I am trying to understand the following automorphism of general linear group in terms of matrices.
(0) Let $V$ be a finite dimensional vector space over a field $k$.
(1) Given $u\in {\rm GL}(V)$, let $\check{u}$ denote contragradient of $u$.
(2) Let $t:V\rightarrow V^*$ denote a correlation map onto dual space.
Then the map $u\mapsto t^{-1}\check{u}t$ is an automorphism of ${\rm GL}(V)$.
Q.1 If we consider ${\rm GL}_n(k)$ instead of ${\rm GL}(V)$, how this map is intrepreted? I didn't understand properly the maps in (2).
Q.2 The map in (1) in terms of matrices is transpose map, am I right?
The automorphism mentioned above appears on first page of the paper A new type of automorphism of general linear group over a ring by Reiner I. (this link)