Difference between revisions of "Canonical linear map"
From Maths
m |
(Marking as stub, added reminder to add example) |
||
| Line 1: | Line 1: | ||
| + | {{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]]}} | ||
| + | __TOC__ | ||
==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> | ||
Latest revision as of 05:54, 7 December 2016
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Important for work, needs work anyway. Be sure to link
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
