Description
Instructions: Read textbook pages 87 to 90 before working on the homework problems. Show all steps to get full credits.
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.
(a) 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.
(b) 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.