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

As a polynomial function

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 ) = a x + b , {\displaystyle 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 < 0 then the gradient is negative and the graph slopes downwards.

For a function f ( x 1 , … , x k ) {\displaystyle f(x_{1},\ldots ,x_{k})} of any finite number of variables, the general formula is

f ( x 1 , … , x k ) = b + a 1 x 1 + ⋯ + a k x k , {\displaystyle 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

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 {\displaystyle f} from a vector space V {\displaystyle \mathbf {V} } to a vector space W {\displaystyle \mathbf {W} } (Both spaces are not necessarily different.) over a same field K such that

f ( x + y ) = f ( x ) + f ( y ) {\displaystyle f(\mathbf {x} +\mathbf {y} )=f(\mathbf {x} )+f(\mathbf {y} )}

f ( a x ) = a f ( x ) . {\displaystyle 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 V {\displaystyle \mathbf {V} }, which might be K itself. Even if the same symbol + {\displaystyle +} is used, the operation of addition between x and y (belonging to V {\displaystyle \mathbf {V} }) is not necessarily same to the operation of addition between f ( x ) {\displaystyle f\left(\mathbf {x} \right)} and f ( y ) {\displaystyle f\left(\mathbf {y} \right)} (belonging to W {\displaystyle \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) f(0, ..., 0) = 0, 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.

See also

Notes

  • Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6
  • Shores, Thomas S. (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-0-387-33195-9.
  • Stewart, James (2012). Calculus: Early Transcendentals (7E ed.). 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