Let A be an m x n singular matrix of rank r with SVD sigma1

Let A be an m x n singular matrix of rank r with SVD sigma_1 Greaterthanorequalto ... Greaterthanorequalto sigma_r > 0, U consists of the first r columns of U, U consists of the remaining m -r columns of U, V consists of the first r columns of V. and V consists of the remaining n - r columns of V. Give bases for the spaces range(A), null(A). range(A^T) and null(A^T) in terms of the components of the SVD of A, and a brief justification.

Solution

U hat will form the bases for range(A)

U tilde will form bases for Null A

V hat will form bases for range A^T

V tilde will form bases for null A^T