- See also linearity (computer and video games)
The word linear comes from the Latin word linearis, which means created by lines.
- Additivity property (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group isomorphism with respect to addition.
- Homogeneity property: f(αx) = αf(x) for all α.
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 adding the solutions up.
Over the reals, a linear function is one of the form:
- f(x) = m x + c
Note that this usage of the term linear is not the same as the above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if c = 0. Hence, if c ≠ 0, the function is often called an affine function (see in greater generality affine transformation).
Namely, linearity of a differential equation means that if two functions f and g are solution of the equation, then their sum f+g is also a solution of the equation.
- Linear medium
- Linear programming
- Linear motor
- Linear A and Linear B scripts.de:Lineare Funktion