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

## Vector spaces

## See also

## External links

In 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

- $f(x)\; =\; mx\; +\; b$

(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 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:

- $f\_\{1\}(x)\; =\; 2x+1$
- $f\_\{2\}(x)\; =\; x/2+1$
- $f\_\{3\}(x)\; =\; x/2-1.$

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

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

- $f(x)\; =\; mathrm\{M\}x$, where M is a matrix.

A function $f(x)\; =\; mx\; +\; b$ 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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Last updated on Thursday October 09, 2008 at 12:54:15 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday October 09, 2008 at 12:54:15 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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