1. Let A ∈ Fm×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 ∈ F6×8 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.
