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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the term linear function refers to two distinct but related notions: * In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 ...

calculus
and related areas, a linear function is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
whose
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
is a
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

straight line
, that is, a
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
of
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
zero or one. For distinguishing such a linear function from the other concept, the term
affine function In Euclidean geometry, an affine transformation, or an affinity (from the Latin, ''affinis'', "connected with"), is a geometric transformation that preserves line (geometry), lines and parallelism (geometry), parallelism (but not necessarily Eucli ...
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 (mat ...
,
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, and
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, a linear function is a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
.


As a polynomial function

In calculus,
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
and related areas, a linear function is a polynomial of degree one or less, including the
zero polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
(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
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
s, often
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

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. 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
of dimension . A
constant function 270px, Constant function ''y''=4 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...

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) may be referred to as a
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about th ...
linear function or a
linear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

linear form
. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued
affine map Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology *Affine cipher, a special case of the more general substitution cipher *Af ...

affine map
s.


As a linear map

In linear algebra, a linear function is a map ''f'' between two
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s s.t. :f(\mathbf + \mathbf) = f(\mathbf) + f(\mathbf) :f(a\mathbf) = af(\mathbf). Here denotes a constant belonging to some
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
of
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
s (for example, the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s) and and are elements of a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, which might be itself. In other terms the linear function preserves
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

vector addition
and
scalar multiplication 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

scalar multiplication
. Some authors use "linear function" only for linear maps that take values in the scalar field;Gelfand 1961 these are more commonly called
linear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

linear form
s. The "linear functions" of calculus qualify as "linear maps" when (and only when) , or, equivalently, when the above constant equals zero. Geometrically, the graph of the function must pass through the origin.


See also

*
Homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar (mathematics), scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneit ...
*
Nonlinear system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
*
Piecewise linear function In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph of a function, graph is composed of straight-line segments. Definition A piecewise linear function is a function ...

Piecewise linear function
*
Linear approximation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Linear interpolation In mathematics, linear interpolation is a method of curve fitting Curve fitting is the process of constructing a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but ...

Linear interpolation
* Discontinuous linear map *
Linear least squares Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for Ordinary least squares, ordinary (unweight ...

Linear least squares


Notes


References

* Izrail Moiseevich Gelfand (1961), ''Lectures on Linear Algebra'', Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. * Thomas S. Shores (2007), ''Applied Linear Algebra and Matrix Analysis'',
Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbook are produced to meet t ...
, Springer. *James Stewart (2012), ''Calculus: Early Transcendentals'', edition 7E, Brooks/Cole. * Leonid N. Vaserstein (2006), "Linear Programming", in
Leslie Hogben Leslie Hogben is an American mathematician specializing in graph theory and linear algebra, and known for her mentorship of graduate students in mathematics. She is a professor of mathematics at Iowa State University, where she holds the Dio Lewis H ...
, ed., ''Handbook of Linear Algebra'', Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. {{Calculus topics Polynomial functions