# Linear function

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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

 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