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Linear function
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication.

Analytic geometry


In analytic geometry
Analytic geometry

Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
, the term linear function is sometimes used to mean a first-degree polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of one variable
Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication.

Analytic geometry


Linear Functions2


In analytic geometry
Analytic geometry

Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
, the term linear function is sometimes used to mean a first-degree polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of one variable
Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
. These functions are called "linear" because they are precisely the functions whose graph
Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian coordinate system, together with Cartesian axes, etc....
 in the Cartesian coordinate plane is a straight line.

Such a function can be written as



(called slope-intercept form), where and are real
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 constants and is a real variable. The constant is often called the slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 or gradient, while is the y-intercept
Y-intercept

In coordinate geometry, the y-intercept is the y-value of the point where the graph of a function or relation intercepts the y-axis of the coordinate system....
, which gives the point of intersection between the graph of the function and the -axis. Changing makes the line steeper or shallower, while changing moves the line up or down.

Examples of functions whose graph is a line include the following:



The graphs of these are shown in the image at right.

Vector spaces


In advanced mathematics, a linear function often means a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 that is a linear map, that is, a map between two vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s that preserves vector addition and scalar multiplication
Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . Note that scalar multiplication is different from scalar product which is an inner product between two vectors....
.

For example, if and are represented as coordinate vector
Coordinate vector

In linear algebra, a coordinate vector is an explicit representation of a vector in an Real_coordinate_space#Intuitive_overview as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
s, then the linear functions are those functions that can be expressed as

, where M is a matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
.

A function is a linear map if and only if . For other values of this falls in the more general class of affine maps.

See also

  • Function (mathematics)
    Function (mathematics)

    The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....


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