Difference between revisions of "Canonical linear map"
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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> | ||
Latest revision as of 05:54, 7 December 2016
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Contents
[hide]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 (V,F) (for some field F) the linear map given by:
- 1V:V→V given by 1V:v↦v is a canonical isomorphism from V to itself.
- because it maps v to v irrespective of basis
Projection of direct sum
Consider the vector space V⊕W where ⊕ denotes the external direct sum of vector spaces. The projections defined by:
- 1V:V⊕W→V with 1V:(v,w)↦v
- PV:V⊕W→V⊕W with PV:(v,w)↦(v,0w)
- 1W:V⊕W→W with 1W:(v,w)↦w
- PW:V⊕W→V⊕W with PW:(v,w)↦(0v,w)
are all canonical linear maps
References
- Jump up ↑ Linear Algebra via Exterior Algebra - Sergei Wintzki
