In mathematics, the term linear function refers to two distinct but related notions:
In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial.
When the function is of only one variable, it is of the form
where and are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. is frequently referred to as the slope of the line, and as the intercept.
For a function of any finite number of independent variables, the general formula is
and the graph is a hyperplane of dimension.
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one independent variable, is a horizontal line.
In this context, the other meaning may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, this meaning is a special kind of affine map.
As a linear mapIn linear algebra, a linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:
Here denotes a constant belonging to some field of scalars and and are elements of a vector space, which might be itself.
Some authors use "linear function" only for linear maps that take values in the scalar field; these are also called linear functionals.
The "linear functions" of calculus qualify as "linear maps" when , or, equivalently, when the constant. Geometrically, the graph of the function must pass through the origin.