# how to find basis of a matrix subspace $W$ of $M_{2\times 2}$

$W = \{\begin{bmatrix} a& b\\ c& d\end{bmatrix} | a+b=c, b+c=d, c+d=a ; a,b,c,d \in F \}$

I know what is a basis for matrix space, but what to do with these equations given here?

Doing an elimination procedure with the homogeneous system of equations $$a+b-c=0,\quad b+c-d=0,\quad -a+c+d=0$$ leads to $$a=3d,\quad b=-d,\quad c=2d, \quad d\ {\rm arbitrary}\ .$$ The space $W$ is therefore one-dimensional, and a basis is obtained by choosing $d=1$. It follows that $$W=\bigl\langle\ \left[\matrix{3&-1\cr 2&1\cr}\right]\ \bigr\rangle\ .$$