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Linear Transformation

Transportation problem: Cost minimization

Three quarries produce lime which is then shipped to five warehouses Quarry Tons Lime 1 200 2 100 3 150 Transport Costs (per ton) Quarry 1 2 3 4 5 1 $5 $1 $6 $3 $1 2 $2 $3 $4 $5 $4 3 $4 $2 $3 $2 $3 Demand 80 90 100 70 60 (tons) Minimize t

Range of Linear Operators

Show that the range of the linear operator defined by the equations w1 = x1 - 2x2 + x3 w2 = 5x1 - x2 + 3x3 w3 = 4x1 + x2 + 2x3 is not all of R3, and find a vector that is not in the range.

Splitting Fields

Let E be an extension of F and f(x) be in F[x]. Also, let Φ be an automorphism of E leaving every element of F fixed. Prove that Φ must take a root of f(x) lying in E into a root of f(x) in E. See attached file for full problem description.

Excel Solver - maximize the annual passenger-carrying capability.

An airline owns an aging fleet of jet airplanes. It is considering a major purchase of up to 17 new model 7a7 and 7b7 jets. The decision must take into account numerous cost and capability factors, including the following: (1) The airline can finance up to $400 million in purchases; (2) each 7a7 jet will cost $35 million and eac

Poisson Kernel and Harmonic Function

Let P_r(t)=R((1+z)/(1-z)), z=re^it be the Poisson kernel for the unit disc |z|<1. Let U(theta) be a continous function of the interval [0,pi] with U(0)=U(pi)=0. Show that the function u(re^itheta)=1/2pi(integral from 0 to pi of {P_r(t-theta)-P_r(t+theta)}U(t)dt is harmonic in the half-disc {re^itheta,0<=r<1, 0<=theta<=pi} and

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

Rational and Jordan Canonical Forms

Find the rational canonical form and the Jordan canonical form(if possible) for the linear tansformations f: R^3->R^3 given by the matrix 1st row: -8 -10 -1 2nd row: 7 9 1 3rd row: 3 2 0 keywords: matrix

Homomorphism and First Isomorphism Theorem

Let R>0 be the group of positive real numbers under multiplication. Let CX be the group of nonzero complex numbers under mu!tiplication. Let S1 = {a + bi such that a^2 + b^2 = 1) be the subgroup of C consisting of all complex numbers of absolute value 1. Note that is normal in Cx since Cx is abelian. Prove that CX/S1 is isomorph

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.

Hamilton Quaternions : Kernels and Images

Show that the map phi: H->M_2(C) defined by phi(a+bi+cj+dk)=(a+b(sqrt -1),c+d(sqrt-1);-c+d(sqrt-1),a-b(sqrt-1)) is a ring homomorphism. Calculate its kernel and describe its image. Ps. Here H is the ring of integral Hamilton Quaternions and M_2(C) are 2x2 matrices with complex coefficients. notation after the equal sign in

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

Surjective homomorphism proofs

See attached file for full problem description. Prove or disprove : Let and be two surjective homomorphism. Then Ker =Ker if and only if there exists an isomorphism such 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

Affine Map Proofs

Let F be an affine map. Prove that the corresponding linear map is unique. See attached file for full problem description.

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

Vector Spaces, Basis and Quotient Spaces

See the attached file. 1. Let and be vector spaces over and let be a subspace of Show that for all is a subspace of and this subspace is isomorphic to . Deduce that if and are finite dimensional, then dim = (dim - dim )dim 2. Let be a linear operator on a finite dimensional vector space Prove

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 .

Linear Combinations, Basis and Transformations

1. Given a basis B = { u1 = [1, 2], u2 = [2, 1] } for R^2, express u = [7, -2] as a linear combination of u1 and u2. How many ways can you do this? (in this problem...the u1 and u2 should actually be u sub 1 and u sub 2...I couldn't do that notation here....also the u1, u2, and u should all be bold to represent vectors) 2. L

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?

Isomorphisms and automorphisms.

This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what they are. Can you give some simple examples?