{{Redirect|Linear transformation|fractional linear transformations|Möbius transformation}}
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)|functionmapping]] {{math|''V'' ↦ ''W''}} 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. 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.
AsLinear a result, itmaps always [[map (mathematics)|maps]] linear subspaces to linear subspaces,such(possibly as straight lines to straight lines or toof a singlelower point.dimension); Theform expressioninstance "linearit operator" is commonly used formaps a linearplane mapthough fromthe a vector spaceorigin to itself (i.e., [[endomorphisms|endomorphism]]). Sometimes the definition of a [[linear function]] coincides with that of a linear mapplane, whilestraight inline [[analyticor geometry]] it does notpoint.
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]].
