Linear function

Linear function
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Linear_function".

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

Usage in elementary mathematics

Main article: linear equation

In elementary algebra and analytic geometry, the term linear function is sometimes used to mean a first degree polynomial function of one variable. These functions are called "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

f (x) = m x + b

(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 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.

Three geometric linear functions — the red and blue ones have the same slope (m), while the red and green ones have the same y-intercept (b).

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

$f 1 (x) = 2 x + 1$
$f 2 (x) = x / 2 + 1$
$f 3 (x) = x / 2 - 1$

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

Usage in advanced mathematics

In advanced mathematics, a linear function often 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 that can be expressed as

f (x) = M x

, where M is a matrix.

A function $f (x) = m x + b$ 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.

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