Let W be subspace of a vector space V. Prove that if V= W \bigoplus W^c. and T is orthogonal projection onto W, then T=T*.

V= W \bigoplus W^c. is direct sum
W^c is W complement
T* is T transpose conjugate.

I know:
V= W \bigoplus W^c means
v= w_1+w_2, st w_1 \in W_1 and w_2 \in W^c
T(w_1+w_2)= T(w_1)=w_1

Not sure how to make T=T*?

Thanks in advance