Let $V$ be an $n$-dimensional [vector space](/page/Vector%20Space) over a field $k$, and let $\mathcal B=(b_1,\ldots,b_n)$ and $\mathcal C=(c_1,\ldots,c_n)$ be ordered bases of $V$. Let $P_{\mathcal C \leftarrow \mathcal B} \in k^{n \times n}$ be the change-of-basis matrix characterized by
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\begin{align*}
[v]_{\mathcal C}=P_{\mathcal C \leftarrow \mathcal B}[v]_{\mathcal B}
\end{align*}
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for every $v \in V$. Then, for each $i \in \{1,\ldots,n\}$, the $i$th column of $P_{\mathcal C \leftarrow \mathcal B}$ is the coordinate vector $[b_i]_{\mathcal C}$. Equivalently,