Lineär

Lineär is an adjective used in several languages, notably German and Swedish, to describe properties associated with linearity. In mathematics, science and engineering, the term denotes proportionality, additivity, and the absence of curvature in a relation or structure. The concept is often contrasted with non-linear and, in some contexts, with affine descriptions.

Mathematically, a function f is linear if it preserves additions and scalar multiplications: f(u+v) = f(u) + f(v) and f(cu) = c f(u) for all vectors u, v and scalars c. Equivalently, a linear map between vector spaces sends the origin to the origin. The graph of a non-constant linear function y = m x crosses the origin and forms a straight line. Linear equations and linear algebra focus on such structures, with linear combinations built from additions and scalar multiples of vectors.

Linearity underpins many areas: linear differential equations, linear time-invariant systems in signal processing and control theory, linear programming in optimization, and linear regression in statistics. In physics and engineering, linearity enables superposition and straightforward analysis through small-signal and linear-response approximations.

Note that in everyday language, lineär or linear is sometimes used loosely to describe simple or direct relationships. Strict mathematical use distinguishes linear from affine, where affine relations have a non-zero constant term. The term derives from Latin linea, “a line,” reflecting the geometric origin of the concept.