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In [[mathematics]], a '''linear map''' (also called a '''linear mapping''', '''linear [[Transformation (function)|transformation]]''' or, in some contexts, '''[[linear function]]''') is a [[function (mathematics)|mapping]] {{math|''V'' ↦ ''W''}} between two [[Module (mathematics)|module]]s (including [[vector space]]s) that preserves (in the sense defined below) the operations of addition and [[scalar (mathematics)|scalar]] multiplication. An important special case is when {{math|''V'' {{=}} ''W''}}, in which case the map is called a '''linear operator''', or an [[endomorphisms|endomorphism]] of {{math|''V''}}. Sometimes the definition of a [[linear function]] coincides with that of a linear map, while in [[analytic geometry]] it does not.
A linear map always [[map (mathematics)|maps]] linear subspaces to linear subspaces (possibly of a lower dimension);
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]].
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