Of course checking all the vector space axioms can be quite tedious, but as a theorem you prove much easier criteria to check. Recall that you already know that
is a vector space, so many of the axioms (associativity, distributive laws etc) are inherited from
Indeed, you prove that to show that
is a subspace of
it is enough to show that the additive identity of
is in
and that
is
closed under the inherited operations of vector addition and scalar multiplication, i.e, whenever
and
we must have
and