# Linear function

In mathematics, the term linear function refers to two different, although related, notions:[1]

## As a kind of polynomial function

In calculus, analytic geometry, and related areas, a linear function is defined by a polynomial of degree zero or one. When there is only one independent variable, these functions are of the form

$f(x)=ax+b,$

where a and b are constants, often real numbers. The graph of such a function of one independent variable is a nonvertical line.

For a function f(x1, ..., xk) of two or more independent variables, the general formula is

f(x1, ..., xk) = b + a1x1 + ... + akxk,

and the the graph is a hyperplane.

A constant function is also considered linear in this context, as it is given by a polynomial of degree zero. Its graph, when there is only one independent variable, is a horizontal line.

In this context, the other meaning (a linear map) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, this meaning (polynomial functions of degree 0 or 1) is a special kind of affine map.

## As a linear map

In linear algebra, a linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:

$f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y})$
$f(a\mathbf{x}) = af(\mathbf{x}).$

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.

Some authors use "linear function" only for linear maps that take values in the scalar field;[4] these are also called linear functionals.

## Notes

1. ^ "The term linear function, which is not used here, means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1
2. ^ Stewart 2012, p. 23
3. ^ Shores 2007, p. 71
4. ^ Gelfand 1961

## References

• Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6
• Thomas S. Shores (2007), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer. ISBN 0-387-33195-6
• James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9
• Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. ISBN 1-584-88510-6