Question about vector space intersection properties

[raymo39]

Its been a while since I've done this stuff, and I don't have a text handy. I know that for sets, intersection distributes over union, I don't remember if the same will hold for vector spaces over addition?

for example does A \cap (B + C) = A \cap B + A \cap C

 

[jbunniii]

Suppose $A,B,C$ are the following subspaces of $\mathbb{R}^2$: $B = \{(b,0) : b \in \mathbb{R}\}$, $C = \{(0,c) : c \in \mathbb{R}\}$, and $A = \{(a,a) : a \in \mathbb{R}\}$.

Then $B + C = \mathbb{R}^2$, so $A \cap (B + C) = A$.

But $A \cap B = \{0\}$ and $A \cap C = \{0\}$, so $(A \cap B) + (A \cap C) = \{0\}$.