Canonical linear map

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• Example:Canonical linear isomorphism between a one dimensional vector space and its field

Definition

A canonical linear map, or natural linear map, is a linear map that can be stated independently of any basis.

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