Let $V = M^{2\times 2}(\bf F),$
$W_1 =\left\{\begin{bmatrix}a & b \\c & a\end{bmatrix}\in V\;:\; a, b, c\in F\right\}$
and
$W_2 =\left\{ \begin{bmatrix}0 & a \\-a & b\end{bmatrix}\in V\;:\; a, b, \in F\right\}$
Prove that $W_1$ and $W_2$ are subspaces of $V$ and find the dimensions of $\,W_1\,,\, W_2\,,\, W_1+W_2\,,\, W_1\cap W_2\,$.
My attempt: Clearly, $W_1$ is of dimension $3$ since it has three independent components, and $W_2$ is of dimension $2$ since it only has $2$. However, does this mean $W_1+W_2$ will have $\dim = 3$ since there will be three independents in total? How do I prove that?