Relationship of dimensional subspace and vector space

Question

Let H be an n-dimensional subspace of a n-dimensional vector space V . Explain why H = V

So we know H lies within V, but why does H itself equal V, that's what I don't understand.

Answer 1

Since $H$ is $n$-dimensional, it means $H$ has a basis of $n$ vectors. Any $n$ linearly independent vectors from an $n$-dimensional space will span that space. Thus the basis of $H$ spans $V$ and

$$ H = \text{span of basis of }H = V $$

Answer 2

As you say, $H \subseteq V$ is clearly implied by $H$ being a subspace. As such, we only need to show that $V \subseteq H$ to conclude that $H=V$.

Let $\vec{v} \in V$ and let $\{\vec{h}_1, ~\vec{h}_2 , ~\dots, ~\vec{h}_n \}$ be a basis for $H$. Assume not: assume that $\vec{v} \not \in H$. Then $\vec{v}$ is not expressible as a linear combination of the basis vectors of $H$, so $\{\vec{h}_1, ~\vec{h}_2 , ~\dots, ~\vec{h}_n , ~\vec{v} \}$ is a linearly independent set. However, $V$ is $n$-dimensional so $n+1$ vectors must be linearly dependent; a contradiction.