Let P = A(AT A) 1AT , where A is an m n matrix of rank n. (a) Show that P2 = P. (b) Prove that Pk = P for k = 1, 2, ... . (c) Show that P is symmetric. [Hint: If B is nonsingular, then (B1) T = (BT ) 1.]

S343 Section 2.1 Notes- First Order Linear Equations and Integrating Factors 8-25-16 Recall equation for motion of falling object: = = − + o Position: = + + 2 0 0 o Velocity: = + 0 Replacing constants and with arbitrary functions of : = − + () o Arrive at general first order form + = ( )