2. Let Si, i = 1, . .., k be k hyperplanes in R" defined by Si = {xEV | (ai, x) = bi}, i=1, ...,k. Let y E R" be given. We consider the projection of y onto ,S, i.e., the solution of min x - y |2. (1) Prove that z is a solution of (1) if and only if z en_15; and (z - y, z - x) =0, Vac ent_Si. (2) 3. Let {(x;, yi)) are given, x; E R" and y; E R. Assume N f(aix -b;) + Alla||2 i= 1 with f : R - R is a differentiable function. (a) Compute VF(x). (b) Compute V2F(x). (c) Prove the minimizer of F must be in the form of ac* = En ca;, with c = [C1, C2, ..., CN]. (d) Re-express the minimization problem (3) in terms of c with fewer unknowns. 4. Compute the gradient and Hessian of the following functions. (a) f(x) = ELVE21+ 12, where a E Ran. (b) f(X) = log det(X), where X c Roxn is symmetric positive definite