# Find a basis of the subspace of R^5 containing all that satisfy 5x_1+9x_2-40x_3=102x_4-76x_5=9x_2+30x_4-40x_5

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### 1 Answer

Let basis of subspace of R^5 be (x_1,x_2,x_3,x_4,x_5) such that

`5x_1+9x_2-40x_3=102x_4-76x_5`

`9x_2+30x_4-40x_5=102x_4-76x_5`

`9x_2+30x_4-40x_5=5x_1+9x_2-40x_3`

Let us write this in matrix form

`[[5,9,-40,-102,-76],[0,9,0,-72,36],[5,0,-40,-30,40]][[x_1],[x_2],[x_3],[x_4],[x_5]]=[[0],[0],[0],[0],[0]]`

write roe echlon form of the coefficient matrix

`[[1,0,-8,-6,0],[0,1,0,-8,0],[0,0,0,0,1]]`

Thus basis of R^5={ (1,0,-8,-6,0),(0,1,0,-8,0),(0,0,0,0,1)}