot 1 \\ 0 & 12 \end{array}]$

Schritt 1.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 1 \\ 0 & 0 \\ 0 & 12 \end{array}]$

Schritt 1.6

Perform the row operation $R_{4} = R_{4} - 12 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Schritt 1.6.1

Perform the row operation $R_{4} = R_{4} - 12 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 7 \\ 0 & 1 \\ 0 & 0 \\ 0 - 12 \cdot 0 & 12 - 12 \cdot 1 \end{array}]$

Schritt 1.6.2

Vereinfache $R_{4}$ .

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

Schritt 1.7

Perform the row operation $R_{1} = R_{1} + 7 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 1.7.1

Perform the row operation $R_{1} = R_{1} + 7 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Schritt 1.7.2

Vereinfache $R_{1}$ .

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

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 $a_{22}$

Pivot Columns: $1$ and $2$

Schritt 3

The rank is the number of pivot columns.

$2$