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).)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2F56455dee-5418-415a-961c-4b48abc8b715%2Faz002yz_processed.jpeg&w=3840&q=75)
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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