linearitate

Linearitate, commonly translated as linearity, is a property of a function, operator, or system that preserves addition and scalar multiplication. In formal terms, a function f between vector spaces over a field F is linear if for all vectors x, y and all scalars α, β in F, f(x + y) = f(x) + f(y) and f(αx) = α f(x). Equivalently, f(αx + βy) = α f(x) + β f(y). Linear maps send the zero vector to zero and preserve linear combinations.

In finite-dimensional spaces the action of a linear map is described by a matrix A relative to chosen bases: f(x) = Ax. Composition of linear maps corresponds to matrix multiplication, and the set of all linear maps between vector spaces forms a vector space itself. Related notions include eigenvalues and eigenvectors, rank, and kernel, which describe how the map stretches, compresses, or annihilates vectors.

Linearity excludes many nonlinear behaviors; an affine map f(x) = Ax + b is linear only if b = 0. For example, f(x) = 2x is linear on the real line, while f(x) = 2x + 1 is not. The concept extends beyond functions on vectors to differential operators, where linear operators satisfy additivity and homogeneity, enabling superposition.

Applications and contexts include solving linear differential equations, physics (superposition principle), computer graphics (linear coordinate transformations), economics and statistics (linear models), and control theory. Linear structure provides a tractable framework for analysis, providing tools such as matrices, bases, and dimensions that illuminate how systems respond to inputs.