Problem 5. (10pts) Let A E Mm.n(k) be a matrix of rank 2. Prove that A...
A is mxn matrix Problem 7 (10pts) Prove any TWO of the following: Let A be a mx n matrix. Then • (AA+)+ = AA+ and (A+A)+ = A+A • A+ = (ATA)+AT = AT (AAT)+ • A+ = (ATA)-IAT and A+A = In if rank(A) = n, • A+ = AT (AAT)-1 and AA+ = Im if rank(A) = m, • A+ = AT, if the columns of A are orthogonal, that is ATA=In
Prove that any m x n matrix A of rank k can be written as A = {k=1 u;v] where {u1, ..., Uk} and {V1, ... , Vk} are linearly independent sets. By SVD, any mxn matrix A of rank k can be written as A = {k=10;U;v] where {u1,..., Uk} and {V1, ... , Vk} are orthonormal sets and 01 > 02 > ... >0k > 0. For this problem, prove without using SVD.
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...
(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...
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.
Let A be an mx n matrix and B be an n xp matrix. (a) Prove that rank(AB) S rank(A). (b) Prove that rank(AB) < rank(B).
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> W be a linear transformation. (a) Prove that for every pair of ordered bases B = exists a unique m x n matrix A such that [T(E)]c = A[r3 for all e V. The matrix A is called the (B,C)-matrix of T, written A = c[T]b. (b) For each n E N, let Pm be the vector space of...
11 (10) [10pts) Let A be the 2 x 2 matrix A-L 5 A = 2 4 The characteristic polynomial of A is (3- A) (1 - ). Compute an invertible matrix P and diagonal matrix D such that A= PDP-
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...