Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \longrightarrow Y$ be a linear operator. (Then by Theorem 2.7-8 in Kreyszig $T$ is bounded since its domain is finite-dimensional). Let $X^\prime$ denote the dual space of $X$ (i.e. the normed space of all the bounded linear functionals with domain $X$); since $X$ is finite-dimensional, $X^\prime$ consists of all the linear functionals with domain $X$, again by Theorem 2.7-8 in Kreyszig. And, the same applies to $Y^\prime$.

Then the adjoint operator $T^\times \colon Y^\prime \longrightarrow X^\prime$ of $T$ is also a bounded linear operator that is defined as follows: Let $g \in Y^\prime$. Then $T^\times (g) = f \in X^\prime$, where $f$ is defined by $$f(x) = g\big( T(x) \big) \mbox{ for all } x \in X.$$
This operator $T^\times$ is linear and bounded with $$ \left\lVert T^\times \right\rVert = \lVert T \rVert.$$

Now let $E = \left( e\_1, \ldots, e\_n \right)$ be an ordered basis for $X$, and let $B = \left( b\_1, \ldots, b\_m \right)$ be an ordered basis for $Y$. Let $A = \left[ \alpha\_{ij} \right]\_{m \times n}$ be the matrix of $T$ with respect to the ordered bases $E$ and $B$.

Let $E^\prime$ and $B^\prime$ denote the dual ordered bases of $E$ and $B$, respectively. Let $A^\times$ be the matrix of $T^\times$ with respect to the dual bases $B^\prime$ and $E^\prime$.

What is the relation between the matrices $A$ and $A^\times$?

By definition, the dual basis $E^\prime$ of $X^\prime$ corresponding to the ordered basis $E$ for $X$ is the ordered $n$-tuple $\left( f\_1, \ldots, f\_n \right)$ of linear functionals with domain $X$ such that, for each $j = 1, \ldots, n$ and for each $k = 1, \ldots, n$, we have
$$ 
f\_j \left( e\_k \right) = \begin{cases} 1 & \mbox{ if } j = k; \\ 0 & \mbox{ if } j \neq k.\end{cases} 
$$

And, similarly the dual basis $B^\prime$ of $B$ is the ordered $m$-tuple $\left( g\_1, \ldots, g\_m \right)$ of linear functionals with domain $Y$ such that, for each $r = 1, \ldots, m$ and for each $s = 1, \ldots, n$, we have
$$f\_r \left( e\_s \right) = \begin{cases} 1 & \mbox{ if } r = s; \\ 0 & \mbox{ if } r \neq s.\end{cases} $$

[Here](https://math.stackexchange.com/questions/2308162/how-is-the-matrix-of-a-linear-operator-affected-by-a-change-of-bases) is a Math Stack Exchange of mine which contains the relevant terminology and notation.