nLab dual linear map

Definition

The operation $V \mapsto V^*$ of forming dual vector spaces extends to a contravariant functor.

Definition (transpose map)

The dual linear map or transpose map of a linear map $A\colon V\to W$, is the linear map $A^* = A^T\colon W^*\to V^*$, given by

$$\langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle$$

for all $w$ in $W^*$ and $v$ in $V$.

Remark

This functor is, of course, the representable functor represented by $K$ as a vector space over itself (a line).

Remark

This construction is the notion of dual morphism applied in the monoidal category Vect with its tensor product monoidal structure.

Related concepts

• Chevalley-Eilenberg algebra

• duality