If $V_1$ and $V_2$ are 4-dimensional subspaces of a 6-dimensional vector space $V$, then the smallest possible dimension of $V_1 \cap V_2$ is _____.

According to the dimension theorem for vector subspaces:

Rearranging for $\dim(V_1 \cap V_2)$:

Given $\dim(V_1) = 4$, $\dim(V_2) = 4$, and $V_1 + V_2$ is a subspace of the 6-dimensional space $V$, the maximum possible dimension of $V_1 + V_2$ is:

To minimize $\dim(V_1 \cap V_2)$, we maximize $\dim(V_1 + V_2)$:

Thus, the smallest possible dimension of $V_1 \cap V_2$ is 2.