Exercise 6.6.9

and so $U^{\text{∗}}U(x)=U^{\text{∗}}(0)=0$. If $x\in N(U^{\text{∗}}U)$, we have

for all $y$. This means that $x\in R{(U^{\text{∗}}U)}^{\text{⊥}}$. Now we know that $V=R(U^{\text{∗}}U)\oplus N(U^{\text{∗}}U)$ and we can write element in $V$ as $p+q$ such that $p\in R(U^{\text{∗}}U)$ and $N(U^{\text{∗}}U)$. Check that

by the definition of $U^{\text{∗}}$ in Exercise 6.5.30(e). Hence it’s an orthogonal projection.

be an orthonormal basis for $W$. Extend it to be an orthonormal basis

for $V$. Now check that

if $i>k$ and

if $i\le k$ by the definition of $U^{\text{∗}}$ in Exercise 6.5.20(e). They meet on a basis and so they are the same.