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.
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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
(called slope-intercept form), where 
and 
are real constants and 
is a real variable. The constant is often called the slope or gradient, while is the y-intercept, which gives the point of intersection between the graph of the function and the 
-axis. Changing makes the line steeper or shallower, while changing moves the line up or down.
Examples of functions whose graph is a line include the following:
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 and 
are represented as coordinate vectors, then the linear functions are those functions 
that can be expressed as
where M is a matrix. A function
is a linear map if and only if = 0. For other values of this falls in the more general class of affine maps.
See also
External links
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