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Let (e1,e2,e3) the standard basis of R3. Let b be a real number and f, the unique endomorphism in R3 such that: fo(e1) = e2 fo(e2) = e1 + b • e3 fo(e3) = b • e2 a) Find dim Im f, for all the possible values of b. b) Find if there exists a value of b such that f, have a eigenvectors base in which the associated mapping matrix is: 0 0 0 A = 0 2 0 0 0 2

Let (e1,e2,e3) the standard basis of R3. Let b be a real number and f, the unique endomorphism in R3 such that: fo(e1) = e2 fo(e2) = e1 + b • e3 fo(e3) = b • e2 a) Find dim Im f, for all the possible values of b. b) Find if there exists a value of b such that f, have a eigenvectors base in which the associated mapping matrix is: 0 0 0 A = 0 2 0 0 0 2

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 33EQ
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Let {e1,e2,e3} the standard basis of R3. Let b be a real number and f, the unique endomorphism in R3 such that: fo(e1) = e2 fo(e2) = e1 + b + e3 fp(e3) = b • e2 a) Find dim Im f, for all the possible values of b. b) Find if there exists a value of b such that f, have a eigenvectors base in which the associated mapping matrix is: 0 0 0 A = 0 2 0 0 0 2 c) Given b = 0, find the unique endomorphism g (not empty/trivial) such that: i) Im g C Ker f, ; ii) g(e1) = g(e2) = g(e3); iii) 1 is an eigenvalue of g
Transcribed Image Text:Let {e1,e2,e3} the standard basis of R3. Let b be a real number and f, the unique endomorphism in R3 such that: fo(e1) = e2 fo(e2) = e1 + b + e3 fp(e3) = b • e2 a) Find dim Im f, for all the possible values of b. b) Find if there exists a value of b such that f, have a eigenvectors base in which the associated mapping matrix is: 0 0 0 A = 0 2 0 0 0 2 c) Given b = 0, find the unique endomorphism g (not empty/trivial) such that: i) Im g C Ker f, ; ii) g(e1) = g(e2) = g(e3); iii) 1 is an eigenvalue of g
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