# Linear Algebra - Orthogonal Projections

1. Homework Statement
Prove that if P in L(V) is such that P2 = P and every vector in Ker(P) is orthogonal to every vector in Im(P), then P is an orthogonal Projection.

2. Homework Equations
Orthogonal projections have the following properties:

1) Im(P) = U
2) Ker(P) = Uperp
3) v - P(v) is in Uperp for every v in V.
4) P2 = P
5) ||P(v)|| <= ||v|| for every v in V.

3. The Attempt at a Solution
Property 4) is given, so that's done. I proved property 3) by using the equation v = P(u) + w, where w is in Uperp. Rearranging, I get v - P(u) = w, which is in Uperp.

Property 1) was proven in class: Clearly Im(P) is a subset of U. But for every u in U, P(u) = u since u = u + 0. Hence every element of U is in the image, and so Im(P) = U.

To prove property 2), I'm not really sure how to do this. My prof said it's basically the same thing as property 1, but I'm not really seeing it.

For property 5), I know that P(v) = <v, e_1>e_1 + ... + <v, e_m>e_m, where (e_1, ... e_m) is an orthonormal basis of U. I know that ||v|| = sqrt(<v, v>). Not sure how to compare these to come up with the inequality.

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Can anyone help on proving 2) and 5)? Does 2) directly follow from 1) since every vector in Ker(P) is orthogonal to every vector in Im(P)? Thanks.

Dick
U is just any subspace of V. V = U $$\oplus$$ Uperp. Thanks for your help.