Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Question

Let $V$ be a finite-dimensional inner product space over field $F$, and let U be a subspace of $V$ . Prove that the orthogonal complement $U_{\perp}$ of $U$ with respect to the inner product $\langle \cdot , \cdot\rangle$ on $V$ satisfies $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Answer 1

Hint: Choose an orthonormal basis of $U$ say $v_1,\cdots,v_k$. Now extend it to an orthonormal basis of $V$ say $v_1,\cdots,v_k,v_{k+1},\cdots,v_n$. Then show that $v_{k+1},\cdots,v_n$ form a basis of $U^{\perp}$