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 $P_{\mathcal C \leftarrow \mathcal B}$ denote 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 $P_{\mathcal C \leftarrow \mathcal B}$ is invertible, and
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\begin{align*}
(P_{\mathcal C \leftarrow \mathcal B})^{-1}=P_{\mathcal B \leftarrow \mathcal C}.
\end{align*}