Let A be an m n matrix of rank n and let P = A(AT A) 1AT . (a) Show that Pb = b for every b R(A). Explain this property 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.

Survey of Mathematics Math 2623 CH 1-3 “Operations Research” GOAL: optimize a result subject to constraints • Optimize- to make best of • Constraints- additional conditions which must be satisfied by something Typically real life conditions • Example: working workers over 40 hours means more money but by law you cannot work workers over 40 hours (constraint) CHAPTER 1: Urban Services • Example: shoveling snow, checking parking permits, etc. Parking Control Officer Problem #1 = at least one parking meter Pool Park Find a route for parking control officer using sides of stree