Linear function

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

• In calculus and related areas, a linear function is a polynomial function of degree zero or one.
• In linear algebra, a linear function is a linear map.

As a kind of polynomial function

Main article: Linear function (calculus)

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, ..., x_k) of two or more independent variables, the general formula is

f(x₁, ..., x_k) = b + a₁x₁ + ... + a_kx_k,

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

Main article: 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; these are also called linear functionals.

See also

• Homogenous function
• Nonlinear system
• Piecewise linear function
• Linear interpolation
• Discontinuous linear map

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