chainlinear

Chainlinear is a term used informally in mathematical and computational contexts to describe a chain or sequence of objects linked together in which each link exhibits linear behavior. It is not a widely standardized term with a single agreed-upon definition, and its precise meaning depends on the discipline and author.

In mathematics, chainlinear often refers to a chain of linear maps. Consider a sequence of vector spaces V0, V1, V2, ... with linear maps f0: V0 → V1, f1: V1 → V2, and so on. The chain is linear in the sense that every map fi is linear, and the overall construction preserves linear structure through the chain. In some discussions, the term may be used to emphasize that the maps are linear at every stage, as opposed to involving nonlinear operations along the chain. Related concepts include chain complexes, where boundary maps connect successive spaces, and chain maps, which respect those structures.

In applied and computational contexts, chainlinear can describe models or systems formed by successive linear transformations. For example, a chain-linear model might apply a sequence of matrix multiplications to an input vector, with each stage performing a linear transformation and no nonlinear activations or nonlinear operations between stages. This viewpoint is common in discussions of layered linear models, signal processing pipelines, or certain control systems composed solely of linear blocks.

Notes: because it is not a standardized term, readers should seek a precise definition within the relevant text. See also chain of linear maps, chain complex, linear map, and linear system.