Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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