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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: 1,394

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.


Postings: 691

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

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Linear programming for Burger Doodle

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Real Life Scenarios: Systems of Linear Equations

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Analytical Models in DSS

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Sensitivity Analysis and LP Solve

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Network Flow using Nodes and Links

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Non-Linear Scatterplot

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Schedule Model

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Correspondence theorem

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Kernel of Phi and Homomorphism

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Finite Abelian Group

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Dirichlet Kernel

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Model Formulation (Minimization)

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linear fractional transformation

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Determining Waiting Times

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network flow 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 Implies Functions

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Modelling Problem Using Excel

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Minimizing travel relocation

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