3. Let A e IRmxn where where x minimizes llAz-bl 2. m 2 n, and A has full rank. Show that A = I a...
-1has a solution . Let AERwhere m 2 n, and A has full rank. Show that T where z minimizes lAr b
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12. 3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12.
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pseudo-inverse of a vector u R" regarded as an m x 1 matrix? 3. (5 pts) Let B AT where A is the matrix in Problem 1. Use Matlab to find the singular value decomposition and the Moore-Penrose pseudo-inverse of B. Then solve minimum-norm least squares problem minl-ll : FE R minimizes IBr-ey where c- [1,2. Compare...
6. Suppose A E Rnxm has full rank, that is, rank(A) min(n, n). Let ơi > > Ơr be the singular values of A. Let B E Rnxm satisfy IA-B 2 < σ'. Then B also has full rank. Suppose A E Rnx'n has full rank, that is, rank(A)-r-min(n, n). Let ơi > > ơr be the singular values of A. Let B E Rnxm satisfy IIA-Blla < ơr. Then B also has full rank 6. Suppose A E Rnxm...
9. (2 pts per part) Let A be an m x n matrix, where m > n, and suppose that the rank of A is n (i.e., A has full column rank). Briefly justify your answers to each question below. a. Which two of the following statements are true? i. There are no vectors in Nul(A). ii. There is no basis for Nul(A). iii. dim(Nul(A)) = 0 iv. dim(Nul(A)) = m – n b. Are the columns of A a...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Please show all work in READ-ABLE way. Thank you so much in advance. Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
3. (15 pts) Let A be an m x n matrix with rank r, and let V = C(A). (a) V CIRP for what p? (b) What is V. in terms of a fundamental subspace for A? (c) How many vectors are in a basis for V, and how many in a basis for v 1? (d) For what m, n, and r docs Ax=b have a solution for every b? (e) Is a set of r vectors in V...
True or False? If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ 1s the same as matrix Σ with singular values ok replaced with 1/0k. Answer: _ If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ...
Let A be an m x n matrix of unspecified rank. Let b e Rm, and let Prove that this infimum is attained. In other words, prove the existence of an ax for which l|Ax bll p. In this problem, the norm is an arbitrary one defined on Rm. Let A be an m x n matrix of unspecified rank. Let b e Rm, and let Prove that this infimum is attained. In other words, prove the existence of an...