Algèbre linéaire Exemples

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 4 & 9 0 & 1 & 1 & - 7 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - 4 & 9 \\ 0 & 1 & 1 & - 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Étape 1

Write as an augmented matrix for

A x = 0

$A x = 0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - 4 & 9 & 0 0 & 1 & 1 & - 7 & 0 0 & 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & - 4 & 9 & 0 \\ 0 & 1 & 1 & - 7 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Étape 2

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

x_{1} - 4 x_{3} + 9 x_{4} = 0

$x_{1} - 4 x_{3} + 9 x_{4} = 0$

x_{2} + x_{3} - 7 x_{4} = 0

$x_{2} + x_{3} - 7 x_{4} = 0$

0 = 0

$0 = 0$

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} 4 x_{3} - 9 x_{4} - x_{3} + 7 x_{4} x_{3} x_{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} 4 x_{3} - 9 x_{4} \\ - x_{3} + 7 x_{4} \\ x_{3} \\ x_{4} \end{array}]$

Étape 4

Write the solution as a linear combination of vectors.

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x_{1} x_{2} x_{3} x_{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = x_{3} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 - 1 10 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + x_{4} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - 9 701 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = x_{3} [\begin{array}{c} 4 \\ - 1 \\ 1 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 9 \\ 7 \\ 0 \\ 1 \end{array}]$

Étape 5

Write as a solution set.

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x_{3} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 - 1 10 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + x_{4} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - 9 701 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ∣ ∣ ∣ ∣ ∣ ∣ x_{3}, x_{4} \in R ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

${x_{3} [\begin{array}{c} 4 \\ - 1 \\ 1 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 9 \\ 7 \\ 0 \\ 1 \end{array}] | x_{3}, x_{4} \in R}$