Difference between revisions of "Canonical linear map"
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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> | ||
Revision as of 18:42, 1 June 2015
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→Vgiven by 1V:v↦vis 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→Vwith 1V:(v,w)↦v
- PV:V⊕W→V⊕Wwith PV:(v,w)↦(v,0w)
- 1W:V⊕W→Wwith 1W:(v,w)↦w
- PW:V⊕W→V⊕Wwith PW:(v,w)↦(0v,w)
are all canonical linear maps
References
- Jump up ↑ Linear Algebra via Exterior Algebra - Sergei Wintzki
