5. Find the kernels and images of the following homomorphisms. Which of the homomorphisms areinjective? Which are surjective?Х(i) The mapping 0 : Z₁₁ → Z₁₁ given by 0(x) = x².Z- 11Х113(ii) The mapping ø : R* → R* given by o(x) = x³.&-1E(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.)

For (i) given by Step 1: Find the kernel.The kernel consists of all such that . Solve This means…

Answered: 5. Find the kernels and images of the…

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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2. Let : RR and y: R+ → R be mappings with 6(x) -2e-x. (a) Find the range of y. = 1. Use your answer to determine whether y is surjective. (b) Use the definition to show that & is bijective. (c) Find & oy. (d) Find the values of x for which 82(x) = 1. - and y(x) = 4 [3] [3] [2] [2]
Please help with both parts b and c: (will leave a like)
Let G = Z16X Z4. (i) Construct a surjective homomorphism : G → Z8. (ii) What is ker()? (iii) Show that there is no surjective homomorphism y: G → Z8 × Z8.
Select the following functions that are bijective. f3: R → R, f(x) = 14 f₂: R → R, f₂(x) 2 f₁:R → R, f₁(x) = x + x + x + x + 4 f4: R → R, f4(x) = -{ = x x², if x 20 X, if x <0
Let L be a linear transformation from P₂ → R2x2 such that L(f) — ƒ(1) 0 Find dim[Range(L)] ƒ(2) [ƒ(2) ƒ(0)
How would you find an isomorphism from f: Z3 X Z14 -> Z21 X Z2. Kindly elborate.
Which of the following functions are homo- morphisms? (a) g: R → M₂(R) given by g(a) = (a ). (b) h: QQ given by f(0) = 0 and 0 0 h() (c) f: Z3 Z3 given by f(x) = x³.
= x2 + 3x + 1 and g(x) = 2x – 3. Find formula Q.17: Let f: R → R and g: R → R be defined by f (x) defining the composition mapping. (i) fog (ii) gof (iii) fof (iv) gog
Let T: R² R² be given by T(x1, x₂) = = [at] It is known that I is an isomorphism. T-¹ is given by the following formula: Select one: O a. T-¹(x₁, x₂) = (x₁ - 2x2, X2) O b. T-¹(x1, x2) = (x₁, x2 - 2x₁) OC. T-¹(x₁, x₂) = (x1, x₂)
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3. (a) Consider the set A = {x² E QxER). Determine whether A is closed under addition and multiplication. (b) Consider the relation R = {(x, y) = Z² : 21x² + y²}. Verify the properties of R.
(c) Find all a e Z10, such that the multiplication mapping ma : Z10 → Z10, ma(x) = a -10 x, is bijection. Do the same thing with Z11.

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