Exercise 9.B.4

Proof. Choose an orthonormal basis of $V$. Then this is also a basis of $V_{}$

C$,see(9.4).Wehave$

C) = M(T) = M(T^*) = M((T^*)_C) = M((T_C)^*),

where the first and third equalities follow from 9.7, the second from 7.10 and the fourth from 9.30 (c) (take $U=V$). This implies that $T_{}$

C = (T_C)^*$.\square$