Find the basis for $W_1 \cap W_2$ $$W_1=\left \{\left[{a \atop a+c} \ {a+b\atop b+c}\right] \mid a,b,c \in F\right\}, W_2=\left \{\left[{x \atop -y} \ {x\atop y}\right] \mid x,y \in F\right\}$$
I know that
$$W_1 \cap W_2 = \left\{\left[{a \atop a+c} \ {a+b\atop b+c}\right] \mid \exists x,y \in F,\left[{a \atop a+c} \ {a+b\atop b+c}\right] = \left[{x \atop -y} \ {x\atop y}\right]\right\}$$
however I am having a hard time making this into something that I can find the basis of by writing it as a linear combination.

Since $a=x$ and $a+b =x$, we have that $b=0$. Then $b+ c = c =y$. Then $a+c = a+y  = -y$, hence $a=-2y$. Thus $x = 2y$. Thus we can see that any element of $W_1 \cap W_2$ must have that form:
$$\begin{pmatrix} -2y & -2y \\ -y & y \end{pmatrix}$$
Further by choosing $a = -2y$, $b =0$, $c=y$, we can see that:
$$\begin{pmatrix} a & a+b \\ a+c & b+c\end{pmatrix} = \begin{pmatrix} -2y & -2y \\ -y & y \end{pmatrix},$$
thus all of these matrices are in $W_1 \cap W_2$. Thus:
$$W_1 \cap W_2 = \left\{ \begin{pmatrix} -2y & -2y \\ -y & y \end{pmatrix} \mid y \in F\right\},$$
which has basis:
$$\left\{ \begin{pmatrix} -2 & -2 \\ -1 & 1 \end{pmatrix}\right\}.$$
