선형 대수 예제

$2 x - y + 3 z - w = - 3$ , $3 x + 2 y - z + w = 13$ , $x - 3 y + z - 2 w = - 4$ , $- x + y + 4 z + 3 w = 0$

단계 1

연립방정식을 행렬 형태로 씁니다.

$[\begin{array}{c} - 1 & 2 & - 1 & 3 & - 3 \\ 1 & 3 & 2 & - 1 & 13 \\ - 2 & 1 & - 3 & 1 & - 4 \\ 3 & - 1 & 1 & 4 & 0 \end{array}]$

단계 2

기약 행 사다리꼴을 구합니다.

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단계 2.1

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

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단계 2.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 2 & - - 1 & - 1 \cdot 3 & - - 3 \\ 1 & 3 & 2 & - 1 & 13 \\ - 2 & 1 & - 3 & 1 & - 4 \\ 3 & - 1 & 1 & 4 & 0 \end{array}]$

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 1 & 3 & 2 & - 1 & 13 \\ - 2 & 1 & - 3 & 1 & - 4 \\ 3 & - 1 & 1 & 4 & 0 \end{array}]$

단계 2.2

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

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단계 2.2.1

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

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 1 - 1 & 3 + 2 & 2 - 1 & - 1 + 3 & 13 - 3 \\ - 2 & 1 & - 3 & 1 & - 4 \\ 3 & - 1 & 1 & 4 & 0 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

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

단계 2.3

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

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단계 2.3.1

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

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 5 & 1 & 2 & 10 \\ - 2 + 2 \cdot 1 & 1 + 2 \cdot - 2 & - 3 + 2 \cdot 1 & 1 + 2 \cdot - 3 & - 4 + 2 \cdot 3 \\ 3 & - 1 & 1 & 4 & 0 \end{array}]$

단계 2.3.2

$R_{3}$ 을 간단히 합니다.

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

단계 2.4

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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단계 2.4.1

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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

단계 2.4.2

$R_{4}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 5 & 1 & 2 & 10 \\ 0 & - 3 & - 1 & - 5 & 2 \\ 0 & 5 & - 2 & 13 & - 9 \end{array}]$

단계 2.5

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

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단계 2.5.1

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

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ \frac{0}{5} & \frac{5}{5} & \frac{1}{5} & \frac{2}{5} & \frac{10}{5} \\ 0 & - 3 & - 1 & - 5 & 2 \\ 0 & 5 & - 2 & 13 & - 9 \end{array}]$

단계 2.5.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & - 3 & - 1 & - 5 & 2 \\ 0 & 5 & - 2 & 13 & - 9 \end{array}]$

단계 2.6

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

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단계 2.6.1

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

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 1 + 3 (\frac{1}{5}) & - 5 + 3 (\frac{2}{5}) & 2 + 3 \cdot 2 \\ 0 & 5 & - 2 & 13 & - 9 \end{array}]$

단계 2.6.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & - \frac{2}{5} & - \frac{19}{5} & 8 \\ 0 & 5 & - 2 & 13 & - 9 \end{array}]$

단계 2.7

Perform the row operation $R_{4} = R_{4} - 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

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단계 2.7.1

Perform the row operation $R_{4} = R_{4} - 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & - \frac{2}{5} & - \frac{19}{5} & 8 \\ 0 - 5 \cdot 0 & 5 - 5 \cdot 1 & - 2 - 5 (\frac{1}{5}) & 13 - 5 (\frac{2}{5}) & - 9 - 5 \cdot 2 \end{array}]$

단계 2.7.2

$R_{4}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & - \frac{2}{5} & - \frac{19}{5} & 8 \\ 0 & 0 & - 3 & 11 & - 19 \end{array}]$

단계 2.8

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

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단계 2.8.1

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

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ - \frac{5}{2} \cdot 0 & - \frac{5}{2} \cdot 0 & - \frac{5}{2} (- \frac{2}{5}) & - \frac{5}{2} (- \frac{19}{5}) & - \frac{5}{2} \cdot 8 \\ 0 & 0 & - 3 & 11 & - 19 \end{array}]$

단계 2.8.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & 1 & \frac{19}{2} & - 20 \\ 0 & 0 & - 3 & 11 & - 19 \end{array}]$

단계 2.9

Perform the row operation $R_{4} = R_{4} + 3 R_{3}$ to make the entry at $4, 3$ a $0$ .

