Примеры

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

Этап 1

Nullity is the dimension of the null space, which is the same as the number of free variables in the system after row reducing. The free variables are the columns without pivot positions.

Этап 2

Приведем матрицу к стандартной форме по строкам.

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Этап 2.1

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

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Этап 2.1.1

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

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

Этап 2.1.2

Упростим $R_{1}$ .

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

Этап 2.2

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

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Этап 2.2.1

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

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

Этап 2.2.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{3} & 0 \\ 0 & - \frac{20}{3} & 0 \\ 1 & 1 & 0 \end{array}]$

Этап 2.3

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

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Этап 2.3.1

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

$[\begin{array}{c} 1 & \frac{5}{3} & 0 \\ 0 & - \frac{20}{3} & 0 \\ 1 - 1 & 1 - \frac{5}{3} & 0 - 0 \end{array}]$

Этап 2.3.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{3} & 0 \\ 0 & - \frac{20}{3} & 0 \\ 0 & - \frac{2}{3} & 0 \end{array}]$

Этап 2.4

Multiply each element of $R_{2}$ by $- \frac{3}{20}$ to make the entry at $2, 2$ a $1$ .

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Этап 2.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{20}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{3} & 0 \\ - \frac{3}{20} \cdot 0 & - \frac{3}{20} (- \frac{20}{3}) & - \frac{3}{20} \cdot 0 \\ 0 & - \frac{2}{3} & 0 \end{array}]$

Этап 2.4.2

Упростим $R_{2}$ .

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

Этап 2.5

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

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Этап 2.5.1

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

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

Этап 2.5.2

Упростим $R_{3}$ .

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

Этап 2.6

Perform the row operation $R_{1} = R_{1} - \frac{5}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Этап 2.6.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Этап 2.6.2

Упростим $R_{1}$ .

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

Этап 3

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 $2$

Этап 4

The nullity is the number of columns without a pivot position in the row reduced matrix.

$1$

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