LinearIn mathematics, a linear function f(x) is one which satisfies the following two properties (but see below for a slightly different usage of the term):
- Superposition: f(x + y) = f(x) + f(y)
- Homogeneity: f(αx) = αf(x) for all α
The concept of linearity can be extended to linear operators which are linear if they satisfy the superposition and homogenity relations. Examples of linear operators are del and the derivative function. 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.
In a slightly different usage to the above, a polynomial of degree 1 is said to be linear.
Over the reals, a linear function is one of the form:
- f(x) = mx + 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 superposition or homogeneity. In fact, they do so if and only if c = 0.