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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: 1,412

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: 706

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

Space Constrained Inventories

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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

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Real Life Scenarios: Systems of Linear Equations

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Formulate the following network problem

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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

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Linear Mapping in Subsets

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Network Flow using Nodes and Links

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Non-Linear Scatterplot

See the attached file. 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

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

Symmetries of a square

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Correspondence theorem

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Problems in Galois Theory

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Homomorphism

Let phi is a homomorphism from Z30 onto a group order 3. Determine the kernel of phi. Find all generators of the kernel of phi.

Abstract Algebra

Suppose that G is a finite Abelian group and G has no element of order 2. Show that the mapping g-->g^2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism (hint: consider Z)

Dirichlet Kernel

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Model Formulation (Minimization)

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linear fractional transformation

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Determining Waiting Times

Question: McBurger's fast-food restaurant has a drive-through window with a single window with a single server who takes orders from an intercom and also is the cashier. The window operator is assisted by other employees who prepare the orders. Customers arrive at the ordering station prior to the drive-through window every 4.5

Transportation Problems

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Minimize the risk of the investment

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Graphical Minimization Method

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Schur's Lemma

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

Module homomorphism proof

Let &#966;: Z "circle +" Z &#8594; Z "circle +" Z be a module homomorphism (of Z-modules). Show that if &#966; is surjective, it must be injective. Give an example to show that the converse is false &#9472; a difference between free Z-modules and vector spaces. (You may, of course, think of &#966; as a 2 x 2 matrix with integer

Consider the linear transformation T : complex numbers^n -> complex numbers^n given by T( z1, z2, ... , zn ) = ( a1z1, a2z2, ... , anzn). What is the dimension of the subspace spanned by the eigenvectors of T? Exhibit a basis for this space, and give the eigenvalues.

Consider the linear transformation T : complex numbers^n -> complex numbers^n given by T( z1, z2, ... , zn ) = ( a1z1, a2z2, ... , anzn). What is the dimension of the subspace spanned by the eigenvectors of T? Exhibit a basis for this space, and give the eigenvalues.