5. Find the kernels and images of the following homomorphisms. Which of the homomorphisms are injective? Which are surjective? Х (i) The mapping 0 : Z₁₁ → Z₁₁ given by 0(x) = x². Z- 11 Х 11 3 (ii) The mapping ø : R* → R* given by o(x) = x³. & -1 E (iii) The mapping : GG given by (x) = g¯¹xg for x = G where G is a group and g = G. (You may assume that this is a homomorphism.)
5. Find the kernels and images of the following homomorphisms. Which of the homomorphisms are injective? Which are surjective? Х (i) The mapping 0 : Z₁₁ → Z₁₁ given by 0(x) = x². Z- 11 Х 11 3 (ii) The mapping ø : R* → R* given by o(x) = x³. & -1 E (iii) The mapping : GG given by (x) = g¯¹xg for x = G where G is a group and g = G. (You may assume that this is a homomorphism.)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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Transcribed Image Text:5. Find the kernels and images of the following homomorphisms. Which of the homomorphisms are
injective? Which are surjective?
Х
(i) The mapping 0 : Z₁₁ → Z₁₁ given by 0(x) = x².
Z- 11
Х
11
3
(ii) The mapping ø : R* → R* given by o(x) = x³.
&
-1
E
(iii) The mapping : GG given by (x) = g¯¹xg for x = G where G is a group and g = G.
(You may assume that this is a homomorphism.)
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