Linear

:See also linearity (computer and video games)

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The word linear comes from the Latin word linearis, which means created by lines.

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In mathematics, a linear function f(x) is one which satisfies the following two properties (but see below for a slightly different usage of the term):

Related Topics:
Mathematics - Function

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• Additivity property (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group isomorphism with respect to addition.
• Homogeneity property: f(αx) = αf(x) for all α.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space.

Related Topics:
Real number - Vector space

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The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up.

Related Topics:
Linear operator - Derivative - Differential operator - Del - Laplacian - Differential equation

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Nonlinear equations and functions are of interest to physicists and mathematicians because they are hard to solve and give rise to interesting phenomena such as chaos.

Related Topics:
Nonlinear - Physicist - Mathematician - Chaos

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Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.

Related Topics:
Linear algebra - Linear transformation

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See also: linear element, linear system, nonlinearity.

Related Topics:
Linear element - Linear system - Nonlinearity

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