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

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 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.

See also

• Nonlinear system
• Piecewise linear function
• Linear interpolation
• Discontinuous linear map

External links

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