A projection matrix is a square matrix P that satisfies P2 = P. (a) Prove that w

Question:

A projection matrix is a square matrix P that satisfies P2 = P.
(a) Prove that w ∈ mg P if and only if P w = w.
(b) Show that mg P and ker P are complementary subspaces, as defined in Exercise 2.2.24, so every v ∈ Rn can be uniquely written as v = w + z where w ∈ mg P, z ∈ ker P.