Đại số tuyến tính Ví dụ

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

Bước 1

Write as an augmented matrix for $A x = 0$ .

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

Bước 2

Use the result matrix to declare the final solution to the system of equations.

$x_{1} + x_{4} = 0$

$x_{2} - 3 x_{4} = 0$

$x_{3} + x_{4} = 0$

Bước 3

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} - x_{4} \\ 3 x_{4} \\ - x_{4} \\ x_{4} \end{array}]$

Bước 4

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = x_{4} [\begin{array}{c} - 1 \\ 3 \\ - 1 \\ 1 \end{array}]$

Bước 5

Write as a solution set.

${x_{4} [\begin{array}{c} - 1 \\ 3 \\ - 1 \\ 1 \end{array}] | x_{4} \in R}$