Thread: Is the intersection of two vector spaces a vector space?

Suppose C and D are vector spaces over a field. Is the intersection of C and D a vector space? Give a proof or counterexample.

Okay, from the simple examples I've tried, it appears to be true, but I cannot figure out exactly how to prove it. My professor says I need to show it contains a zero element, is closed under scalar multiplication, and is closed under addition. I know that since both are vector spaces, they both must contain the zero element, so their intersection does as well. However, I don't know where to start with the other two requirements.

i will get you started:

suppose u,v are in C∩D. is u+v in C∩D?

well, u+v is in C, since C is a vector space, and both u,v are in C.

similarly, u+v is in D, since D is a vector space, and both u,v are in D.

so...now you continue. do the same thing for scalar multiplication.....