In mathematics, the term linear function refers to two distinct but related notions:^[1]

In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero).

where a and b are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept.

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 variable, is a horizontal line.

In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.

In linear algebra, a linear function is a map f between two vector spaces s.t.

Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be K itself.

In other terms the linear function preserves vector addition and scalar multiplication.

Some authors use "linear function" only for linear maps that take values in the scalar field;^[6] these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) f(0, ..., 0) = 0, or, equivalently, when the above constant b equals zero. Geometrically, the graph of the function must pass through the origin.