Lineaire

Lineaire is a term used primarily in French-language mathematical and scientific literature to denote a linear relationship, function, or model. It describes relations that can be characterized by proportionality and additivity, with properties that preserve structure under combination and scaling. In English, the corresponding term is linear.

Etymology and orthography: the standard French spelling is linéaire, with an acute accent on the e. In non-accented contexts such as computer code or simplified typography, lineaire may be used. The word ultimately derives from the Latin linea, meaning line.

Mathematical usage: in mathematics, lineaire commonly refers to a linear transformation, a function T between vector spaces that satisfies T(u+v)=T(u)+T(v) and T(cu)=cT(u). It also describes linear functions in one variable, f(x)=ax, and more generally functions that satisfy additivity and homogeneity. A key distinction in French notation is between linear (linéaire) and affine (affine or linéairement déviée) relations, where affine includes a constant term and linear maps must map the origin to itself.

Other fields: lineaire is used to describe direct proportionality or linear relations in physics and economics. In statistics, models termed modèles linéaires express a response as a linear combination of predictors. In computer science and algorithm analysis, a lineaire time complexity (complexité linéaire) indicates an operation count proportional to the input size, O(n).

Notes: in English-language texts, the concept is typically described as linear rather than lineaire. The term remains common in French mathematics, physics, computer science, and statistics, where linéaire and related terms form a standard vocabulary.