Weighted least squares. in least squares, the objective (to be minimized) is kax bk2 = xm i=1 (~ati x bi)2; where ~ati are the rows of a, and the n-vector x is to chosen. in the weighted least squares problem, we minimize the objective xm i=1 wi(~ati x bi)2; where wi are given positive weights. the weights allow us to assign di erent weights to the di erent components of the residual vector. (the objective of the weighted least squares problem is the square of the weighted norm, kax bk2 w, as de ned in exercise 3.28.) (a) show that the weighted least squares objective can be expressed as kd(ax b)k2 for an appropriate diagonal matrix d. this allows us to solve the weighted least squares problem as a standard least squares problem, by minimizing kbx dk2, where b = da and d = db. (b) show that when a has linearly independent columns, so does the matrix b. (c) the least squares approximate solution is given by ^x = ( b. give a similar formula for the solution of the weighted least squares problem. you might want to use the matrix w = diag(w) in your formula.