Two subspaces of different dimensions and the orthogonal complement

Question

I need help getting started with this problem. Please just give me a nudge in the right direction.

Suppose $U_1$ and $U_2$ are subspaces of the Euclidean space $V$ such that $\dim(U_1) < \dim(U_2)$.
Show that there is a nonzero vector $u \in U_2$ such that $u$ is in the orthogonal complement of $U_1$.

Answer 1

If $U_1^\perp \cap U_2 = \{0\}$ then $$ \dim(V) \ge \dim(U_1^\perp) + \dim(U_2) = \dim(V) - \dim(U_1) + \dim(U_2) > \dim(V) $$ by the hypothesis $\dim(U_1) < \dim(U_2)$.

Answer 2

Here's a nudge: this can be proved easily using the codimension inequality: $$\mathop{\mathrm{codim}}(U_1^{\perp}\cap U_2)\le\mathop{\mathrm{codim}}(U_1^{\perp})+\mathop{\mathrm{codim}}(U_2)$$