## Vector Subspace Question (GRE 0568 Q3)

This is an interesting question on vector subspaces (a topic from linear algebra):

Question:
If V and W are 2-dimensional subspaces of $\mathbb{R}^4$, what are the possible dimensions of the subspace $V\cap W$?

(A) 1 only
(B) 2 only
(C) 0 and 1 only
(D) 0, 1, and 2 only
(E) 0, 1, 2, 3, and 4

To begin this question, we would need this theorem on the dimension of sum and intersection of subspaces (for finite dimensional subspaces):

$\dim (M+N)=\dim M+\dim N-\dim (M\cap N)$

Note that this looks familiar to the Inclusion-Exclusion principle, which is indeed used in the proof.

Hence, we have $\dim(M\cap N)=\dim M+\dim N-\dim (M+N)=4-\dim (M+N)$.

$\dim (M+N)$, the sum of the subspaces M and N, is at most 4, and at least 2.

Thus, $\dim (M\cap N)$ can take the values of 0, 1, or 2.