Determine the dimension of the subspace $W$ of $\mathbb{R}^4$ spanned by the vectors $v_1 = (1, 0, 1, 0)$, $v_2 = (0, 1, 0, 1)$, $v_3 = (1, 1, 1, 1)$, and $v_4 = (1, 2, 3, 4)$.