I have this problem set. It is long and I do not know how to do it.

3. Constrained Least-Squares Optimization In this problem, you'll go through aprocess of guided discovery to solve the following optimization problem: Consider a matrix A E IRMXN, of full column rank, where M > N. Determine a unit vector if that minimizes \"M\and vEATA = 129% . (5) Premultiply Equation 4 with g, postmultiply Equation 5 with 9}, compare the two, and explain how one may then infer that if}; and g are orthogonal. (c) Since the N eigenvectors of ATA aJe mutually orthogonaland each has unit lengththey form an orthonormal basis in B\". This means that we can express an arbitrary vector if E RN as a linear combination of the eigenvectors 31,. . . , ifN, as follows: N if = 2 an if". 11:] i. Determine the nth coefcient can in terms of if and one or more of the eigenvectors if] , . . . ,VN. ii. Suppose if is a unit-length vector (i.e., a unit vector) in R\". Show that N 2 (13:1. n=l (d) N ow you're well-positioned to tackle the grand challenge of this problemdetermine the unit vector fx'that minimizes MAJf\