Linearity

See also linearity (computer and video games). For the early 90s pop trio, see Linear (pop group).

The word linear comes from the Latin word linearis, which means created by lines.

Mathematics

Linear functions are gay and fat

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 property (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
Homogeneity property: f(αx) = αf(x) for all α. It turns out that homogeneity follows from the additivity property in all cases where α is rational. In that case if the linear function is continuous, homogeneity is not an additional axiom to establish if the additivity property is established.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space.

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.

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.

Linear polynomials

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).

Physics

In physics, linearity is a property of the differential equations governing a lot of systems (like, for instance Maxwell equations or the diffusion equation).

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.

Electronics

In electronics, the linear operating region of a transistor is where the collector-emitter current is related to the base current by a simple scale factor, enabling the transistor to be use as an amplifier that preserves the fidelity of audio signals. Linear is similarly used to describe regions of any function, mathematical or physical, that follow a straight line with arbitrary slope.

Military tactical formations

In military tactical formations, "linear formations" were adapted from phalanx-like formations of pike protected by handgunners towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation would get thinner until its extreme in the age of Wellington with the 'Thin Red Line'. It would eventually be replaced by skirmish order at the time of the invention of the breech-loading rifle that allowed soldiers to move and fire independently of the large scale formations and fight in small, mobile units.

Music

In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect.