Lemma 7.3.1. Let V and W be finite-dimensional inner product spaces and let T: VW be a linear transformation. Then Im(7*) = Ker(T)¹ and Ker(T*) = Im(7)¹ 4 (2) dim(Ker(T*)) = dim(Ker(T)) If dim(W) = dim(V) then dim(Im(7*)) dim(Im(T)).
Lemma 7.3.1. Let V and W be finite-dimensional inner product spaces and let T: VW be a linear transformation. Then Im(7*) = Ker(T)¹ and Ker(T*) = Im(7)¹ 4 (2) dim(Ker(T*)) = dim(Ker(T)) If dim(W) = dim(V) then dim(Im(7*)) dim(Im(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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![Lemma 7.3.1. Let V and W be finite-dimensional inner product spaces
and let T : V → W be a linear transformation. Then
Im(7*) = Ker(T)+ and Ker(T*) = Im(7)+
(2) dim(Ker(7*)) = dim(Ker(T))
(3) If dim(W) = dim(V) then dim(Im(7*)) = dim(Im(7)).
ਦਾ sa dimil]-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc774bb4e-5e91-450b-91f0-dc972d0bae97%2Fca01ad8c-7a71-43d8-9f8c-9682b369547a%2Fr8y0tu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Lemma 7.3.1. Let V and W be finite-dimensional inner product spaces
and let T : V → W be a linear transformation. Then
Im(7*) = Ker(T)+ and Ker(T*) = Im(7)+
(2) dim(Ker(7*)) = dim(Ker(T))
(3) If dim(W) = dim(V) then dim(Im(7*)) = dim(Im(7)).
ਦਾ sa dimil]-
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