Algebra lineare Esempi

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

Passaggio 1

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 1.1

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

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Passaggio 1.1.1

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

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

Passaggio 1.1.2

Semplifica $R_{1}$ .

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

Passaggio 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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Passaggio 1.2.1

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

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

Passaggio 1.2.2

Semplifica $R_{2}$ .

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

Passaggio 1.3

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

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Passaggio 1.3.1

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

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

Passaggio 1.3.2

Semplifica $R_{3}$ .

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

Passaggio 1.4

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

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Passaggio 1.4.1

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

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

Passaggio 1.4.2

Semplifica $R_{2}$ .

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

Passaggio 1.5

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

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Passaggio 1.5.1

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

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & 3 & - 1 \\ 0 + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 & \frac{7}{2} + \frac{3}{2} \cdot 3 & \frac{3}{2} + \frac{3}{2} \cdot - 1 \end{array}]$

Passaggio 1.5.2

Semplifica $R_{3}$ .

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

Passaggio 1.6

Multiply each element of $R_{3}$ by $\frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 1.6.1

Multiply each element of $R_{3}$ by $\frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

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

Passaggio 1.6.2

Semplifica $R_{3}$ .

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

Passaggio 1.7

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

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Passaggio 1.7.1

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

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

Passaggio 1.7.2

Semplifica $R_{2}$ .

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

Passaggio 1.8

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

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Passaggio 1.8.1

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

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

Passaggio 1.8.2

Semplifica $R_{1}$ .

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

Passaggio 1.9

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

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Passaggio 1.9.1

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

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

Passaggio 1.9.2

Semplifica $R_{1}$ .

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

Passaggio 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 $3$

Passaggio 3

The rank is the number of pivot columns.

$3$