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Created page with 'File:lighterstill.jpgright|frame The word '''linear''' comes from the Latin word linearis, which means created by lines. I...'
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The [[word]] '''linear''' comes from the [[Latin]] [[word]] linearis, which means created by lines. In [[mathematics]], a linear map or [[function]] f(x) is a function which satisfies the following two properties...
* Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
* Homogeneity of degree 1: f(αx) = αf(x) for all α. It turns out that homogeneity follows from the additivity property in all cases where α is [[rational]]. ([[proof]]) In that case, provided that the function is continuous, it becomes useless to establish the condition of homogeneity as an additional axiom.
In this definition, x is not necessarily a real [[number]], but can in general be a member of any [[vector]] [[space]]. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.
The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a [http://en.wikipedia.org/wiki/Differential_operator differential operator]], and many constructed from it, such as [http://en.wikipedia.org/wiki/Del del] and the [http://en.wikipedia.org/wiki/Laplacian Laplacian]. When a [http://en.wikipedia.org/wiki/Differential_equation 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.
[http://en.wikipedia.org/wiki/Linear_algebra Linear algebra] is the branch of [[mathematics]] concerned with the [[study]] of [[vectors]], vector spaces (also called linear spaces), linear transformations (also called linear maps), and [[systems]] of linear equations.
[http://en.wikipedia.org/wiki/Nonlinear Nonlinear equations] and [[functions]] are of interest to [[physicists]] and mathematicians because they can be used to represent many natural [[phenomena]], including [[chaos]].[http://en.wikipedia.org/wiki/Linear]
[[Category: Mathematics]]
[[Category: Philosophy]]
The [[word]] '''linear''' comes from the [[Latin]] [[word]] linearis, which means created by lines. In [[mathematics]], a linear map or [[function]] f(x) is a function which satisfies the following two properties...
* Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
* Homogeneity of degree 1: f(αx) = αf(x) for all α. It turns out that homogeneity follows from the additivity property in all cases where α is [[rational]]. ([[proof]]) In that case, provided that the function is continuous, it becomes useless to establish the condition of homogeneity as an additional axiom.
In this definition, x is not necessarily a real [[number]], but can in general be a member of any [[vector]] [[space]]. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.
The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a [http://en.wikipedia.org/wiki/Differential_operator differential operator]], and many constructed from it, such as [http://en.wikipedia.org/wiki/Del del] and the [http://en.wikipedia.org/wiki/Laplacian Laplacian]. When a [http://en.wikipedia.org/wiki/Differential_equation 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.
[http://en.wikipedia.org/wiki/Linear_algebra Linear algebra] is the branch of [[mathematics]] concerned with the [[study]] of [[vectors]], vector spaces (also called linear spaces), linear transformations (also called linear maps), and [[systems]] of linear equations.
[http://en.wikipedia.org/wiki/Nonlinear Nonlinear equations] and [[functions]] are of interest to [[physicists]] and mathematicians because they can be used to represent many natural [[phenomena]], including [[chaos]].[http://en.wikipedia.org/wiki/Linear]
[[Category: Mathematics]]
[[Category: Philosophy]]