The word linear comes from the Latin word linearis, which means created by lines. In advanced mathematics, a linear map or function f(x) is a function which satisfies the following two properties:

• Additivity (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 of degree 1: 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, provided that the function is continuous, it becomes useless to establish the condition of homogeneity as an additional axiom.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.

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.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations (or linear maps), and systems of linear equations.

Nonlinear equations and functions are of interest to physicists and mathematicians because they can be used to represent many natural phenomena, including chaos.

Integral linearity

For a device that converts a quantity to another quantity there are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.

Many times a device's specifications will simply refer to linearity, with no other explanation as to which type of linearity is intended. In cases where a specification is expressed simply as linearity, it is assumed to imply independent linearity.