Chapter 5.3, Problem 9E

LetAbean $m\times n$matrixofranknandlet $P=A{\left(A^{T}A\right)}^{-1}{A}^T.$
(a) Show that $Pb=b$ for every $b\in R\left(A\right).$ Explain this property in terms of projections.
(b) If $b\in R{\left(A\right)}^{\perp },$ showthat $Pb=0.$
(c) Give a geometric illustration of parts (a)and (b) if $R\left(A\right)$ is a plane through the origin in ${ℝ}^3.$