What is the relation between transition functions of a principal fiber bundle and its dual?
As an example, consider the transition map of the frame bundle on a manifold of dimension $n$ as the following:
$$g:U_{a}\cap U_{b}\to GL(n,R)\qquad x\mapsto A_{x}$$ where locally frames $(v_{1},...,v_{n})$ and $(w_{1},...,w_{n})$ attached to $x\in U_{a}\cap U_{b} $ $$(w_{1},...,w_{n})_{x}=(v_{1},...,v_{n})_{x}A_{x} \qquad\forall x\in U_{a}\cap U_{b} $$ Let $(\alpha_{1},...,\alpha_{n})$ and $(\beta_{1},...,\beta_{n})$ be the dual co-frame $(v_{1},...,v_{n})$ and $(w_{1},...,w_{n})$ attached to $x\in U_{a}\cap U_{b} $. In the above the group $GL(n,R)$ acts from the right on frame bundle, I would like to know how $Gl(n,R)$ acts on dual co-frame bundle? What is relation between action $Gl(n,R)$ on frame bnndle and action $Gl(n,R)$ on its dual co-frame?
Thanks.