How can I prove that f is one-to-one (injective)? Question 1 Consider the groups Z² = Z × Z and Z³ = Z × Z × Z and the map ƒ : Z³ → Z²defined by the rulef((a,b,c))=(a+b+c,a+b).

We know, a function/map f is one to one (injective) iff For a ≠ b, f(a) ≠ f(b) for any two…

Answered: How can I prove that f is one-to-one…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 10EQ: In Exercises 1-12, determine whether T is a linear transformation. T:FF defined by T(f)=f(x2)

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45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
3. For each of the following mappings, write out and for the given and, where.
Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.
Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.
22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. b. in Exercise 36 of section 3.1. c. in Exercise 35 of section 3.1. d., the general linear group of order over. Exercise 34 of section 3.1. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. Exercise 36 of section 3.1 Consider the matrices in , and let . Given that is a group of order 8 with respect to multiplication, write out a multiplication table for. Exercise 35 of section 3.1. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. Given that is a group of order with respect to matrix multiplication, write out a multiplication table for .
-EN- 4. Consider the the linear maps f: R² → R² and g: R² R² given by f((x, y))=(x-y, 2+ y) and g((x, y)) = (3x + 2y, 5x + 3y) Describe g of and fog.
) The linear transformation T is given by the following formula T(x1, x2, x3, x4) = (x1 - x₂ + x3 x4,2x1 Determine for what values of b it is onto and for what values of b it is one-to-one. Explain briefly why. 5. 1. - x2 + bx4, x1 - 2x2 + 3x3 - 4x4).
- 3) Given the mapping T: R? → R defined by T(x),x2) = (3x1 – 2x2, 4.x1 + 5x2, x1 -x2) a) Prove that T is a linear transformation. b) Determine if (4,-1,9) is in the range of T. Explain your answer.
Suppose T: V --> W is a linear and surjective map and V and W are of finite dimension. Show that T has a linear right inverse, i.e., there is a linear map S: W --> V so that the composition T of S = Id. Explain your answer.
2. Consider the groups (R, +) and (Rx R, +). Define the map RX (RX R) → Rx R defined by r (x, y) = (x + ry.y). a Show that this map is a group action. (b). Find Orb ((1, 0)), Orb((1, 1)) and Stab((0,0)). primes and ne
Consider maps π1 : G1 × G2 → G1 given by π1(g1, g2) = g1 and π2 : G1 × G2 → G2 given byπ2(g1, g2) = g2. Show that π1 and π2 are homomorphisms (You can just show one of them is a homomorphism and say the other one follows similarly.)

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