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.