In mathematics, the term linear function can refer to either of two different but related concepts:
 a firstdegree polynomial function of one variable;
 a map between two vector spaces that preserves vector addition and scalar multiplication.
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Analytic geometry
In analytic geometry, the term linear function is sometimes used to mean a firstdegree 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
(called slopeintercept 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 yintercept, which gives the point of intersection between the graph of the function and the yaxis. 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
where M is a matrix. A function
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
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