Must a vector subspace contain the null vector? If it must, why? If it mustn't, why?
I was able to prove that a certain set of vectors doesn't have the zero vector as a member and normally, I've been told to conclude that it is not a vector subspace from this point. I got confused and wondered if this always shows that the set isn't an empty set since that is one of the conditions a vector subspace must satisfy.

Let $V$ be a vector space and $U\subset V$ be a subspace. Since $U$ is non-empty there is a $u\in U$. Since $U$ is a subspace we get $-u\in U$. Hence also $0=u+(-u)\in U$.
Alternatively also $0=0_K\cdot u\in U$, where $0_K$ denotes the zero in the ground field $K$ (assuming $V$ is a $K$-vector space).
