# Finding base of a subspace

Find base of a subspace and expand it to the base of $\mathbb{R}^4$

subspace is given by the following system of eqiuations:

$\begin{cases} x_1+2x_2+2x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$

(Vectors that expand base of W to the base of $\mathbb{R}^4$ span space $W^\prime \subset \mathbb{R}^4$ such that $W\bigoplus W^ \prime= \mathbb{R}^4$

Not know how to do this detailed explanations appreciated.

$$-x_1+x_3+x_4=0\iff x_1=x_3+x_4$$ so the first equation gives
$$2x_2+3x_3+5x_4=0\iff x_2=-\frac32 x_3-\frac52x_4$$
hence $(x_1,x_2,x_3,x_4)$ is in the given subspace if it's equal to $$\left(x_3+x_4,-\frac32 x_3-\frac52x_4,x_3,x_4\right)=x_3\left(1,-\frac32 ,1,0\right)+x_4\left(1,-\frac52,0,1\right)\\=x_3v_1+x_4v_2$$ hence we see that $$W=\operatorname{span}(v_1,v_2)$$