containing zero is important for vector spaces and are hence important for subspaces. so the subspaces must have 0 in them. it is not to say that they have to have 0 to be non-empty per sae, but it is usually easy to verify that the zero vector is in there. so you always check that. and if 0 is not in there, then you're in trouble. you wouldn't have a vector space at all. just some other kind of set.