Linear function

related topics
{math, number, function}
{island, water, area}

In mathematics, the term linear function can refer to either of two different but related concepts:

  • a first-degree polynomial function of one variable;
  • a map between two vector spaces that preserves vector addition and scalar multiplication.

Contents

Analytic geometry

In analytic geometry, the term linear function is sometimes used to mean a first-degree polynomial function of one variable. These functions are known as "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.

Such a function can be written as

(called slope-intercept form), where m and b are real constants and x is a real variable. The constant m is often called the slope or gradient, while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.

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

  • f1(x) = 2x + 1
  • f2(x) = x / 2 + 1
  • f3(x) = x / 2 − 1.

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

Vector spaces

In advanced mathematics, a linear function means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.

For example, if x and f(x) are represented as coordinate vectors, then the linear functions are those functions f that can be expressed as

where M is a matrix. A function

is a linear map if and only if b = 0. For other values of b this falls in the more general class of affine maps.

See also

External links

Full article ▸

related documents
Regular graph
Inner automorphism
Elias gamma coding
Subring
Earley parser
Connectedness
Parse tree
Subset
Hausdorff maximal principle
Row and column spaces
Unitary matrix
Axiom of power set
Injective function
Profinite group
Identity matrix
Unit interval
Additive function
Random sequence
Inequation
Kleene star
Euler's identity
Specification language
Sharp-P
Disjunctive normal form
Class (set theory)
Inverse functions and differentiation
Inverse transform sampling
Just another Perl hacker
Discrete probability distribution
Endomorphism