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The word linear comes from the Latin word linearis, which means created by lines.
In 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):
- Additivity: f(x + y) = f(x) + f(y)
- Homogeneity: f(αx) = αf(x) for all α
Systems that satisfy both additivity and homogeneity are considered to be linear systems. These two rules, taken together, are often referred to as the principle of superposition.
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 of scalars is the rationals or a finite field, superposition is enough to imply that they are from the same kind. 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. 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.
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.
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
See also: linear element, linear system, nonlinearity.
In a slightly different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line.
Over the reals, a linear function is one of the form:
- f(x) = m x + c
m is often called the slope or gradient; c the y-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 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).
In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect.
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