Editing Nonlinear system (section)

==Definition==
In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:
*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math>
*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math>

Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|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
:<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math>

An equation written as
:<math>f(x) = C</math>

is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math> and <math>f(x)</math> is a [[homogeneous function]].

The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].