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Linear Transformation

Group Homomorphism and Abelian Groups

Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi). Proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel. Please show how to do the reverse (<=) and show that phi is

Homomorphism Subgroup Proofs

Let G be a group and let H be a normal subgroup of G. Let m be the index of H in G (that is, the number of cosets of H). Prove that for any a we have am H. (b) Give an example of group G, a subgroup H of index in, and an element a G such that am is not in H. (Of course, your subgroup H had better not be normal.) (4) (a) Suppos

Linear Transformations, Rotations, Submodules and Subspaces

(7) Let F=R, let V=R^2 and let T be the linear transformation from V to V which is rotation clockwise about the origin by pi-radians. Show that every subspace of V is an F[X]-submodule for this T. Here F[X] is a polynomial domain where the coefficient ring is a field F.

Linear Transformations : The composition of any two reflections

Matrix Theory - Homework 7 Prove the following in several stages: The composition of any two reflections,whose lines of reflection are orthogonal,is a half-turn. We will work in the vector space &#8477; .... First Stage Refer to the diagram provided.The line l makes some angle &#57534;with the x-axis. Let us suppose that

Automorphisms and Conjugation

Show that if H is any group then there is a group G that contains H as a normal subgroup with the property that for every automorphism f of H there is an element g of G such that the conjugation by g when restricted to H is the given automorphism f, i.e every automorphism of H obtained as an inner automorphism of G restricted to

Homomorphisms and Surjections

Let f:G->H be a group homomorphism. Prove or disprove the following statement. 1.Let a be an element of G. If f(a) is of finite order, then a is also of finite order. 2.Let f be a surjection. Then f is an isomorphism iff the order of the element f(a) is equal to the order of the element a , for all a belong to G. F

Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups

Problem 1. Let a,b be elements of a group G Show a) the conjugate of the product of a and b is the product of the conjugate of a and the conjugate of b b) show that the conjugate of a^-1 is the inverse of the conjugate of a c)let N=(S) for some subset S of G. Prove that the N is a normal subgroup of G if gSg^-1<=N for

Maximizing Revenue: Example

A cable TV company has 12,000 customers and charges $16 per month for basic service. The managers believe they will lose 400 customers for every dollar they raise the monthly charge. Determine the monthly charge that will maximize their revenue.

Composition of Functions and Isomorphisms

I can't prove the following statements about functions f:A->B and g:B->C 1. If gof is one-to-one then so is f. 2. If gof is onto then so is g. Furthermore I don't know how to show that f: A->B is an isomorphism of sets if and only if there is a function g: B->A such that gof=1A and fog=1B. Here fog and gof are compossiti

Maximizing Profit Problem

Suppose that a laundry service determines that to attract x customers per day, its price of service must be 3.20 - 0.02x dollars. If their total cost to serve x customers is 0.05x^2 + 1.10x + 120 dollars, how many customers should be served to maximize their profit? (Hint: Use the price to help get the revenue)

Linear Transformations and Subspaces

B1) This question concerns the following two subsets of : (a) Show that , and find a vector in that does not belong to T. [3] (b) Show that T is a subspace of . [4] (c) Show that S is a basis for T, and write down the dimension of T. [7] (d) Find an orthogonal basis for T that contains the vector .

Maximizing the Sustainable Yield

3.17) A lake has a carrying capacity of 10,000 fish. At the current level of fishing, 2000 fish per year are taken and the fish population seems to hold fairly steady at about 4000. If you wanted to maximize the sustainable yield, what would you suggest in terms of population size and yield?


Please help. I only need answers with brief explanations. No need of detailed working. (See attached file for full problem description) --- State whether the following are true or false with reasons: 1. If a in S6, then an =1 for some n greater than or equal to 1. 2. If axa-1=bxb-1, then a=b 3. The function e

Cyclic Vector

Suppose that T is a linear operator on a two dimensional vector space V, and that T is not multiple of the identity transformation. Show that T has a cyclic vector (i.e., there exists v V such that {v, Tv } is a base for V). Please see the attached file for the fully formatted problems.

Isomorphism proof

Show that a homomorphism from a field onto a ring with more than one element must be an isomorphism. Recall The the function f is an isomorphism if and only if f is onto and Kernel ={ 0}. Please explain step by step with reasons in every step.

