Definition
In mathematics, a linear function (or map) is one which satisfies both of the following properties:
- additivity,
- homogeneity,
(Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity; for example, an antilinear map is additive but not homogeneous.) The conditions of additivity and homogeneity are often combined in the superposition principle
An equation written as
is called linear if is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if .
The definition is very general in that can be any sensible mathematical object (number, vector, function, etc.), and the function can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If contains differentiation of, the result will be a differential equation.
Read more about this topic: Nonlinear System
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