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단계 2.9.1

Perform the row operation $R_{4} = R_{4} + 3 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & 1 & \frac{19}{2} & - 20 \\ 0 + 3 \cdot 0 & 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 11 + 3 (\frac{19}{2}) & - 19 + 3 \cdot - 20 \end{array}]$

단계 2.9.2

$R_{4}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & 1 & \frac{19}{2} & - 20 \\ 0 & 0 & 0 & \frac{79}{2} & - 79 \end{array}]$

단계 2.10

Multiply each element of $R_{4}$ by $\frac{2}{79}$ to make the entry at $4, 4$ a $1$ .

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단계 2.10.1

Multiply each element of $R_{4}$ by $\frac{2}{79}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & 1 & \frac{19}{2} & - 20 \\ \frac{2}{79} \cdot 0 & \frac{2}{79} \cdot 0 & \frac{2}{79} \cdot 0 & \frac{2}{79} \cdot \frac{79}{2} & \frac{2}{79} \cdot - 79 \end{array}]$

단계 2.10.2

$R_{4}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 & 0 & 1 & \frac{19}{2} & - 20 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

단계 2.11

Perform the row operation $R_{3} = R_{3} - \frac{19}{2} R_{4}$ to make the entry at $3, 4$ a $0$ .

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단계 2.11.1

Perform the row operation $R_{3} = R_{3} - \frac{19}{2} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & \frac{2}{5} & 2 \\ 0 - \frac{19}{2} \cdot 0 & 0 - \frac{19}{2} \cdot 0 & 1 - \frac{19}{2} \cdot 0 & \frac{19}{2} - \frac{19}{2} \cdot 1 & - 20 - \frac{19}{2} \cdot - 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

단계 2.11.2

$R_{3}$ 을 간단히 합니다.

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

단계 2.12

Perform the row operation $R_{2} = R_{2} - \frac{2}{5} R_{4}$ to make the entry at $2, 4$ a $0$ .

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단계 2.12.1

Perform the row operation $R_{2} = R_{2} - \frac{2}{5} R_{4}$ to make the entry at $2, 4$ a $0$ .

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

단계 2.12.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & - 3 & 3 \\ 0 & 1 & \frac{1}{5} & 0 & \frac{14}{5} \\ 0 & 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

단계 2.13

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 4$ a $0$ .

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단계 2.13.1

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 3 \cdot 0 & - 2 + 3 \cdot 0 & 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 3 + 3 \cdot - 2 \\ 0 & 1 & \frac{1}{5} & 0 & \frac{14}{5} \\ 0 & 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

단계 2.13.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 2 & 1 & 0 & - 3 \\ 0 & 1 & \frac{1}{5} & 0 & \frac{14}{5} \\ 0 & 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

단계 2.14

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

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단계 2.14.1

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

$[\begin{array}{c} 1 & - 2 & 1 & 0 & - 3 \\ 0 - \frac{1}{5} \cdot 0 & 1 - \frac{1}{5} \cdot 0 & \frac{1}{5} - \frac{1}{5} \cdot 1 & 0 - \frac{1}{5} \cdot 0 & \frac{14}{5} - \frac{1}{5} \cdot - 1 \\ 0 & 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

단계 2.14.2

$R_{2}$ 을 간단히 합니다.

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

단계 2.15

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

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단계 2.15.1

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

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

단계 2.15.2

$R_{1}$ 을 간단히 합니다.

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

단계 2.16

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

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단계 2.16.1

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

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

단계 2.16.2

$R_{1}$ 을 간단히 합니다.

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

단계 3

결과 행렬을 이용해 연립방정식의 최종 해를 구합니다.

$w = 4$

$x = 3$

$y = - 1$

$z = - 2$

단계 4

해는 연립방정식을 참이 되게 하는 순서쌍의 집합입니다.

$(4, 3, - 1, - 2)$

단계 5

각 행의 종속 변수에 대해 식을 풀고 기약행 사다리꼴 형태의 첨가 행렬로 표현된 각 방정식을 재정렬함으로써 벡터해를 분해하여 벡터 등식을 구합니다.

$X = [\begin{array}{c} w \\ x \\ y \\ z \end{array}] = [\begin{array}{c} 4 \\ 3 \\ - 1 \\ - 2 \end{array}]$