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I need help getting started with this problem. Please just give me a nudge in the right direction.
Suppose $U_1,U_2$ are subspaces of the euclidean space $V$ such that $dimU_1<dimU_2$.
Show that there is a nonzero vector $u\in U_2$ such that $u$ is in the orthogonal complement of $U_1$.
Abridged solution. If $U_1^\perp \cap U_2 = \{0\}$ then $U_1 \subset U_2.$ Hence, the dimension hypothesis completes the result. Q.E.D.
