How to find a basis for the following vector space

Question

How to find the basis for the vector space $W=\{(x_1,x_2,x_3,x_4,x_5)\in\Bbb R^5:2x_1+3x_2+x_5=0\}$. Thanks in advance.

Answer 1

Quite easy.

Your space $W=\{(x_1,x_2,x_3,x_4,x_5)\in\Bbb R^5:2x_1+3x_2+x_5=0\}\\=\{(x_1,x_2,x_3,x_4,-2x_1-3x_2):x_1,x_2,x_3,x_4\in\Bbb R\}\\=span\{(1,0,0,0,-2),(0,1,0,0,-3),(0,0,1,0,0),(0,0,0,1,0)\}$

Thanks to Paolo: To check for linear independency just note that

$$rank\left(\begin{bmatrix} 1 &0&0&0&-2 \\ 0&1&0&0&-3 \\0&0&1&0&0\\ 0&0&0&1&0 \end{bmatrix}\right)=4$$