# Question about vector space intersection properties

1. Aug 5, 2013

### 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 dont remember if the same will hold for vector spaces over addition?

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

2. Aug 5, 2013

### 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 $(A \cap B) + (A \cap C) = \{0\}$.