Algèbre linéaire Exemples

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

Étape 1

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

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

Étape 2

Déterminez la forme d’échelon en ligne réduite.

Appuyez ici pour voir plus d’étapes...

Étape 2.1

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

Appuyez ici pour voir plus d’étapes...

Étape 2.1.1

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

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

Étape 2.1.2

Simplifiez $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 8 & 8 & 0 \\ 0 & 1 & 5 & 5 & 0 \end{array}]$

Étape 3

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

$x_{1} + 8 x_{3} + 8 x_{4} = 0$

$x_{2} + 5 x_{3} + 5 x_{4} = 0$

Étape 4

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

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} - 8 x_{3} - 8 x_{4} \\ - 5 x_{3} - 5 x_{4} \\ x_{3} \\ x_{4} \end{array}]$

Étape 5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = x_{3} [\begin{array}{c} - 8 \\ - 5 \\ 1 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 8 \\ - 5 \\ 0 \\ 1 \end{array}]$

Étape 6

Write as a solution set.

${x_{3} [\begin{array}{c} - 8 \\ - 5 \\ 1 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - 8 \\ - 5 \\ 0 \\ 1 \end{array}] | x_{3}, x_{4} \in R}$