# 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.

Last updated on Aug 22, 2013

#### Categories within Linear Transformation:

Linear Programming

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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

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

Linear Programs and Inequalities

I'm looking to see if someone could explain this answer - please see the attached spreadsheet that goes along with the below problem. One way to solve a linear program is to graph the inequalities to find the feasible region. Next, find the value of the objective function at each of the corner points (i.e., enumerating the c

Linear Algebra: Gaussian Elimination

Finding Solutions for Inequalities

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Formulate the following network problem

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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

Setup a network Flow for this Problem using nodes and links but do not solve for optimal solution

A manufacturer must produce a certain product in sufficient quantity to meet contracted sales in the next four months. The production facilities available for this product are limited, but by different amounts in the respective months. The unit cost of production also varies according to the facilities and personnel available. T

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.

Linear Transformations Multiple Choice

Determine which of the functions in the attached document are linear transformations.

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

Solve the following LP problem graphically using Excel

Solve the following LP problem graphically using Excel with the computations in the cells: Minimize cost = 24X + 15Y 7X + 11Y ? 77 16X + 4Y ? 80 X,Y ? 0

Prove a relationship between the vector spaces involved in the composition of an injective and a surjective linear map.

Let U, V, and W be vector spaces over a field F. Suppose that T : U --> V and S : V --> W are linear transformations and that Im(T) = Ker(S). If T is injective and S is surjective, prove that dim(V) = dim(U) + dim(W). Here, dim denotes the dimension of a vector space over the field F.

Linear Fractional Transformation is a Mapping f: C --> C

A fixed point of a transformation is a point z_0 so that f(z_0) = z_0. Show that every linear fractional transformation which is not the identity has at most two fixed points.

Automorphism

Please see the attachment. Only part (d).

Isomorphisms