= Solution
The <dimension formula for a sum of subspaces> and the zero intersection imply
$$
\dim(U+U^\perp)=\dim U+\dim U^\perp\geq n.
$$
But $U+U^\perp$ is a <vector subspace> of the $n$-dimensional space $V$, so its dimension is at most $n$. Equality follows, and the sum is all of $V$. Since the intersection is zero, it is a <direct sum>:
$$
\boxed{V=U\oplus U^\perp.}
$$
The argument remains valid for a degenerate ambient <symmetric bilinear form>; it uses the stated intersection hypothesis rather than assuming nondegeneracy everywhere.
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