Linear function

In mathematics, the term linear function refers to two distinct but related notions:

In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing such a linear function from the other concept, the term affine function is often used.
In linear algebra, mathematical analysis, and functional analysis, a linear function is a kind of function between vector spaces.

As a polynomial function

thumb|Graphs of two linear functions.

In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial. (The latter is a polynomial with no terms, and it is not considered to have degree zero.)

When the function is of only one variable, it is of the form

:<math>f(x)=ax+b,</math>

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.

If a > 0 then the gradient is positive and the graph slopes upwards.

If a < 0 then the gradient is negative and the graph slopes downwards.

For a function <math>f(x_1, \ldots, x_k)</math> of any finite number of variables, the general formula is

:<math>f(x_1, \ldots, x_k) = b + a_1 x_1 + \cdots + a_k x_k ,</math>

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 variable, is a horizontal line.

In this context, a function that is also a linear map (the other meaning of linear functions, see the below) 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.

As a linear map

thumb|An [[integral of an integrable function is a linear map from a vector space of integrable functions to real numbers (that is also a vector space).]]

In linear algebra, a linear function is a map <math>f</math> from a vector space <math>\mathbf{V}</math> to a vector space <math>\mathbf{W}</math> (Both spaces are not necessarily different.) over a same field such that

:<math>f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) </math>

:<math>f(a\mathbf{x}) = af(\mathbf{x}). </math>

Here denotes a constant belonging to the field of scalars (for example, the real numbers), and and are elements of <math>\mathbf{V}</math>, which might be itself. Even if the same symbol <math>+</math> is used, the operation of addition between and (belonging to <math>\mathbf{V}</math>) is not necessarily same to the operation of addition between <math>f\left( \mathbf{x} \right)</math> and <math>f\left( \mathbf{y} \right)</math> (belonging to <math>\mathbf{W}</math>).

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; these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) , or, equivalently, when the constant equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

Linear function

As a polynomial function

As a linear map

See also

Notes