# Canonical linear map

(Redirected from Natural linear map)
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## 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:

• $1_V:V\rightarrow V$ given by $1_V:v\mapsto v$ 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:

• $1_V:V\oplus W\rightarrow V$ with $1_V:(v,w)\mapsto v$
• $P_V:V\oplus W\rightarrow V\oplus W$ with $P_V:(v,w)\mapsto (v,0_w)$
• $1_W:V\oplus W\rightarrow W$ with $1_W:(v,w)\mapsto w$
• $P_W:V\oplus W\rightarrow V\oplus W$ with $P_W:(v,w)\mapsto (0_v,w)$

are all canonical linear maps

## References

1. Linear Algebra via Exterior Algebra - Sergei Wintzki