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{{short description|Linear map or polynomial function of degree one}}

In [[mathematics]], the term '''linear function''' refers to a function that satisfies the following two properties:

{{for|the use of the term in calculus|Linear function (calculus)}}

In [[mathematics]], the term '''linear function''' refers to two distinct but related notions:<ref>"The term ''linear function'' means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1</ref>

* In [[calculus]] and related areas, a linear function is a [[function (mathematics)|function]] whose [[graph of a function|graph]] is a [[straight line]], that is, a [[polynomial function]] of [[polynomial degree|degree]] zero or one.<ref>Stewart 2012, p. 23</ref> For distinguishing such a linear function from the other concept, the term ''[[affine function]]'' is often used.<ref>{{cite book|author=A. Kurosh|title=Higher Algebra|year=1975|publisher=Mir Publishers|page=214}}</ref>

* In [[linear algebra]], [[mathematical analysis]],<ref>{{cite book|author=T. M. Apostol|title=Mathematical Analysis|year=1981|publisher=Addison-Wesley|page=345}}</ref> and [[functional analysis]], a linear function is a [[linear map]].<ref>Shores 2007, p. 71</ref>

== As a polynomial function ==

<math>f(x + y) = f(x) + f(y) </math>

{{main article|Linear function (calculus)}}

[[File:Linear Function Graph.svg|thumb|Graphs of two linear functions.]]

In calculus, [[analytic geometry]] and related areas, a linear function is a polynomial of degree one or less, including the [[zero polynomial]] (the latter not being considered to have degree zero).

<math>f(a*x) = a*f(x) </math>

When the function is of only one [[variable (mathematics)|variable]], it is of the form

The linear functions are often confused with [[affine functions| affine function]]. Affine functions can be written as <math>f(x) = mx + b</math>. Although affine functions make lines when graphed, they do not satisfy the properties of linearity.

:<math>f(x)=ax+b,</math>

where {{mvar|''a''}} and {{mvar|''b''}} are [[constant (mathematics)|constant]]s, often [[real number]]s. The [[graph of a function|graph]] of such a function of one variable is a nonvertical line. {{mvar|''a''}} is frequently referred to as the slope of the line, and {{mvar|''b''}} as the intercept.

If ''a > 0'' then the [[Slope|gradient]] is positive and the graph slopes upwards.

==Vector spaces==

If ''a < 0'' then the [[Slope|gradient]] is negative and the graph slopes downwards.

In advanced mathematics, a ''linear function'' means a [[function (mathematics)|function]] that is a [[linear map]], that is, a map between two [[vector space]]s that preserves vector addition and [[scalar multiplication]].

For a function <math>f(x_1, \ldots, x_k)</math> of any finite number of variables, the general formula is

For example, if <math>x</math> and <math>f(x)</math> are represented as [[coordinate vector]]s, then the linear functions are those functions <math>f</math> that can be expressed as

:<math>f(x_1, \ldots, x_k) = b + a_1 x_1 + \cdots + a_k x_k ,</math>

and the graph is a [[hyperplane]] of dimension {{nowrap|''k''}}.

A [[constant function]] is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.

:<math>f(x) = \mathrm{M}x,</math>

In this context, a function that is also a linear map (the other meaning) may be referred to as a [[homogeneous function|homogeneous]] linear function or a [[linear form]]. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued [[affine map]]s.

where M is a [[matrix (mathematics)|matrix]]. A function

== As a linear map ==

:<math>f(x) = mx + b</math>

{{main article|Linear map}}

[[File:Integral as region under curve.svg|thumb|The [[integral]] of a function is a linear map from the vector space of integrable functions to the real numbers.]]

In linear algebra, a linear function is a map ''f'' between two [[vector space]]s such that

is a linear map if and only if <math>b</math> = 0. For other values of <math>b</math> this falls in the more general class of [[affine map]]s.

:<math>f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) </math>

:<math>f(a\mathbf{x}) = af(\mathbf{x}). </math>

Here {{math|''a''}} denotes a constant belonging to some [[field (mathematics)|field]] {{math|''K''}} of [[Scalar (mathematics)|scalar]]s (for example, the [[real number]]s) and {{math|'''x'''}} and {{math|'''y'''}} are elements of a [[vector space]], which might be {{math|''K''}} itself.

In other terms the linear function preserves [[vector addition]] and [[scalar multiplication]].

Linear functions form the basis of [[linear algebra]].

Some authors use "linear function" only for linear maps that take values in the scalar field;<ref>Gelfand 1961</ref> these are more commonly called [[linear form]]s.

==See also==

The "linear functions" of calculus qualify as "linear maps" when (and only when) {{math|1=''f''(0, ..., 0) = 0}}, or, equivalently, when the constant {{mvar|b}} equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

== See also ==

* [[Homogeneous function]]

* [[Nonlinear system]]

* [[Piecewise linear function]]

* [[Linear approximation]]

* [[Linear interpolation]]

* [[Discontinuous linear map]]

* [[Linear least squares]]

== Notes ==

<references/>

==External linksReferences ==

* Izrail Moiseevich Gelfand (1961), ''Lectures on Linear Algebra'', Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. {{isbn|0-486-66082-6}}

*[http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/linearFunctions.html Linear Functions on Id Mind]

* {{cite book

*[http://www.mathopenref.com/linearexplorer.html Interactive tool to explore linear functions]

| first = Thomas S.

{{Polynomials}}

| last = Shores

| title = Applied Linear Algebra and Matrix Analysis

| publisher = Springer

| year = 2007

| series = [[Undergraduate Texts in Mathematics]]

| isbn = 978-0-387-33195-9

}}

* {{cite book

| first = James

| last = Stewart

| title = Calculus: Early Transcendentals

| publisher = Brooks/Cole

| year = 2012

| edition = 7E

| isbn = 978-0-538-49790-9

}}

* Leonid N. Vaserstein (2006), "Linear Programming", in [[Leslie Hogben]], ed., ''Handbook of Linear Algebra'', Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. {{isbn|1-584-88510-6}}

{{Calculus topics}}

[[Category:Polynomials]]

[[Category:Polynomial functions]]

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