Linear map: Difference between revisions

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In [[mathematics]], and more specifically in [[linear algebra]], a '''linear map''' (also called a '''linear mapping''', '''vector space homomorphism''', or in some contexts '''linear function''') is a [[Map (mathematics)|map]] <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]].

A linear map whose [[___domain of a function|___domain]] and [[codomain]] are the same vector space over the same [[field (mathematics)|field]] is called a '''linear transformation''' or '''linear endomorphism'''. Note that the codomain of a map is not necessarily identical the [[range of a function|range]] (that is, a linear transformation is not necessarily [[surjective function|surjective]]), allowing linear transformations to map from one vector space to another with a lower [[dimension (vector space)| dimension]], as long as the range is a [[linear subspace]] of the ___domain. The terms 'linear transformation' and 'linear map' are often used interchangeably, and one would often used the term 'linear endomorphism' in its stict sense.

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>: