Let $W$ be the subspace of $\mathbb{R}^4$ spanned by the vectors $v_1 = (1, 1, 0, 0)^T$, $v_2 = (0, 0, 1, 1)^T$, and $v_3 = (1, 1, 1, 1)^T$. What is the dimension of $W$?