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 $(V,F)$ (for some field $F$) the linear map given by:

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