dual representation map as a linear map

Question

For $\rho\colon G\to GL(V)$, it dual representation is given by $\rho^*(g) = \rho(g^{-1})^T$.

How to understand this transposition in the definition $\rho(g^{-1})^T$ when we consider as a linear map $V^* \to V^*$ instead of a matrix?

Answer 1

Every linear map $f:V\to W$ induces a linear map $W^*\to V^*$. This linear map is denoted by any of the following symbols: $f^*,f^t,f’$, and is called the dual map or adjoint map or transpose map or pullback map. For our purposes here, I’ll denote it as $f^t$

The definition is that $f^t:W^*\to V^*$ sends $\omega\in W^*$ to $f^t(\omega):=\omega\circ f\in V^*$. As for why it is sometimes called and denoted as the transpose map, it’s because if you fix bases $\beta,\gamma$ for $V,W$ respectively, and let $\beta^*,\gamma^*$ be the dual bases for $V^*,W^*$ respectively, then the matrix representations of $f$ and $f^t$ are transposes: $[f^t]_{\gamma^*}^{\beta^*}=\left([f]_{\beta}^{\gamma}\right)^{t}$, where on the right we have a matrix transpose.