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Exercise 2.18 For all a ER, let fa : R³ R³ be the linear map defined by fa(x, y, z) = (x+(2-a)y+z, x-y+z, x-y+ (4- a) z). a) Write the matrix of fa in the canonical bases. b) Determine the values of a such that fa is injective. c) Determine the values of a such that (1, 1, 1) belongs to Im(fa). d) For a = 1, determine the kernel of f1. e) Construct, if possible, a linear map g: R² R³ such that Img = Im fo. → f) Construct, if possible, a surjective linear map h: R³ R2 such that Ker(h) Ker(fi). =

Exercise 2.18 For all a ER, let fa : R³ R³ be the linear map defined by fa(x, y, z) = (x+(2-a)y+z, x-y+z, x-y+ (4- a) z). a) Write the matrix of fa in the canonical bases. b) Determine the values of a such that fa is injective. c) Determine the values of a such that (1, 1, 1) belongs to Im(fa). d) For a = 1, determine the kernel of f1. e) Construct, if possible, a linear map g: R² R³ such that Img = Im fo. → f) Construct, if possible, a surjective linear map h: R³ R2 such that Ker(h) Ker(fi). =

Advanced Engineering Mathematics
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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Exercise 2.18 For all a € R, let fa : R³ R³ be the linear map defined by fa(x, y, z)=(x +(2-a)y+z,x=y+z,x=y+(4- a)z). a) Write the matrix of fo in the canonical bases. b) Determine the values of a such that fa is injective. c) Determine the values of a such that (1, 1, 1) belongs to Im(fa). d) For a = 1, determine the kernel of f1. e) Construct, if possible, a linear map g: R². →→R³ such that Im g = Im fo. R³ f) Construct, if possible, a surjective linear map h : Ker(fi). →R2 such that Ker(h) =
Transcribed Image Text:Exercise 2.18 For all a € R, let fa : R³ R³ be the linear map defined by fa(x, y, z)=(x +(2-a)y+z,x=y+z,x=y+(4- a)z). a) Write the matrix of fo in the canonical bases. b) Determine the values of a such that fa is injective. c) Determine the values of a such that (1, 1, 1) belongs to Im(fa). d) For a = 1, determine the kernel of f1. e) Construct, if possible, a linear map g: R². →→R³ such that Im g = Im fo. R³ f) Construct, if possible, a surjective linear map h : Ker(fi). →R2 such that Ker(h) =
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