Exercise 6.C.3

Proof. By 6.31, we have $\mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(u_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,u_m)=\mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(e_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,e_m)$. Therefore $e_1,\dots ,e_m$ is indeed an orthonormal basis of $U$. Clearly $\mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(f_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,f_n)\subset U^{\text{⊥}}$ (because each of the $f$’s are orthogonal to the basis of $U$). Moreover,

Hence $f_1,\dots ,f_n$ is linearly independent list of length $\dim <!--\; FUNCTION\; APPLICATION\; -->U^{\text{⊥}}$, that is, a basis of $U^{\text{⊥}}$. □