Linear function

In mathematics, the term linear function can refer to either of two different but related concepts.

Usage in elementary mathematics

Main article: linear equation

In elementary algebra and analytic geometry, the term linear function is sometimes used to mean a first degree polynomial function of one variable. These functions are called "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.

Such a function can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = mx + b

(called slope-intercept form), where Failed to parse (Missing texvc executable; please see math/README to configure.): m

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
are real constants and Failed to parse (Missing texvc executable; please see math/README to configure.): x
is a real variable. The constant Failed to parse (Missing texvc executable; please see math/README to configure.): m
is often called the slope while Failed to parse (Missing texvc executable; please see math/README to configure.): b
is the y-intercept, which gives the point of intersection between the graph of the function and the Failed to parse (Missing texvc executable; please see math/README to configure.): y

-axis. Changing Failed to parse (Missing texvc executable; please see math/README to configure.): m

makes the line steeper or shallower, while changing Failed to parse (Missing texvc executable; please see math/README to configure.): b
moves the line up or down.

Examples of functions whose graph is a line include the following:

• Failed to parse (Missing texvc executable; please see math/README to configure.): f_{1}(x) = 2x+1

• Failed to parse (Missing texvc executable; please see math/README to configure.): f_{2}(x) = x/2+1

• Failed to parse (Missing texvc executable; please see math/README to configure.): f_{3}(x) = x/2-1

The graphs of these are shown in the image at right.

Usage in advanced mathematics

In advanced mathematics, a linear function often means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.

For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): x

and Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
are represented as coordinate vectors, then the linear functions are those functions that can be expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \mathrm{M}x

, where M is a matrix.

A function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = mx + b

is a linear map if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): b = 0

. For other values of Failed to parse (Missing texvc executable; please see math/README to configure.): b

this falls in the more general class of affine maps.

External links

• Linear Functions on Id Mindals:Lineare Funktion

bg:Линейна функция ca:Funció lineal de:Lineare Funktion es:Función lineal fr:Fonction linéaire it:funzione lineare he:פונקציה לינארית ms:Fungsi lelurus nl:Lineaire functie ja:一次関数 pl:Funkcja liniowa pt:Função linear pt:Função Polinomial#Função do Primeiro Grau ru:Линейная функция sl:Linearna funkcija sr:Линеарна функција fi:Lineaarinen funktio