Definition 4.3.1. Change of basis matrix.
Let \(B=(\boldv_1, \boldv_2, \dots, \boldv_n)\) and \(B'\) be two ordered bases for the vector space \(V\text{.}\) The change of basis from \(B\) to \(B'\) is the \(n\times n\) matrix \(\underset{B\rightarrow B'}{P}\) defined as
\begin{equation}
\underset{B\rightarrow B'}{P}=
\begin{bmatrix}
\vert \amp \vert \amp \amp \vert \\
\phantom{v}[\boldv_1]_{B'} \amp \phantom{v}[\boldv_2]_{B'}\amp \dots \amp \phantom{v}[\boldv_n]_{B}\\
\vert \amp \vert \amp \amp \vert
\end{bmatrix}\text{.}\tag{4.3.1}
\end{equation}
In other words, the \(j\)-th column of \(\underset{B\rightarrow B'}{P}\) is obtained by computing the coordinate vector of the \(j\)-th element of the original basis \(B\) with respect to the new basis \(B'\text{.}\)