Let $V$ be the vector space of $2 \times 2$ matrices. Consider the subspace $W$ spanned by $A_1 = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$, $A_2 = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}$, and $A_3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. What is the dimension of $W$?