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4. Let T: V→ W 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) = N(T) by Homework 9 Textbook 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).)

4. Let T: V→ W 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) = N(T) by Homework 9 Textbook 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).)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 4EQ
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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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