Linear Transformation

A linear transformation or linear map is a function between two modules that preserves the operations of module addition and scalar multiplication. As a result, it always maps linear subspaces to linear subspaces. For example it maps straight lines to straight lines or a single point. The expression linear operator is often used to refer to a linear map from a vector space to itself. A linear map is a homomorphism of modules. It is a morphism in the category of modules over a given ring.

For example, let V and W be vector spaces over the same field K. A function f: V→W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, then the following two conditions are satisfied.

F(x + y) = f(x) + f(y)

F(αx) = αf(x)

This is equivalent to requiring the same for any linear combination of vectors.

Occasionaly, V and W can be considered to be vector spaces over different fields. It is necessary in these situations to specify which of these ground fields are being used in the definition of linear. 

Categories within Linear Transformation

Linear Programming

Postings: 1,515

Linear programming is a method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships.

Matrices

Postings: 745

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Quadratic expressions

Give examples of Quadratic relationships in either nature or business, Remember, a quadratic relationship the form y=a*x^2+b*x+c.

Space Constrained Inventories

A grocer has exactly 1,000 square feet available to display and sells 3 kinds of vegetables. The space consumed by each kind of vegetable is proportional to its cost, and tomatoes consume 0.5 square feet per pound. There is a $100 setup cost for replenishing any of the vegetables, and the interest rate is 25% per annum. The 3

Operations Research for Arizona Plumbing and Widgetco

Problem 1 - Arizona Plumbing Arizona Plumbing, which makes, among other products, a full line of bathtubs must decide which of its factories should supply which of its warehouses. Relevant data for Arizona Plumbing are presented in Table 1 and Table 2. Table 1 show, for example, that it costs Arizona Plumbing $5 to ship one

Linear programming for Burger Doodle

This must all be done in Excel. (Very Important) I have also put a link of an example problem for the first part. This how I would like it formatted. Please take a look at it for reference. http://documents.saintleo.edu/docs/AVP/MBA550/MBA550_ch9.mp4 The manager of a Burger Doodle franchise wants to determine how many sa

Formulating a nonlinear program representing profit maximization

A bakery produces muffins and doughnuts. Let x1 be the number of doughnuts produced and x2 be the number of muffins produced. The profit function for the bakery is expressed by the following equation: profit = 4x1 + 2x2 + 0.003x12 + 0.004x22. The bakery has the capacity to produce 800 units of muffins and doughnuts combined and

Linear Programming Using a Demand Function

I am trying to use Linear Programming to solve the attached equation. Accordingly, I used a Demand Function and determined the variable coefficients. I have been receiving comments stating that the problem is unclear and ambiguous. Thus, I tried cleaning it up a bit. The idea is to show the method not a precise answer. So,

Real Life Scenarios: Systems of Linear Equations

Determine a simple real-life example scenario to solve systems of linear equations - Discuss how the math concept applies to your scenario. Scenarios or examples should clearly demonstrate how the math principle can be used to solve a real-life problem. - Research and include references formatted consistent with APA guidel

Formulate the following network problem

(See the attached file for the diagram) - In the above network, a given flow Q is transferred through the network to a demand node N. The goal is to route the flows in such a way that total channel loss is minimized, where channel loss coefficients clij per unit length are given for each link or arc of length Lij. Ignore the

Maximizing Purchases & Minimizing Costs

You need to purchase a combination of three items, item x ($50 ea), item y ($150 ea), and item z ($100 ea). The purchased collection of items must meet the following constraint 5x + 5y + 5z >= 2,500 5x + 10y + 15z >= 3,500 3x - y + 3z <= 0 x, y, and z >=0 (a) How many of each should you purchase to minimize the total cost?

Operations Research: Duals

Q. Write the dual of this problem, sketch the feasible region of the dual, and find the dual solution graphically Minimize -X1 - X2 Subjet to 2X1 + X2 <= 4 X1,X2 >=0

Analytical Models in DSS

A company that assembles electronic alarm systems requires three component parts: C1, C2, and C3. In-house production costs are estimated to be $15 per unit for part C1, $18 per unit for part C2, and $ 20 per unit for part C3. It requires 0.16 hours of machining time and 0.1 hours of finishing time to produce to each unit of pa

Sensitivity Analysis and LP solve

Transportaton Problem (Minimal Cost) There are three warehouses at different cities: Tauranga, Wanganui and Wellington. They have 180, 100 and 150 tons of paper available over the next week respectively. There are four publishers in Auckland, Palmerston North, Hamilton and Wellington. They have ordered 190, 70, 120 and 50 ton

Linear Mapping in Subsets

Question 1. 1) Suppose (V, | * |) is a normed space. If x, y E V and r is a positive real number, show that the open r-balls Br(x) and Br(x + y) in V are homeomorphic. 2) Suppose V and W are two normed spaces. If A : V ---> W is a linear map, then show that it is continuous at every point v E V if and only if it is continuou

LU factorisation

Form an LU factorisation of the following symmetric matrix to show that it is not positive definite. 4 1 -1 2 1 3 -2 -1 -1 -2 1 6 2 -1 6 1 Using little ingenuity we can find a non-zero vector such as x^T = ( 0 1 1 0) that does not satisfy the requirement x^T Ax > 0.

Non-Linear Scatterplot

An experiment is conducted to determine the relationship between initial speed and stopping distance of automobiles. A sample of twelve cars is tested and the following data are recorded: Initial speed in mph (x) 20 20 30 30 40 40 50 50 60 60 70 70 Stopping distance in ft (y) 15.9 24 41.2 58.7 74.8 88.8 112.6 12

Schedule Model

Suppose you are waiting in line to check out at a grocery store and there are 7 other customers in front of you (so you are customer 8). By inspecting the amount of items in their baskets, you estimate the following check-out time in minutes: Customer 1 2 3 4 5 6 7 8 Checkout time 10 5 3 7 5 10 2 5 a) What would

Solver: Maximize Total Return

The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue producing investments together with annual rates of return are as

Calculating Production and Material Consumption Rates

The Ottawa Coat company manufactures winter coats and spring jackets. The winter coat requires 4m of material while the spring jacket requires just 1m. The company wants to minimize the consumption of material. Each winter coat has one zipper while each spring jacket has two. The seamstresses can sew in zippers at a top rate

Math: Linear Transformation of the Space of Polynomials

Which of the following is a linear transformation T of the space of polynomials? Circle the letters corresponding to correct answers. A. T(p(x)) = p(x) + 1. B. T(p(x)) = (x^2 - 1)p(x). C. T(p(x)) = p(0)x^2. D. T(p(x))= p^3(x). E. T(p(x)) = p(x^3).

Linear Transformation of w = iz

State why the transformation w = iz is a rotation in the z plane through the angle pi/2. Then find the image of the infinite strip 0<x<1.

Maximizing Profit: Demand for Steak Dinners

The unit cost of producing a steak dinner at the Smalltown Inn is $6. If a restaurant charges p dollars for a steak dinner, customers will demand 200 - 5p steak dinners per week. To maximize the profit earned on steak dinners, what price should the inn charge? A. $20 B. $21 C. $22 D. $23 E. $25