Eilineaarisuus

Eilineaarisuus is a term used in theoretical mathematics and related fields to describe a generalized form of linearity. It refers to a property whereby a process preserves additive structure and scalar scaling along a specified set of subspaces or scales, even when the transformation is not globally linear.

Informally, a map is eilineaarinen if it behaves linearly within a chosen collection of subspaces that are stable under a given symmetry or operator. Concretely, suppose T is a map between vector spaces. If there exists a family of subspaces {E_s} such that, for every s, T restricted to E_s is linear (T(a u + b v) = a T(u) + b T(v) for all u,v in E_s and scalars a,b), and the subspaces are preserved by a fixed operator, then T is said to be eilineaarinen with respect to that family. Outside the union of these subspaces, T may fail to be linear.

Origins and usage: the term is employed in speculative or theoretical discussions to capture a middle ground between linearity and nonlinearity, especially in systems governed by symmetries or multi-scale structure. It provides a language for describing transformations that are linear in one or more constrained contexts but not in full generality.

Examples: a transformation that is linear on each invariant subspace of a symmetry group, yet non-linear when acting across different subspaces; a data processing pipeline that is linear within specific frequency bands but nonlinear when combining bands.

Relation to other concepts: eilineaarisuus is related to piecewise linearity, subspace analysis, and eigenstructure concepts. It is distinct from full linearity but shares the core idea of preserving linear relations within restricted domains. See also linearity, nonlinearity, subspaces, eigenstructure.