Suppose $V$ is finite dimensional and $U$ is a subspace of $V$ such that $\dim U = \dim V$. Prove that $U = V$.

Let $u_1,\dots,u_m$ be a basis of $U$, extend this to a basis of $V$ using 2.33. Since a basis of $V$ has the same length this "extension" is the same as doing nothing. Thus $u_1,\dots,u_m$ is a basis for $U$ and $V$ so