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# Linear Transformation

### Fields and homomorphisms

Not sure why, but I am having trouble with this one. I'd really appreciate it if someone is willing to help here. Please see the attached file. Thank you

### Homomorphisms and subgroups

I need a detailed explanation for this problem. Any help is greatly appreciated. Thanks!

### Projection onto a Line

Please solve problem 22 from the attached file.

### Graphically maximize P = 50X1 + 80X2

Graphically maximize P = 50X1 + 80X2 subject to: X1 + 2X2 <= 32 3X1 + 4X2 <= 84 X1, X2 >= 0

### 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

### Standard Inner Product and Linear Operators

Let ( | ) be the standard inner product on C^2. Prove that there is no non-zero linear operator on C^2 such that (alpha|Talpha) = 0 for every alpha in C^2. Generalize.

### Groups: Isomorphisms and Mappings

Let G be the additive group of all real numbers, and G' be the group of all positive real numbers under multiplication. Verify that the mapping theta:G'->G defined by Theta(x)=log x is an isomorphism from G' to G.

### Solve: Onto Homomorphisms

1. Let R* be the set of nonzero real numbers. Then R* is a group with the usual multiplication. a) Let {}: R* --> R* be defined by {}(x) = x^-3, for any x is a member of R*. Show that {} is a group isomorphism. b) Let theta: R* --> R* be defined by theta(x) = 2^x, for any x is a member of R*. Show that theta is a group homom

### 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

### Linear Transformations : Hilbert Space and Inner Product

Please see the attached file for the fully formatted problems. Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product Let P5 V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let , be its natural basis. Define a linear transf

### 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

### Linear Transformations : Coordinate Vectors

Please do part b. Please see the attached file for the fully formatted problems.

### Abelianization of a Group

Let G be a group. An Abelianization of G is a pair (A, f), where A is an Abelian group and f: G --> A is a group homomorphism, which satisfies the following mapping property: given any Abelian group B and any group homomorphism h: G --> B, there is a unique group homomorphism h_a: A --> B such that h_a * f = h. i) Explain wh

### Graph theory: Representing the Linear Transformations

Please see attached file for full problem description. Let C0 ={SUM (i = 1 to p) &#949;ivi&#9474;&#949;i 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) &#949;iei&#9474;&#949;i is an element of F2, ei is an element of E(G)} be the vector space of 1-chains Re

### Center of Mass of a 3D Pyramid

I need three different algebraic solutions of the center of mass of a square-based pyramid. Finding the center of mass of a square-based 3D pyramid is usually done in calculus, but our professor wants us to work it out in algebra. One hint he gave us (for one way of approaching this problem) is to split the pyramid into small bl

### Convex Sets and Linear Transformations

Prove that the following sets are convex, where E stands for the sum symbol: *S = {x E R^n : x = Ay, A E R^(n x m), y E R^m, y >= 0}

### 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

### Economics : Equations, Quantity and Profit

A company manufacturers and sells a product. The estimated demand and cost functions are as below: Demand P = 16000 -2Q(squared) 0< Q < 85 Total cost 1000q = 100000 Where p is unit price (in £'s) q is quantity, tc is total cost (in £'s) A) Find the equation for total revenue hence an equation for profit. B) Find t

### 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

### Prove that the Kernel of a homomorphism is a subspace.

Prove that the Kernel of a homomorphism is a subspace.

### Transformation of Formula

R sin (10t + 0.8977) = 2 Transform for t =

### 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

### Kernel Proof of an Ideal Ring

Show that if @:R -> S is a ring homomorphism, then the ker(@) is an ideal of R and that @ is injective if. and only if, the kernel is (0).

### Linear Mapping Proof

Let phi be a linear mapping from M into N let a be from AnnM and b from AnnN Let c=gcd(a,b). Show that c belongs to AnnIm(phi). See the attached file.

### 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

### Linearity of Transformations; Basis and Dimensions for Four Subspaces

22. Which of these transformations is not linear? The input is v = (v1, v2). (a) T(v) = (v2, v1). (b) T(v) = (V1, v1). (c) T(v) = (0, v1). (d) T(v) = (0, 1). 22. Without elimination, find dimensions and bases for the four subspaces for A = [0 3 3 3 0 0 0 0 0 1 0 1] and B = [1 1 4 4