Linéaire

Linéaire describes a property in mathematics in which a relation or function preserves the structure of addition and scalar multiplication. An operation is linear if f(x + y) = f(x) + f(y) and f(αx) = α f(x) for all vectors x, y and scalars α. A linear map is a function between vector spaces that satisfies these conditions; linearity is independent of the chosen coordinate system.

A key distinction is between linear and affine. A linear function has the origin as a fixed point: f(0) = 0. By contrast, an affine function may include a constant term, such as f(x) = mx + b; this is linear only when b = 0. Linear equations have variables appearing to the first power and no products of variables. In one variable, ax + b = 0; in several variables, a1x1 + ... + anxn = b.

In linear algebra, linear maps are represented by matrices relative to chosen bases. The kernel (solutions of f(x) = 0) and the image (range) describe the structure; the rank equals the dimension of the image. A map is invertible if it has a trivial kernel and full rank. These concepts underpin many algorithms and theories in mathematics and applied disciplines.

In calculus, linearity underpins linear approximation. The differential at a point is a linear map that approximates a function near that point; the Jacobian matrix collects all first-order partial derivatives and serves as the best linear approximation.

Linéaire concepts appear across disciplines: linear models in statistics, linear programming in optimization, and linear-time algorithms in computer science. The term derives from Latin linea, meaning line, and entered French and English through mathematical usage to emphasize straight-line relations.