Since you are given that H is a subspace of V, to show that H= V, you need only show that any vector in V is also in H.

Let v be any vector in V and select a basis, for H. If v were not in H, then it could not be written as a linear combination of those basis vectors and so would be a linearly independent set of n+ 1 vectors in V. Do you seewhythey are linearly independent and why that contradicts the fact that V has dimension n?