Linear function

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Linear map or polynomial function of degree one

title: "Linear function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polynomial-functions"] description: "Linear map or polynomial function of degree one" topic_path: "general/polynomial-functions" source: "https://en.wikipedia.org/wiki/Linear_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Linear map or polynomial function of degree one ::

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

As a polynomial function

Main article: Linear function (calculus)

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/0e/Linear_Function_Graph.svg" caption="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 :f(x)=ax+b, 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.

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

If *a

For a function f(x_1, \ldots, x_k) of any finite number of variables, the general formula is :f(x_1, \ldots, x_k) = b + a_1 x_1 + \cdots + a_k x_k , 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 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

Main article: Linear map

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f2/Integral_as_region_under_curve.svg" caption="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 f from a vector space \mathbf{V} to a vector space \mathbf{W} (Both spaces are not necessarily different.) over a same field K such that :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 the field K of scalars (for example, the real numbers), and x and y are elements of \mathbf{V}, which might be K itself. Even if the same symbol + is used, the operation of addition between x and y (belonging to \mathbf{V}) is not necessarily same to the operation of addition between f\left( \mathbf{x} \right) and f\left( \mathbf{y} \right) (belonging to \mathbf{W}).

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 b equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

Notes

References

  • Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989.
  • {{cite book | first = Thomas S. | last = Shores | title = Applied Linear Algebra and Matrix Analysis | publisher = Springer | year = 2007 | series = Undergraduate Texts in Mathematics | isbn = 978-0-387-33195-9
  • {{cite book | first = James | last = Stewart | title = Calculus: Early Transcendentals | publisher = Brooks/Cole | year = 2012 | edition = 7E | 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.

References

  1. "The term ''linear function'' means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1
  2. Stewart 2012, p. 23
  3. A. Kurosh. (1975). "Higher Algebra". Mir Publishers.
  4. T. M. Apostol. (1981). "Mathematical Analysis". Addison-Wesley.
  5. Shores 2007, p. 71
  6. Gelfand 1961

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