Outermorphism

In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.

Let ${\displaystyle f}$ be an ${\displaystyle \mathbb {R} }$-linear map from ${\displaystyle V}$ to ${\displaystyle W}$. The extension of ${\displaystyle f}$ to an outermorphism is the unique map ${\displaystyle \textstyle {\underline {\mathsf {f}}}:\bigwedge (V)\to \bigwedge (W)}$ satisfying

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