Dimension of The Intersection of Subspaces

[darkchild]

Homework Statement

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, 2 only
(E) 0,1,2,3, and 4

Homework Equations

dim(V + W) = dim V + dim W - dim(V $$\cap$$ W)

dim (V + W) $$\leq$$ dim $$\mathbb{R}^{4}$$

The Attempt at a Solution

V + W $$\leq \mathbb{R}^{4}$$ implies dim (V + W) $$\leq$$ dim $$\mathbb{R}^{4}$$, so

dim (V + W) $$\leq$$ 4

dim V + dim W - dim(V $$\cap$$ W) $$\leq$$ 4

4 - dim(V $$\cap$$ W) $$\leq$$ 4

- dim(V $$\cap$$ W) $$\leq$$ 0

dim(V $$\cap$$ W) $$\geq$$ 0

V $$\cap$$ W is a subspace of $$\mathbb{R}^{4}$$, therefore,

dim(V $$\cap$$ W) $$\leq$$ 4

put both inequalities together to get:

4 $$\geq$$ dim(V $$\cap$$ W) $$\geq$$ 0

so I would have chosen (E), but the correct answer is (D). What did I do incorrectly?

 

[darkchild]

Oh, I think I've figured it out...the intersection of two subspaces cannot have a greater dimension than the subspaces themselves.

 

[tiny-tim]

Yup!