Let A be an m x n matrix of rank n and let P = A(ATA)-1 AT.

Question:

Let A be an m x n matrix of rank n and let P = A(ATA)-1 AT.
(a) Show that Pb - b for every b ∈ R(A). Explain this in terms of projections.
(b) If b ∈ R(A)⊥, show that Pb = 0.
(c) Give a geometric illustration of parts (a) and (b) if R(A) is a plane through the origin in R3.