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

Analytic geometry

Three geometric linear functions — the red and blue ones have the same slope (m), while the red and green ones have the same y-intercept (b).

Main article: linear equation

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(x) = 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:

• f₁(x) = 2x + 1
• f₂(x) = x / 2 + 1
• f₃(x) = 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(x) are represented as coordinate vectors, then the linear functions are those functions that can be expressed as

f(x) = Mx, 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.

See also

• Function (mathematics)

External links

• Linear Functions on Id Mind
• Interactive tool to explore linear functions