mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term ''

affine function In Euclidean geometry, an affine transformation or affinity (from the Latin, ''wikt:affine, affinis'', "connected with") is a geometric transformation that preserves line (geometry), lines and parallel (geometry), parallelism, but not necessarily ...

'' is often used. * In

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, a linear function is a

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...

As a polynomial function

In calculus,

analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...

and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When the function is of only one variable, it is of the form :

f(x)=ax+b,

where and are constants, often

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s. The graph of such a function of one variable is a nonvertical line. is frequently referred to as the slope of the line, and as the intercept. If ''a > 0'' then the

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

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

is negative and the graph slopes downwards. 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 In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

of dimension . A

constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...

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) may be referred to as a

homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...

linear function or a

linear form In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mat ...

. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.

As a linear map

In linear algebra, a linear function is a map ''f'' between two

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

s such that :

f(\mathbf + \mathbf) = f(\mathbf) + f(\mathbf)

f(a\mathbf) = af(\mathbf).

Here denotes a constant belonging to some field of scalars (for example, the

s) and and are elements of a

, which might be itself. In other terms the linear function preserves

vector addition Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

and

scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...

. Some authors use "linear function" only for linear maps that take values in the scalar field;Gelfand 1961 these are more commonly called

s. The "linear functions" of calculus qualify as "linear maps" when (and only when) , or, equivalently, when the constant 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. * * * 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. {{Calculus topics Polynomial functions

As a polynomial function

As a linear map

See also

Notes

References