Linear function

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

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 whose graph is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial).<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 kind of function between vector spaces.<ref>Shores 2007, p. 71</ref>

== As a polynomial function == {{main article|Linear function (calculus)}} 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 is a polynomial with no terms, and it is not considered to have degree zero.)

When the function is of only one variable, it is of the form :<math>f(x)=ax+b,</math> where {{mvar|''a''}} and {{mvar|''b''}} are constants, often real numbers. The 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 gradient is positive and the graph slopes upwards.

If ''a < 0'' then the gradient is negative and the graph slopes downwards.

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

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 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 maps.

== As a linear map == {{main article|Linear map}} [[File:Integral as region under curve.svg|thumb|An 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 <math>f</math> from a vector space <math>\mathbf{V}</math> to a vector space <math>\mathbf{W}</math> (Both spaces are not necessarily different.) over a same field {{math|''K''}} such that :<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 the field {{math|''K''}} of scalars (for example, the real numbers), and {{math|'''x'''}} and {{math|'''y'''}} are elements of <math>\mathbf{V}</math>, which might be {{math|''K''}} itself. Even if the same symbol <math>+</math> is used, the operation of addition between {{math|'''x'''}} and {{math|'''y'''}} (belonging to <math>\mathbf{V}</math>) is not necessarily same to the operation of addition between <math>f\left( \mathbf{x} \right)</math> and <math>f\left( \mathbf{y} \right)</math> (belonging to <math>\mathbf{W}</math>).

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

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 forms.

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.

== Notes == <references/>

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

Category:Polynomial functions