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5. Let A be an m x n matrix with real or complex entries. Use the results in the previous Problem 4 to show that rank(A* A) = rank(AA*) = rank(A), and rank(A) = n if and only if A*A is invertible.

5. Let A be an m x n matrix with real or complex entries. Use the results in the previous Problem 4 to show that rank(A* A) = rank(AA*) = rank(A), and rank(A) = n if and only if A*A is invertible.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
Problem 63EQ
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5. Let A be an m x n matrix with real or complex entries. Use the results in the previous Problem 4 to show that rank(A*A) and rank(A) = n if and only if A*A is invertible. = rank(AA*) = rank(A),
Transcribed Image Text:5. Let A be an m x n matrix with real or complex entries. Use the results in the previous Problem 4 to show that rank(A*A) and rank(A) = n if and only if A*A is invertible. = rank(AA*) = rank(A),
4. Let T: VW be a linear transformation between finite-dimensional inner product spaces, and let T*: W→ V be the adjoint of T. Prove the following statements. 1 = (a) rank(T*T) = rank(T). (Hint: N(T*T) = Sec. 6.3 Exercise 15(e).) (b) T is injective if and only if T*T is invertible. (Hint: use (a).) (c) rank(TT*) rank(T). (Hint: interchange the roles of T and T* in (a), and recall that rank(T*) = rank(T).) = N(T) by Homework 9 Textbook
Transcribed Image Text:4. Let T: VW be a linear transformation between finite-dimensional inner product spaces, and let T*: W→ V be the adjoint of T. Prove the following statements. 1 = (a) rank(T*T) = rank(T). (Hint: N(T*T) = Sec. 6.3 Exercise 15(e).) (b) T is injective if and only if T*T is invertible. (Hint: use (a).) (c) rank(TT*) rank(T). (Hint: interchange the roles of T and T* in (a), and recall that rank(T*) = rank(T).) = N(T) by Homework 9 Textbook
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