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Question 1. Let f : VV be an endomorphism of a finite-dimensional inner product space. Prove that (a) the map f is injective if and only if its adjoint f* is surjective, and that (b) the map f is surjective if and only if its adjoint f* is injective.

Question 1. Let f : VV be an endomorphism of a finite-dimensional inner product space. Prove that (a) the map f is injective if and only if its adjoint f* is surjective, and that (b) the map f is surjective if and only if its adjoint f* is injective.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.5: Permutations And Inverses
Problem 10E: 10. Let and be mappings from to. Prove that if is invertible, then is onto and is...
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Question 1. Let f : VV be an endomorphism of a finite-dimensional inner product space. Prove that (a) the map f is injective if and only if its adjoint f* is surjective, and that (b) the map f is surjective if and only if its adjoint f* is injective.
Transcribed Image Text:Question 1. Let f : VV be an endomorphism of a finite-dimensional inner product space. Prove that (a) the map f is injective if and only if its adjoint f* is surjective, and that (b) the map f is surjective if and only if its adjoint f* is injective.
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