[SOLVED] MATH307-Individual Homework 18

Description

1. Let A ∈ F^m×n with F = R or C, find a basis for both range(A) and range(A^∗) and then prove that the column rank of A is the same as the row rank of A.
2. Assume matrix A ∈ F^6×8 has singular value decomposition A = UΣV ^∗ with singular values 21,11,6,6,0.2,0.
   • Find the row rank of A, i.e, the dimension of range(A^∗) and find an orthonormal basis of range(A^∗) in terms of the SVD of A and prove it.
   • Find the nullity A^∗, i.e., the dimension of null(A^∗) and find an orthonormal basis of null(A^∗) in terms of the SVD of A and prove it. You may use the rank-nullity theorem without proving it.