As we saw in Chapter 1, it's often useful to change from one set of
coordinates (say, corresponding to the basis $\mathcal{B}$) to a different
set of coordinates (say, corresponding to a different basis $\mathcal{D}$.)
This is accomplished with a change-of-basis matrix $P_{\mathcal{DB}}$.
The definition of the matrix is that, for any vector ${\bf v}\in V$,
$$ [{\bf v}]_\mathcal{D} = P_{\mathcal{DB}} [{\bf v}]_\mathcal{B}.$$
Note that, in our notation, $P_{\mathcal{DB}}$ converts from the
$\mathcal{B}$ basis
to the $\mathcal{D}$ basis, not the other way around.
The formula for the matrix is
$$P_{\mathcal{DB}} = \begin{pmatrix} \phantom{a} &&& \cr
[{\bf b}_1]_\mathcal{D} &
[{\bf b}_2]_\mathcal{D} &
\cdots &
[{\bf b}_n]_\mathcal{D} \cr\phantom{a} &&&
\end{pmatrix}.$$
In particular, if $V={\bf R}^n$ and $\mathcal{E}$ is the standard basis, then
$$P_{\mathcal{EB}} = \begin{pmatrix} \phantom{a}&&& \cr
{\bf b}_1 &
{\bf b}_2 &
\cdots &
{\bf b}_n \cr \phantom{a}&&&
\end{pmatrix}.$$
If you think of $P_{\mathcal{DB}}$ as converting from $\mathcal{B}$ to
$\mathcal{D}$, then it's easy to see how to combine change-of-basis
matrices:
$$ P_{\mathcal{BD}} = \left ( P_{\mathcal{DB}} \right )^{-1}; \qquad
P_{\mathcal{BD}} = P_{\mathcal{BC}} P_{\mathcal{CD}}.$$