So I have to prove the following, and not sure where to go with it:
Prove that if $\displaystyle V=W\oplus W^{\perp}$ and T is the projection on W along $\displaystyle W^{\perp}$ then T=T*.
Any help? THanks
let $\displaystyle v_1=w_1+w_1', \ v_2=w_2+w_2'$ where $\displaystyle w_i \in W, \ w_i' \in W^{\perp}.$ then $\displaystyle <Tv_1,v_2>=<w_1,w_2+w_2'>=<w_1,w_2>$ and $\displaystyle <v_1,Tv_2>=<w_1+w_1',w_2>=<w_1,w_2>.$
so $\displaystyle <Tv_1,v_2>=<v_1,Tv_2>$ and thus $\displaystyle T=T^*.$