, a linear function f
) 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
In this definition, x
is not necessarily a real number
, but can in general be a member of any vector space
. In the case that the field
is the rationals
or a finite field
, superposition is enough to imply homogeneity. However, in the case of the reals
or complex numbers
, both relations are needed. We are often concerned with bounded linear functions[?]
, which is equivalent to continuous
ones. Although it is possible for a function to be linear and unbounded, these functions are usually of little practical importance.
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.
Nonlinear equations and functions are of interest to physicists and mathematicians because they are hard to solve and give rise to interesting phenomena such as chaos.
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 of the form:
- f(x) = m'x + c
m is often called the slope or gradient; c the intercept, which gives the point of intersection between the graph of the function and the y-axis.
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.
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