View source for Canonical linear map

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* [[Example:Canonical linear isomorphism between a one dimensional vector space and its field]]}}
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==Definition==
A ''canonical'' [[Linear map|linear map]], or ''natural'' linear map, is a linear map that can be stated independently of any [[Basis|basis]].<ref>Linear Algebra via Exterior Algebra - Sergei Wintzki</ref>
===Examples===
====Identity====
Given a [[Vector space|vector space]] {{M|(V,F)}} (for some [[Field|field]] {{M|F}}) the linear map given by:
* <math>1_V:V\rightarrow V</math> given by <math>1_V:v\mapsto v</math> is a ''canonical [[Isomorphism#Linear isomorphism|isomorphism]]'' from {{M|V}} to itself.
*: because it maps {{M|v}} to {{M|v}} irrespective of basis
====Projection of direct sum====
Consider the vector space {{M|V\oplus W}} where {{M|\oplus}} denotes the [[External direct sum|external direct sum]] of vector spaces. The [[Projector|projections]] defined by:
* <math>1_V:V\oplus W\rightarrow V</math> with <math>1_V:(v,w)\mapsto v</math> 
* <math>P_V:V\oplus W\rightarrow V\oplus W</math> with <math>P_V:(v,w)\mapsto (v,0_w)</math>
* <math>1_W:V\oplus W\rightarrow W</math> with <math>1_W:(v,w)\mapsto w</math> 
* <math>P_W:V\oplus W\rightarrow V\oplus W</math> with <math>P_W:(v,w)\mapsto (0_v,w)</math>
are all ''canonical'' linear maps

==References==
<references/>

{{Definition|Linear Algebra|Abstract Algebra}}