Algèbre linéaire Exemples

[\begin{array}{c} 1 & 3 & 0 & 2 \\ 0 & 0 & 1 & 9 \end{array}]

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

Étape 1

Write as an augmented matrix for

A x = 0

$A x = 0$ .

[\begin{array}{c} 1 & 3 & 0 & 2 & 0 \\ 0 & 0 & 1 & 9 & 0 \end{array}]

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

Étape 2

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

x_{1} + 3 x_{2} + 2 x_{4} = 0

$x_{1} + 3 x_{2} + 2 x_{4} = 0$

x_{3} + 9 x_{4} = 0

$x_{3} + 9 x_{4} = 0$

Étape 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} - 3 x_{2} - 2 x_{4} \\ x_{2} \\ - 9 x_{4} \\ x_{4} \end{array}]

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} - 3 x_{2} - 2 x_{4} \\ x_{2} \\ - 9 x_{4} \\ x_{4} \end{array}]$

Étape 4

Write the solution as a linear combination of vectors.

[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = x_{2} [\begin{array}{c} - 3 \\ 1 \\ 0 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 2 \\ 0 \\ - 9 \\ 1 \end{array}]

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = x_{2} [\begin{array}{c} - 3 \\ 1 \\ 0 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 2 \\ 0 \\ - 9 \\ 1 \end{array}]$

Étape 5

Write as a solution set.

{x_{2} [\begin{array}{c} - 3 \\ 1 \\ 0 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 2 \\ 0 \\ - 9 \\ 1 \end{array}] | x_{2}, x_{4} \in R}

${x_{2} [\begin{array}{c} - 3 \\ 1 \\ 0 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 2 \\ 0 \\ - 9 \\ 1 \end{array}] | x_{2}, x_{4} \in R}$