linearity

Linearity is a property of a mathematical object that preserves the operations of addition and scalar multiplication. In the context of vector spaces, a function T: V → W is linear if for all vectors u and v in V and all scalars α in the underlying field F, T(u+v) = T(u) + T(v) and T(αu) = α T(u). Equivalently, T(a u + b v) = a T(u) + b T(v). A consequence is T(0) = 0.

Linear maps (or linear operators) include matrices acting on finite-dimensional spaces, and linear functionals map vectors to the underlying field. A common example is f(x) = 2x on R → R. Functions like f(x) = x^2 or f(x) = sin x fail linearity.

Linearity is often contrasted with affine maps, which have the form f(x) = A x + b with a nonzero constant offset b. Affine maps preserve straight lines but not necessarily the origin.

Linearity also extends to differential equations: a differential operator is linear if it satisfies additivity and homogeneity with respect to the unknown function. Linear differential equations obey the superposition principle: the sum of solutions is again a solution, and scaled solutions are solutions. This leads to linear systems theory, eigenvector decompositions, and easy analysis of stability.

In physics and engineering, many fundamental models are linear or approximated as linear because they exhibit superposition and analytic tractability. Examples include linear circuits obeying Ohm's law, Maxwell's equations in linear media, and the Schrödinger equation in quantum mechanics.

Nonlinear objects fail these properties, typically due to products or powers of the unknown, such as f(x) = x^2. Linearity thus provides a framework for decomposing problems into simpler, additive components.