Can you please help me in answering problem 3 part (a). I am confused for what is (i) and how to compute || Avi ||

(15) Two species model - Lotka-V X @ Assignment 2 - MATH-2740-A01 X TED INTERSPECIFIC COMPETITION * * Final-Review-sol X + X C A universityofmanitoba.desire2learn.com/d21/le/content/341936/viewContent/1695420/View?ou=341936 G . .. (a) Find a QR factorization of A. (b) Use your answer from (a) to find the least squares solution. Problem 2. Let 4= (3 7 1) (a) Compute a singular value decomposition of A. (b) Let b = (,). Compute At and find a least squares solution to Ax = b. (c) Compute projcol(A) (b) (Hint: you don't need the formula from Lab 2). Problem 3. Suppose that A is m xn matrix with rank k, singular value decomposition A = USVT, and reduced singular value decomposition A = USVI, where U = (u u2 . .. um) and V = (V1 V2 ... Vn). Suppose that the eigenvalues of A" A satisfy X1 2 12 2 ... > >.. (a) For all i E [n] (not just ), compute ||4vill. What happens if i > k? (Hint: Look at your calculations for Problems 2a and 2b in Lab 3) (b) Let y E R" such that there is a w ( R" such that y = Aw (i.e. y E Col(A)). Show that there is a z ( R* such that y = Uz (that is, show that y E Col(U)) (Hint: w = CIVI + C2V2 + . .. + (nVn for some constants ci, C2, . .., (n)- (c) It follows from Lab 3 (Problem 2) and (b) above that Col(A) = Col(U). Show that UU"b = projcol(A)(b) (Hint: Use the "column-row expansion" of UUT). (d) Use (c) above to verify your answer from Problem 2 (c). Type here to search M 9 ? FAXENG 7:59 PM 2019-10-04