Editing Linear map (section)

{{Short description|Mathematical function, in linear algebra}}
{{Redirect|Linear transformation|fractional linear transformations|Möbius transformation}}
{{Redirect|Linear Operators|the textbook by Dunford and Schwarz|Linear Operators (book)}}
{{Distinguish|linear function}}
{{More footnotes needed|date=December 2021}}
In [[mathematics]], and more specifically in [[linear algebra]], a '''linear map''' (also called a '''linear mapping''', '''linear transformation''', '''vector space homomorphism''', or in some contexts '''linear function''') is a [[Map (mathematics)|mapping]] <math>V \to W</math> between two  [[vector space]]s that preserves the operations of [[vector addition]] and [[scalar multiplication]]. The same names and the same definition are also used for the more general case of [[module (mathematics)|modules]] over a [[ring (mathematics)|ring]]; see [[Module homomorphism]].

If a linear map is a [[bijection]] then it is called a '''{{visible anchor|Linear isomorphism|text=linear isomorphism}}'''. In the case where <math>V = W</math>, a linear map is called a '''linear endomorphism'''. Sometimes the term '''{{visible anchor|Linear operator|text=linear operator}}''' refers to this case,<ref>"Linear transformations of {{mvar|V}} into {{mvar|V}} are often called ''linear operators'' on {{mvar|V}}." {{harvnb|Rudin|1976|page=207}}</ref> but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that <math>V</math> and <math>W</math> are [[Real number|real]] vector spaces (not necessarily with <math>V = W</math>),{{citation needed|date=November 2020}} or it can be used to emphasize that <math>V</math> is a [[function space]], which is a common convention in [[functional analysis]].<ref>Let {{mvar|V}} and {{mvar|W}} be two real vector spaces. A mapping a from {{mvar|V}} into {{mvar|W}} Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from {{mvar|V}} into {{mvar|W}}, if <br /> <math display="inline">a(\mathbf u + \mathbf v) = a \mathbf u + a \mathbf v</math> for all <math display="inline">\mathbf u,\mathbf v \in V</math>, <br /> <math display="inline"> a(\lambda \mathbf u) = \lambda a \mathbf u </math> for all <math>\mathbf u \in V</math> and all real {{mvar|λ}}. {{harvnb|Bronshtein|Semendyayev|2004|page=316}}</ref> Sometimes the term ''[[linear function]]'' has the same meaning as ''linear map'', while in [[mathematical analysis|analysis]] it does not.

A linear map from <math>V</math>  to <math>W</math> always maps the origin of <math>V</math>  to the origin of <math>W</math>. Moreover, it maps  [[linear subspace]]s in <math>V</math> onto linear subspaces in <math>W</math> (possibly of a lower [[Dimension (vector space)|dimension]]);<ref>{{harvnb|Rudin|1991|page=14}}<br />Here are some properties of linear mappings <math display="inline">\Lambda: X \to Y</math> whose proofs are so easy that we omit them; it is assumed that <math display="inline">A \subset X</math> and <math display="inline">B \subset Y</math>:
{{ordered list|<math display="inline">\Lambda 0 = 0.</math>|If {{mvar|A}} is a subspace (or a [[convex set]], or a [[balanced set]]) the same is true of <math display="inline">\Lambda(A)</math>|If {{mvar|B}} is a subspace (or a convex set, or a balanced set) the same is true of <math display="inline">\Lambda^{-1}(B)</math>|In particular, the set: <math display="block">\Lambda^{-1}(\{0\}) = \{\mathbf x \in X: \Lambda \mathbf x = 0\} = {N}(\Lambda)</math> is a subspace of {{mvar|X}}, called the ''null space'' of <math display="inline">\Lambda</math>.|list-style-type=lower-alpha}}</ref> for example, it maps a [[Plane (geometry)|plane]] through the [[Origin (geometry)|origin]] in <math>V</math> to either a plane through the origin in <math>W</math>, a [[Line (geometry)|line]] through the origin in <math>W</math>, or just the origin in <math>W</math>. Linear maps can often be represented as [[matrix (mathematics)|matrices]], and simple examples include [[Rotations and reflections in two dimensions|rotation and reflection linear transformations]].

In the language of [[category theory]], linear maps are the [[morphism]]s of vector spaces, and they form a category [[equivalence of categories|equivalent]] to [[category of matrices|the one of matrices]].