Automorphism of a group

Modern Algebra Group Theory (LXI) Automorphism of a Group Is the mapping given below an automorphism of the group ? G group of integers under addition, T:x --> -x

Isomorphisms and path classes

(See attached file for full problem description with proper symbols) --- a) Under what conditions will two path classes, and , from to , give rise to the same isomorphism of onto ? b) Let be an arcwise-connected space. Under what conditions is the following true: For any two points , all path classes from


We are working on the proof of showing G (the group of rigid motions of a regular dodecahedron) is isomorphic to the alternating group A_5. Lemma: Let H be a normal subgroup of a finite group G, and let x be an element of G. If o(x) and [G:H] are relatively prime, then x is in H. Theorem: Any 60 element group having 24 el

Linear Mapping, Linear Space, Differentiability and Continuity

In each of Exercises 40 through 46 following, a linear space V is given and a mapping T : V&#8594;V is defined as indicated. In each case determine whether T is a linear mapping. If T is linear, determine the kernel (or null space) and range, and compute the dimension of each of these subspaces wherever they are finite-dimension

Linear Transformations, Change of Basis and Conjugation

Let V = Q3 and let ' be the linear transformation from V to itself: '(x, y, z) = (9x + 4y + 5z,&#8722;4x &#8722; 3z,&#8722;6x &#8722; 4y &#8722; 2z), x, y, z E Q With respect to the standard basis B find the matrix representing this linear transformation. Take the basis E = {(2,&#8722;1,&#8722;2), (1, 0,&#8722;1), (3,&#8722;2

Linear Transformations : Basis, Kernel, Image, Onto, One-to-one and Matrices

1) Define a linear transformation.... a) Find a basis for Ker T. b) Find a basis for Im T. c) Is T an onto map? d) Is T a one-to-one map? 2) Define a linear transformation... a) Find the matrix for T with respect to the standard basis. b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and t


I need help and an explanation for the following: Using the matrix A = 1 -1 0 0 -1 1 -1 2 -1 to compute TsubA(x), for x = (1, 2, 3)^T. Here TsubA: R^3 into R^3 is defined by TsubA(x) = Ax. Also describe the kernel of the transformation TsubA (that is state what a typical vector in ker T looks like).

Order of Elements, Factor Groups and Homomorphisms

Help with this linear transformation problem. Please help with the following problem. Provide step by step calculations. A. What is the order of the element 14 + <8> in the factor group Z24 / <8>? I know that if I let G = Z24 and H = <8> , then H = <8> = {0,8,16}. So Z24/<8> = {0+H,1+H,2+H,3+H,4+H,5+H,6+H,7+H}.

Job Shop Problem

Suppose you have N jobs that have to be processed on a single machine. For i = 1, 2, . . . ,N, job i requires pi units of time on the machine, and has weight wi. The objective is to schedule these jobs so as to minimize the sum of the weighted completion time of all the jobs, where the completion time of job i is the time at w

Matrices, Reflection Matrices and Linear Transformation

From the cubics P3 to the fourth degree polynomials P4, what matrix represents multiplication by 2 + 3t? The columns of the 5 by 4 matrix A come from applying the transformation to each basis vector x1 = 1, x2 = t, x3 = t2, x4 = t3. Verify directly from c2 + s2 = 1 that the reflection matrices satisfy H2 = I. c = cosine s

Using Excel Solver : Maximizing Interest and Defining Cash-Flow Constraints

At the start of the year a company wants to invest excess cash in one-month, three-month and six-month CD's. The company is somewhat conservative and wants to make sure it has a safety margin of cash on hand each month. (left over from previous month/ available at the outset, plus principal and interest from CD's that have becom

Using Excel Solver : Maximizing Interest and Cash Flow

At the start of the year a company wants to invest excess cash in one-month, three-month and six-month CD's. The company is somewhat conservative and wants to make sure it has a safety margin of cash on hand each month. (left over from previous month/ available at the outset, plus principal and interest from CD's that have becom

Topology : Homomorphism Formatted

Please see the attached file for the fully formatted problems. B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant. (b) State what it means for a map f X -?> Y to be open. Show that a continuous open bijection is a homomorphism. (c) (i) Recall that Fr E, the fron