Linear function

Linear function

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

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 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.

Vector spaces

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.

ee also

*Function (mathematics)

External links

* [http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/linearFunctions.html Linear Functions on Id Mind]
* [http://www.mathopenref.com/linearexplorer.html Interactive tool to explore linear functions]


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • linear function — noun : a mathematical function in which the variables appear only in the first degree, multiplied by constants, and combined only by addition and subtraction * * * Math. See linear transformation. [1855 60] * * * linear function, Mathematics. a… …   Useful english dictionary

  • linear function — noun Date: 1853 1. a mathematical function in which the variables appear only in the first degree, are multiplied by constants, and are combined only by addition and subtraction 2. linear transformation …   New Collegiate Dictionary

  • linear function — noun a) Any function whose graph is a straight line b) Any function of the sum of two variables whose value is the same as the sum of the values of the same function of the two variables singly …   Wiktionary

  • linear function — Math. See linear transformation. [1855 60] * * * …   Universalium

  • Piecewise linear function — In mathematics, a piecewise linear function :f: Omega o V, where V is a vector space and Omega is asubset of a vector space, is any function with the property that Omega can be decomposed into finitely many convex polytopes, such that f is equal… …   Wikipedia

  • Linear programming — (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.… …   Wikipedia

  • Linear phase — is a property of a filter, where the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at pmpi. In a causal system, perfect linear phase can be achieved with a discrete time FIR filter.Since a… …   Wikipedia

  • Linear prediction — is a mathematical operation where future values of a discrete time signal are estimated as a linear function of previous samples.In digital signal processing, linear prediction is often called linear predictive coding (LPC) and can thus be viewed …   Wikipedia

  • linear programming — n. Math. a procedure for minimizing or maximizing a linear function of several variables, subject to a finite number of linear restrictions on these variables …   English World dictionary

  • Linear — The word linear comes from the Latin word linearis , which means created by lines .In advanced mathematics, a linear map or function f ( x ) is a function which satisfies the following two properties:* Additivity (also called the superposition… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”