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 at most one.
- In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map. In this case, and in case of possible ambiguity, the name affine function is often used for the concept above.
As a polynomial function
When the function is of only one variable, it is of the form
For a function of any finite number of independent variables, the general formula is
and the graph is a hyperplane of dimension k.
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 (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
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 (and only when) , or, equivalently, when the constant . Geometrically, the graph of the function must pass through the origin.
- "The term linear function means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1
- Stewart 2012, p. 23
- T. M. Apostol (1981). Mathematical Analysis. Addison-Wesley. p. 345.
- Shores 2007, p. 71
- A. Kurosh (1975). Higher Algebra. Mir Publishers. p. 214.
- Gelfand 1961
- 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