Definitions

Linear function

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

Analytic geometry

In 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

$f\left(x\right) = mx + b$

(called slope-intercept form), where $m$ and $b$ are real constants and $x$ is a real variable. The constant $m$ is often called the slope or gradient, while $b$ is the y-intercept, which gives the point of intersection between the graph of the function and the $y$-axis. Changing $m$ makes the line steeper or shallower, while changing $b$ moves the line up or down.

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

• $f_\left\{1\right\}\left(x\right) = 2x+1$
• $f_\left\{2\right\}\left(x\right) = x/2+1$
• $f_\left\{3\right\}\left(x\right) = x/2-1.$

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

Vector spaces

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 $x$ and $f\left(x\right)$ are represented as coordinate vectors, then the linear functions are those functions that can be expressed as

$f\left(x\right) = mathrm\left\{M\right\}x$, where M is a matrix.

A function $f\left(x\right) = mx + b$ is a linear map if and only if $b = 0$. For other values of $b$ this falls in the more general class of affine maps.