선형 대수 예제

$[\begin{array}{c} 1 & 0 & 1 & 0 & 6 \\ 1 & 1 & 2 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 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.

단계 2

기약 행 사다리꼴을 구합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 2.1

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

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 2.1.1

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

$[\begin{array}{c} 1 & 0 & 1 & 0 & 6 \\ 1 - 1 & 1 - 0 & 2 - 1 & 2 - 0 & 0 - 6 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 2.1.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 1 & 0 & 6 \\ 0 & 1 & 1 & 2 & - 6 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 2.2

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

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 2.2.1

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

$[\begin{array}{c} 1 & 0 & 1 & 0 & 6 \\ 0 - 0 & 1 - 0 & 1 - 1 & 2 - 0 & - 6 - 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 1 & 0 & 6 \\ 0 & 1 & 0 & 2 & - 6 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 2.3

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

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 2.3.1

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

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & 0 - 0 & 6 - 0 \\ 0 & 1 & 0 & 2 & - 6 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 2.3.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 0 & 0 & 6 \\ 0 & 1 & 0 & 2 & - 6 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

단계 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: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$

단계 4

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

$2$