# Linear function

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Linear map or polynomial function of degree one

Not to be confused with [Linear functional](/source/Linear_functional).

In [mathematics](/source/Mathematics), the term **linear function** refers to two distinct but related notions:[1]

- In [calculus](/source/Calculus) and related areas, a [*linear function*](/source/Linear_function_(calculus)) is a [function](/source/Function_(mathematics)) whose [graph](/source/Graph_of_a_function) is a [straight line](/source/Straight_line), that is, a [polynomial function](/source/Polynomial_function) of [degree](/source/Polynomial_degree) zero (a constant polynomial) or one (a linear polynomial).[2] For distinguishing such a linear function from the other concept, the term *[affine function](/source/Affine_function)* is often used.[3]

- In [linear algebra](/source/Linear_algebra), [mathematical analysis](/source/Mathematical_analysis),[4] and [functional analysis](/source/Functional_analysis), a *[linear function](/source/Linear_function_(linear_algebra))* is a kind of function between [vector spaces](/source/Vector_space).[5]

## As a polynomial function

Main article: [Linear function (calculus)](/source/Linear_function_(calculus))

Graphs of two linear functions.

In calculus, [analytic geometry](/source/Analytic_geometry) and related areas, a linear function is a polynomial of degree one or less, including the [zero polynomial](/source/Zero_polynomial). (The latter is a polynomial with no terms, and it is not considered to have degree zero.)

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

- f ( x ) = a x + b , {\displaystyle f(x)=ax+b,}

where *a* and *b* are [constants](/source/Constant_(mathematics)), often [real numbers](/source/Real_number). The [graph](/source/Graph_of_a_function) of such a function of one variable is a nonvertical line. *a* is frequently referred to as the slope of the line, and *b* as the intercept.

If *a > 0* then the [gradient](/source/Slope) is positive and the graph slopes upwards.

If *a < 0* then the [gradient](/source/Slope) is negative and the graph slopes downwards.

For a function f ( x 1 , … , x k ) {\displaystyle f(x_{1},\ldots ,x_{k})} of any finite number of variables, the general formula is

- f ( x 1 , … , x k ) = b + a 1 x 1 + ⋯ + a k x k , {\displaystyle f(x_{1},\ldots ,x_{k})=b+a_{1}x_{1}+\cdots +a_{k}x_{k},}

and the graph is a [hyperplane](/source/Hyperplane) of dimension *k*.

A [constant function](/source/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.

In this context, a function that is also a linear map (the other meaning of linear functions, see the below) may be referred to as a [homogeneous](/source/Homogeneous_function) linear function or a [linear form](/source/Linear_form). In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued [affine maps](/source/Affine_map).

## As a linear map

Main article: [Linear map](/source/Linear_map)

An [integral](/source/Integral) of an integrable function is a linear map from a vector space of integrable functions to real numbers (that is also a vector space).

In linear algebra, a linear function is a map f {\displaystyle f} from a [vector space](/source/Vector_space) V {\displaystyle \mathbf {V} } to a vector space W {\displaystyle \mathbf {W} } (Both spaces are not necessarily different.) over a same [field](/source/Field_(mathematics)) *K* such that

- f ( x + y ) = f ( x ) + f ( y ) {\displaystyle f(\mathbf {x} +\mathbf {y} )=f(\mathbf {x} )+f(\mathbf {y} )}

- f ( a x ) = a f ( x ) . {\displaystyle f(a\mathbf {x} )=af(\mathbf {x} ).}

Here *a* denotes a constant belonging to the field *K* of [scalars](/source/Scalar_(mathematics)) (for example, the [real numbers](/source/Real_number)), and **x** and **y** are elements of V {\displaystyle \mathbf {V} } , which might be *K* itself. Even if the same symbol + {\displaystyle +} is used, the operation of addition between **x** and **y** (belonging to V {\displaystyle \mathbf {V} } ) is not necessarily same to the operation of addition between f ( x ) {\displaystyle f\left(\mathbf {x} \right)} and f ( y ) {\displaystyle f\left(\mathbf {y} \right)} (belonging to W {\displaystyle \mathbf {W} } ).

In other terms the linear function preserves [vector addition](/source/Vector_addition) and [scalar multiplication](/source/Scalar_multiplication).

Some authors use "linear function" only for linear maps that take values in the scalar field;[6] these are more commonly called [linear forms](/source/Linear_form).

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

## See also

- [Homogeneous function](/source/Homogeneous_function)

- [Nonlinear system](/source/Nonlinear_system)

- [Piecewise linear function](/source/Piecewise_linear_function)

- [Linear approximation](/source/Linear_approximation)

- [Linear interpolation](/source/Linear_interpolation)

- [Discontinuous linear map](/source/Discontinuous_linear_map)

- [Linear least squares](/source/Linear_least_squares)

## Notes

1. **[^](#cite_ref-1)** "The term *linear function* means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1

1. **[^](#cite_ref-2)** Stewart 2012, p. 23

1. **[^](#cite_ref-3)** A. Kurosh (1975). *Higher Algebra*. Mir Publishers. p. 214.

1. **[^](#cite_ref-4)** T. M. Apostol (1981). *Mathematical Analysis*. Addison-Wesley. p. 345.

1. **[^](#cite_ref-5)** Shores 2007, p. 71

1. **[^](#cite_ref-6)** Gelfand 1961

## References

- Izrail Moiseevich Gelfand (1961), *Lectures on Linear Algebra*, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. [ISBN](/source/ISBN_(identifier)) [0-486-66082-6](https://en.wikipedia.org/wiki/Special:BookSources/0-486-66082-6)

- Shores, Thomas S. (2007). *Applied Linear Algebra and Matrix Analysis*. [Undergraduate Texts in Mathematics](/source/Undergraduate_Texts_in_Mathematics). Springer. [ISBN](/source/ISBN_(identifier)) [978-0-387-33195-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-33195-9).

- Stewart, James (2012). *Calculus: Early Transcendentals* (7E ed.). Brooks/Cole. [ISBN](/source/ISBN_(identifier)) [978-0-538-49790-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-538-49790-9).

- Leonid N. Vaserstein (2006), "Linear Programming", in [Leslie Hogben](/source/Leslie_Hogben), ed., *Handbook of Linear Algebra*, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. [ISBN](/source/ISBN_(identifier)) [1-584-88510-6](https://en.wikipedia.org/wiki/Special:BookSources/1-584-88510-6)

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Adapted from the Wikipedia article [Linear function](https://en.wikipedia.org/wiki/Linear_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Linear_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.