(a) Let V = {A is a 4 × 4 matrix | Tr (A) = 1}. Determine whether V is a vector space. Jus-tify your answer.(b) Let A be a 5 × 5 matrix. Assume that the nullspace of A has dimension 2, show that 0 isan eigenvalue of AT with algebraic multiplicity at least 2.(c) Let A be a 5 × 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 × n matrix, show that A and AT A have the same nullspace.

According to our guidelines we can answer only three subparts, or first question and rest can be…

(a) Let V = {A is a 4 x 4 matrix | Tr (A) = 1}. Determine whether V is a vector space. Jus- 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 AT 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- 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 AT 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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