[Solved] MATH307 Individual Homework18

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

1. Let A F^mn 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 F⁶⁸ has singular value decomposition A = UV 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.