(a) Let V = {A is a 4 x 4 matrix | Tr (A) = 1}. Determine whether V is a vector space. Jus- %3D tify your answer. (b) Let A be a 5 x 5 matrix. Assume that the nullspace of A has dimension 2, show that 0 is an eigenvalue of A" with algebraic multiplicity at least 2. (c) Let A be a 5 x 7 matrix. (i) What is the maximum and minimum possible values of rank (A)? (ii) What is the maximum and minimum possible values of rank (AT)? Justify your answers. (d) If A is an m x n matrix, show that A and AT A have the same nullspace.
(a) Let V = {A is a 4 x 4 matrix | Tr (A) = 1}. Determine whether V is a vector space. Jus- %3D tify your answer. (b) Let A be a 5 x 5 matrix. Assume that the nullspace of A has dimension 2, show that 0 is an eigenvalue of A" with algebraic multiplicity at least 2. (c) Let A be a 5 x 7 matrix. (i) What is the maximum and minimum possible values of rank (A)? (ii) What is the maximum and minimum possible values of rank (AT)? Justify your answers. (d) If A is an m x n matrix, show that A and AT A have the same nullspace.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.2: Diagonalization
Problem 32E
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