dont need to solve question d)

hand writing solution please

Problem 3. Suppose that A is mxn matrix with rank k, singular value decomposition A= USVT, and reduced singular value decomposition A = ST, where U = (u, uz ... um) and V = (v1 V2 ... vn). Suppose that the eigenvalues of A? A satisfy li > 12 > ... > In. (a) For all i E [n] (not just [k]), compute || Avi|l. What happens if i > k? (b) Let y E RM such that there is a w E Rsuch that Aw (i.e. y e Col(A)). Show there is a z e Rk such that y = z (that is, show y e Col()) (Hint: w = CV1 + C2V2 + . for some constants C1, C2, ..., Cn). (c) It follows from Lab 4 (Problem 2) and (b) above that Col(A) = Col(). Show that Ub = projcol(A)(b) (Hint: Use the column-row expansion of T). (d) Use (c) above to verify your answer from Problem 2 (c). + Vn Problem 3. Suppose that A is mxn matrix with rank k, singular value decomposition A= USVT, and reduced singular value decomposition A = ST, where U = (u, uz ... um) and V = (v1 V2 ... vn). Suppose that the eigenvalues of A? A satisfy li > 12 > ... > In. (a) For all i E [n] (not just [k]), compute || Avi|l. What happens if i > k? (b) Let y E RM such that there is a w E Rsuch that Aw (i.e. y e Col(A)). Show there is a z e Rk such that y = z (that is, show y e Col()) (Hint: w = CV1 + C2V2 + . for some constants C1, C2, ..., Cn). (c) It follows from Lab 4 (Problem 2) and (b) above that Col(A) = Col(). Show that Ub = projcol(A)(b) (Hint: Use the column-row expansion of T). (d) Use (c) above to verify your answer from Problem 2 (c). + Vn