Do subspaces obey the axioms for a vector space? If I have a subspace $W$, then would elements in $W$ obey the 8 axioms for a vector space $V$ such as:
$u + (-u) = 0$ ; where $u ∈ V$ 

Yes, indeed. A subspace of a vector space is also a vector space, restricted to the operations of the vector space of which it is a subspace. And as such, the axioms for a vector space are all be satisfied by a subspace of a vector space.
As stated in my comment below: In fact, a subset of a vector space is a subspace if and only if it satisfies the axioms of a vector space.
