Problem 6. We want to prove the following statement: If A e Rmxn is full rank, then ATA is nonsingular (i.e., invertible). Let's prove this step-by-step. (a) Let x, z E R", bE RM, a E R, and consider the following: 1| A(a + Oz) - 6/13. Write this quantity as a quadratic polynomial in terms of a. (b) Suppose a solves the least squares problem, i.e., Ax - b|5 is minimized while a + oz is not a solution to the lease squares problem. Then, using Part (a), deduce A (Ax - b) = 0, i.e., a is a solution to the normal equation AT Ax = ATb. (c) Suppose both ac and ac + oz are both solutions to the same least squares problem. Show that z E null(A). (d) If A is full rank, then show that such z must be 0. This concludes the proof of the state- ment we wanted to prove, i.e., the normal equation has a unique solution, i.e., A A is nonsingular