I'm trying to understand this passage of a book:
Why this last expressions shows that the $i$th column of $\bar{A}$ is the representation of $Aq_i$ with respect to the basis $\{q_1,\ldots q_n\}$? I can't understand how the author reached this conclusion.
This approach may help. I believe if you explicitly extract out the $i$'th column of the left hand side and compare to $Q$ multiplying the $i$'th column of $\bar{A}$ on the right hand side, you will see the claim.
For example, let $i=1$, then we have:
$$ A\mathbf{q}_{1} = \begin{bmatrix}\mathbf{q}_{1} & \mathbf{q}_{2} & \cdots & \mathbf{q}_{n} \end{bmatrix} \begin{bmatrix} \bar{A}_{11} \\ \bar{A}_{21} \\ \vdots \\ \bar{A}_{n1} \\ \end{bmatrix} $$
$$ A\mathbf{q}_{1} = \bar{A}_{11} \mathbf{q}_{1} + \bar{A}_{21}\mathbf{q}_{2} + \ldots + \bar{A}_{n1}\mathbf{q}_{n} $$
I hope this helps.