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

**Question:**

If V and W are 2-dimensional subspaces of , what are the possible dimensions of the subspace ?

(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):

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

Hence, we have .

, the sum of the subspaces M and N, is at most 4, and at least 2.

Thus, can take the values of 0, 1, or 2.

Answer: **Option D**

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