Linear function

In mathematics, the term linear function can refer to either of two different but related concepts:
 * a first-degree polynomial function of one variable;
 * a map between two vector spaces that preserves vector addition and scalar multiplication.

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 known as "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


 * $$f(x) = mx + b$$
 * $$(y-y_1) = m(x-x_1)$$
 * $$0= Ax + By + C$$

(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:


 * $$f_{1}(x) = 2x+1$$
 * $$f_{2}(x) = x/2+1$$
 * $$f_{3}(x) = x/2-1.$$

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

Vector spaces
In advanced mathematics, a linear function 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 $$f(x)$$ are represented as coordinate vectors, then the linear functions are those functions $$f$$ that can be expressed as


 * $$f(x) = \mathrm{M}x,$$

where M is a matrix. A function


 * $$f(x) = mx + b$$

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.