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- Linearity abstract "In common usage, linearity refers to a function or relationship which can be graphically represented as a straight line, as in two quantities that are directly proportional to each other, such as voltage and current in a simple DC circuit, or the mass and weight of an object.A crude but simple example of this concept can be observed in the volume control of an audio amplifier. While our ears may (roughly) perceive a relatively even gradation of volume as the control goes from 1 to 10, the electrical power consumed in the speaker is rising geometrically with each numerical increment. The "loudness" is proportional to the volume number (a linear relationship), while the wattage is doubling with every unit increase (a non-linear, parabolic relationship).In mathematics, a linear map or linear function f(x) is a function which satisfies the following two properties: Additivity : f(x + y) = f(x) + f(y). Homogeneity of degree 1: f(αx) = αf(x) for all α. The homogeneity and additivity properties together are called the superposition principle. It can be shown that additivity implies the homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well then this can be extended to show that homogeneity for α any real number, using the fact that rationals form a dense subset of the reals.In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A more specific definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations (also called linear maps), and systems of linear equations. The word linear comes from the Latin word linearis, which means pertaining to or resembling a line. For a description of linear and nonlinear equations, see linear equation. Nonlinear equations and functions are of interest to physicists and mathematicians because they can be used to represent many natural phenomena, including chaos.".
- Linearity wikiPageExternalLink dictL.html.
- Linearity wikiPageExternalLink l0180100.html.
- Linearity wikiPageExternalLink l0180300.html.
- Linearity wikiPageID "91591".
- Linearity wikiPageRevisionID "606262203".
- Linearity hasPhotoCollection Linearity.
- Linearity subject Category:Concepts_in_physics.
- Linearity subject Category:Elementary_algebra.
- Linearity comment "In common usage, linearity refers to a function or relationship which can be graphically represented as a straight line, as in two quantities that are directly proportional to each other, such as voltage and current in a simple DC circuit, or the mass and weight of an object.A crude but simple example of this concept can be observed in the volume control of an audio amplifier.".
- Linearity label "Lineal".
- Linearity label "Linearidade".
- Linearity label "Lineariteit".
- Linearity label "Linearity".
- Linearity label "Linearità (matematica)".
- Linearity label "Linearität".
- Linearity label "Linéarité".
- Linearity label "خطية".
- Linearity label "線型性".
- Linearity label "線性關係".
- Linearity sameAs Linearität.
- Linearity sameAs Lineal.
- Linearity sameAs Linéarité.
- Linearity sameAs Linearità_(matematica).
- Linearity sameAs 線型性.
- Linearity sameAs 선형성.
- Linearity sameAs Lineariteit.
- Linearity sameAs Linearidade.
- Linearity sameAs m.0mrtk.
- Linearity sameAs Q1753188.
- Linearity sameAs Q1753188.
- Linearity wasDerivedFrom Linearity?oldid=606262203.
- Linearity isPrimaryTopicOf Linearity.