Linear map: Difference between revisions

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In [[mathematics]], a '''linear map''' (also called a '''linear mapping''', '''linear [[Transformation (function)|transformation]]''', '''linear operator''' or, in some contexts, '''[[linear function]]''') is a [[function (mathematics)|function]] between two [[Module (mathematics)|module]]s (including [[vector space]]s) that preserves (in the sense defined below) the operations of module (or vector) addition and [[scalar (mathematics)|scalar]] multiplication.

As a result, it always [[map (mathematics)|maps]] linear subspaces to linear subspaces, likesuch as straight lines to straight lines or to a single point. The expression "linear operator" is commonly used for a linear map from a vector space to itself (i.e., [[endomorphisms|endomorphism]]). Sometimes the definition of a [[linear function]] coincides with that of a linear map, while in [[analytic geometry]] it does not.

In the language of [[abstract algebra]], a linear map is a [[homomorphism]] of modules. In the language of [[category theory]] it is a [[morphism]] in the [[category of modules]] over a given [[Ring (mathematics)|ring]].