Difference between revisions of "Canonical linear map"
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| + | {{Stub page|grade=A*|msg=Important for work, needs work anyway. Be sure to link | ||
| + | * [[Example:Canonical linear isomorphism between a one dimensional vector space and its field]]}} | ||
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==Definition== | ==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> | 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> | ||
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*: because it maps {{M|v}} to {{M|v}} irrespective of basis | *: because it maps {{M|v}} to {{M|v}} irrespective of basis | ||
====Projection of direct sum==== | ====Projection of direct sum==== | ||
| − | Consider the vector space {{M|V\oplus W}} where {{M|\oplus}} denotes the [[ | + | 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>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>P_V:V\oplus W\rightarrow V\oplus W</math> with <math>P_V:(v,w)\mapsto (v,0_w)</math> | ||
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Definition
A canonical linear map, or natural linear map, is a linear map that can be stated independently of any basis.[1]
Examples
Identity
Given a vector space [ilmath](V,F)[/ilmath] (for some field [ilmath]F[/ilmath]) 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 from [ilmath]V[/ilmath] to itself.
- because it maps [ilmath]v[/ilmath] to [ilmath]v[/ilmath] irrespective of basis
Projection of direct sum
Consider the vector space [ilmath]V\oplus W[/ilmath] where [ilmath]\oplus[/ilmath] denotes the external direct sum of vector spaces. The 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
- ↑ Linear Algebra via Exterior Algebra - Sergei Wintzki
