If $V, W$ are vector spaces with bases $\beta, \gamma$ respectively, let $\beta^*, \gamma^*$ be dual bases.
Suppose $S \in \mathcal{L}(V, W)$. Then the dual map of $S$, $S^* \in \mathcal{L}(W^*, V^*)$, is defined $$ S^*: W^* \to V^*; \quad \phi \mapsto \phi S. $$ This is my first exposure to dual maps, and I am completely stuck on the theorem that states $[S^*]_{\beta^*}^{\gamma^*} = ([S]_\gamma^\beta)^T$.
I would be glad to post my attempts so far, but they are a mess so I will just outline my strategy.
First I would let $\beta = \{ \mathrm{v_1}, ..., \mathrm{v_n} \}$ and $\gamma = \{ \mathrm{w_1}, ..., \mathrm{w_m} \}$.
As I understand it, the dual bases of $\beta$ and $\gamma$ are defined as $\beta^* = \{ \varphi_1, ..., \varphi_n \}$ and $\gamma^* = \{ \pi_1, ... \pi_m \}$ where $$ \varphi_j(\mathbf{v_i}) = \begin{cases} 1, & i = j\\ 0, & i \neq j \end{cases} $$ and $$ \pi_j(\mathbf{w_i}) = \begin{cases} 1, & i = j\\ 0, & i \neq j \end{cases}. $$
So next, I suppose $$S(\mathbf{v_j}) = \sum_{i=1}^{m}{a_{ij} \mathbf{w_i}}$$ so that I can construct $[S]_\beta^\gamma$.
It turns out $([S]_\beta^\gamma)^T$ is the following matrix. $$ \begin{bmatrix} a_{11} & a_{21} & \cdots & a_{m1}\\ a_{12} & a_{22} & \cdots & a_{m2}\\ \vdots & \vdots & \ddots & \vdots\\ a_{1n} & a_{2n} & \cdots & a_{mn} \end{bmatrix} $$
Now the part I have trouble with is how to begin forming $[S^*]_{\gamma^*}^{\beta^*}$ and showing it is the same matrix as above.
I know that $[S^*]_{\gamma^*}^{\beta^*} = \left[ [S^*(\pi_1)]^{\beta^*} ... [S^*(\pi_m)]^{\beta^*} \right]$ but I have no idea how to express, say, $S^*(\pi_j)$ in terms of transformations in $\beta^*$.
So I am completely stuck. If someone could shed light on how this works, it would be much appreciated! Cheers.