# Contents

## 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.

Last revised on July 31, 2018 at 09:42:23. See the history of this page for a list of all contributions to it.