Linearity

(mathematics) The property whereby a mathematical system is well behaved (in the context of the given system) with regard to addition and scalar multiplication.
(physics) The relationship that exists between two quantities when a change in one of them produces a directly proportional change in the other.

A relationship between two or more quantities which can be expressed in terms of linear algebraic, differential, or integral equations. A system in which all quantities (or variables) can be described in terms of such equations is said to be a linear system. By definition, linear systems satisfy the principle of superposition. By this principle, the response of a linear system to multiple inputs is given simply by the sum of the responses due to each individual input. In addition, if all inputs are multiplied by a common constant factor, the resulting response is multiplied by the same factor. See also Differential equation; Integral equation; Linear algebra.

As an example of a simple linear relationship, the voltage V across an ideal (ohmic) resistor is directly proportional to the current I through it, as given by the equation below, where R is the constant of proportionality. Here, the customary notation V(I) is used to denote V as a function of I.

Linearity is a desirable characteristic of all systems where an output response is required to be a faithful reproduction (except for a constant scale factor) of one or more inputs. For example, electronic amplifiers used in measurement and signal transmission and reproduction systems are designed with linearity as a primary goal. Although physical systems are generally nonlinear to some degree, in practice the objective is to realize a good approximation to an ideal linear system by minimizing nonlinearities as far as possible. Any departure from linearity in these systems causes unwanted distortion of the original signal and results in a degraded and erroneous response. See also Amplifier; Distortion (electronic circuits).

1. In Linear Programming (LP) the requirement that the measure of effectiveness, such as contribution margin or cost and resource usage, must be proportional to the level of each activity conducted individually. For example, each unit of a resource must make the same contribution to the objective function that every other unit of that resource makes.

2. In Linear Regression, the requirement that the relationship between a mixed cost and an activity variable such as machine-hours is straight line in the form of y = a + bx. This means that the variable portion changes by a constant amount per hour no matter how many hours are used.