1. The problem statement, all variables and given/known data Suppose P ∈ L(V) is such that P2 = P. Prove that P is an orthogonal projection if and only if P is self-adjoint. 2. Relevant equations 3. The attempt at a solution Let v be a vector in V. Let w be a vector in W and u be a vector in U and let U and W be subspaces of V where dim W+dim U=dimV. Let v=u+w. Now let P be an orthogonal projection onto u. We know that u and w are orthogonal so <u, w>=0. So P2=Pu=u. Since Pu=u, <Pu, w>=0. Now we know that Pw=0 since PP(v)=PP(u+w)=P(Pu+Pw)=P(u+0)=Pu therefore <u, Pw>=<u, 0>=0. Thus <Pu, w>=0=<u, Pw>.