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Linear Transformation

Understanding Kernel

Please help me to understand the attached homework problem on finding the kernel. Find the Kernel Consider the groups of nonzero complex numbers, *, and positive real number +, both with multiplication. Let f: *  + be defined by f(z)= |z|. Prove that f is a homomorphism and find

Category Theory - Morphisms, Uniqueness & Equivalence

If f:A-->B is an equivalence in a category C and g:B-->A is a morphism such that gf=1_A (the identity on A) and fg=1_B, show that g is unique. Prove that any two universal (initial) objects in a category C are equivalent. Prove that any two couniversal (terminal) objects in a category C are equivalent. In the category o

Schur's Lemma Implies Functions

I have some trouble understanding the solution to the attached problem (solution included). Could you please provide some clarification of the solution. I have indicated what my points of concern are. Show that if M1 and M2 are irreducible R modules, then any nonzero R-module homomorphism from M1 to M2 is an isomorphism. De

Universal pair proofs

Definition: A pair (U, epsilon) is universal for a group G, with respect to abelian homomorphic images, if U is an abelian group, epsilon:G --> U, an epimorphism, such that for any other abelian group A and surjection f:G --> A, there exists a unique g:U --> A such that f=g composed with epsilon. In this case, we say f can be

Rigorous Subgroup Proofs

I need to see rigorous proofs of these propositions. 1.) Suppose [G:H] is finite. Show that there is a normal subgroup K of G with K, a subgroup of H, such that [G:K] is finite. 2.) Suppose H is a subgroup of S_n but H is not a subgroup of A_n. Show that [H:A_n intersect H]=2. 3.) Prove that if H,K are

Linear transformation

Please show work. Consider R^p with the standard inner product....Prove that the orthogonal projection...is a linear transformation...Is the transformation one-to-one (injective)? Is the transformation onto (surjective)? Justify your answers.

Automorphism proof

Please show all work. Thanks See attached Task: Prove the following theorem: is an automorphism of under componentwise addition. Additional Notes on the Task: Be careful not to confuse the homomorphism's action as a function with the group operation's action as componentwise addition. is componentwise ad

Modelling Problem Using Excel

A company that makes bikes wants to maximize profit over the next five months. Materials for each bike costs $600. Both humans and machines are needed to produce each bike. Each human can work on up to 100 bikes per month. Each machine can work up to 200 bikes per month. TetraCon, Inc. has 4 humans and no m

Minimizing travel relocation

A quarry consists of 8 quarry pits along a 4000 yard north-south section of land. Each pit is 500 yards away from the other. Rock must be removed from some quarry pits and relocated to others quarry pits as listed below. Quarry 1: 0 to 500 yards - 7,000 tons of rock are needed Quarry 2: 500 to 1000 yards - 3,000 to

Converting temperature between Celsius and Fahrenheit

Temperature can be given in two ways: Celsius and Fahrenheit. Using the Library, web resources, and other course materials, find the equations that show the relationship between Celsius and Fahrenheit. Give both equations (one that solves for F and one that solves for C) and explain why each can be beneficial to use. The re

Maximizing the Volume of a Rectangular Box

You are planning to make an open rectangular box from a 12-by 14-cm piece of cardboard by cutting congruent squares from the corners and folding up the sides. a) What are the dimensions of the box of larges volume you can make this way? b) What is its volume?

Transformation matrices

See attached file. Consider the relationship between the matrices A and B, explore the relationship between A and B and matrices representing other linear transformations derived from F and G. Explore 2 examples for example F+G for various  and , or the composite function FG/ Check th

Linear Transformation: Minimization

Boise, Idaho is about 300 miles inland from the nearest point on the Pacific coast; San Diego is about 1000 miles south of that point down the coast. Assuming the coast is a straight line going north-south, C is the point along the coast directly west of Boise. It costs 2 cents per mile to transport a ton of potatoes by truck an

Maximizing the Revenue for an Airline Company

An airline has a new airplane that will be fitted out for a combination of first and second class passengers. A first class seat will cost $120 on a certain one-way trip and a second class seat $80. The seating capacity of the plane is 200 second class seats. A first class seat takes 1.2 times the floor area of a second-class se

Linear Programming Model for Maximizing Profit of a Production Schedule

Ms. Olsen, a coffee processor, markets three blends of coffee. They are Brand X, Minim and Taster's Reject. Ms. Olsen uses two types of coffee beans, Columbian and Mexican, in her coffee. The following chart lists the compositions of the blends. Blend Columbian Beans Mexican Beans Brand X 80% 20% Minim 50

Maximizing Revenue: Example Problem

The manager of a bicycle shop has found that, at a price (in dollars) of p(x) = 150 - x/4 per bicycle, x bicycles will be sold. a. Find an expression for the total revenue from the sale of x bicycles.(revenue = demand x price) b. Find the number of bicycle sales that leads to maximum revenue. c. Find the maximum revenue

Linear Transformations, Hilbert Space, Inner Product and Matrix Adjoint

Please see the attached file for the fully formatted problems. Find a 5x5 matrix M>0 such that if and x(t)= then Can we use this definition to find the adjoint of T (T is given at the end)? This part is the additional information to solve the question above; Let V=L2[-1,1] be the Hilbert space of func

Transformation of State Variables and State-Space Representations

Consider the system . It can be shown that there is other state equation possible for this system. Let us define a new set of state-variables z1 , z2 , z3 by the transformation: Obtain the state-space representation of the system in terms of the new variables. Please see the attached file for the fully formatted problem

Graph theory: Representing the Linear Transformations

Please see attached file for full problem description. Let C0 ={SUM (i = 1 to p) εivi│εi is an element of F2, vi is an element of V(G)} be the vector space of 0-chains and Let C1 ={SUM (i = 1 to q) εiei│εi is an element of F2, ei is an element of E(G)} be the vector space of 1-chains Re

Linear Spaces, Mappings and Dimensional Spaces

1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X. 2) Let U and V be finite dimensional linear spaces and S belong to L(V,W), T belong to L(U,V). Show that the dimension of the null space of ST is less than or equal to the sum of the dimensions of the null spaces of S and T. 3)

Linear Mappings, Differentiation and Linear Spaces

Please help with the following problems. Provide step by step calculations for each. 1) Show that this mapping is linear: T: P5 -> P8 defined as Tp(t)=p(t+1)-p(t)+integral(t-1 to t) s^2 p(s) ds 2) Prove the following is true, or give a counterexample: If l is a nonzero scalar linear function on linear space X (which may

Maximizing the Volume of an Open Top Box..

A square sheet of cardboard 24 inches on a side is made into a box by cutting squares of equal size from each corner of the sheet and folding the projecting flaps into an open-top box. What should be the length of the edge of any of the cutout squares to give the box maximum volume? 4 inches 4.5 inch

Solve: Inventory Management

1. Cooper Automotive Products manufactures components used in the automotive industry. The company purchases parts for use in its manufacturing operation from a variety of different suppliers. One supplier provides a part where the assumptions of the EOQ model are realistic. The annual demand is 5000 units, the ordering cost i

The Decision Variables Explanation

I need a physical explanation as to why X37 = 400, X78 = 0, and X48 = 400. I need for these computer generated numbers to make sense and be able to justify them , if they are justifiable. these problems come from Management Science (Anderson,Sweeney, and Williams) 11th edition on page 327-330 figure 7.13, The management scientis

Simulation-Assigning Random Numbers

A researcher wants to simulate sunny and rainy days in her town for a 3-week period. What is the minimum number of digits the student must obtain from a random number table for each observation if it rained on two-fifths of the days over the past several years at this time of the year? Assume that days can be classified historic

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