Linear Algebra Question - Vector Spaces and Vector Subspaces

Question

Can I have help with the following question related to the concept vector 'subspaces'?

Question : Let X and Y be vector subspaces of a vector space V over a field F, (V,+,•,F).

Prove that the intersection of X and Y is also a vector subspace of V.

(1)So when we take any two vectors u,v€(X^Y) (€denote belongs to a set ans ^ denote intersection),

Then we can show that the vector u+v€ (X^Y)

2) If we consider any scalar A€F, any vector u€(X^Y), 

we can show that the vector Au€(X^Y)

My question is, do we need to show that the identity element of V(say 0), also belong to X^Y, to show that X^Y is a vector subspace of V?

Or is proving the conditions (1) & (2) above would be sufficient to prove that X^Y is a vector subspace of V?

Thank you!

Answer 1

If X is a subspace of V then the 0 element lies in X. It follows because since X is nonempty then there exists an x in X and since X is a vector space the element -x is also in X and since X is a vector space x-x=0 is also in X. Therefore, the set (X intersection Y) contains the 0 element since 0 lies in X and 0 lies in Y.

So proving 1 and 2 is sufficient.