linearity
linearity (n)
linearity (n)
- plural
- linearities
English Definitions:
one-dimensionality, linearity (noun)
the property of having one dimension
linearity (Noun)
The state of being linear.
linearity (Noun)
A relationship between several quantities which can be considered as proportional and expressed in terms of linear algebra, or any mathematical property of a relationship, operation or function that is analogous to such proportionality, satisfying additivity and homogeneity.
Linearity
In mathematics, a linear map or linear function f(x) is a function which satisfies the following two properties: ⁕Additivity: f = f(x) + f. ⁕Homogeneity of degree 1: f = αf(x) for all α. 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.
Linearity
Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include rectilinear motion, the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. The word linear comes from Latin linearis, "pertaining to or resembling a line".
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