Difference between revisions of "Canonical linear map"
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* <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> | ||
| − | * <math> | + | * <math>1_W:V\oplus W\rightarrow W</math> with <math>1_W:(v,w)\mapsto w</math> |
| − | * <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 | are all ''canonical'' linear maps | ||
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Linear Algebra|Abstract Algebra}} | {{Definition|Linear Algebra|Abstract Algebra}} | ||
Revision as of 18:15, 1 June 2015
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 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
