Algèbre linéaire Exemples

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 1 0 & 1 & 0 & - 3 0 & 0 & 1 & 1 \end{matrix} ⎤ ⎥ ⎦

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

Étape 1

Write as an augmented matrix for

A x = 0

$A x = 0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 1 & 0 0 & 1 & 0 & - 3 & 0 0 & 0 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Étape 2

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

x_{1} + x_{4} = 0

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

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

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

x_{3} + x_{4} = 0

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

Étape 3

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

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x_{1} x_{2} x_{3} x_{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - x_{4} 3 x_{4} - x_{4} x_{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Étape 4

Write the solution as a linear combination of vectors.

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x_{1} x_{2} x_{3} x_{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = x_{4} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - 1 3 - 1 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Étape 5

Write as a solution set.

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x_{4} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - 1 3 - 1 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ∣ ∣ ∣ ∣ ∣ ∣ x_{4} \in R ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

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