Let $k$ be a field, let $V$ be a [finite-dimensional vector space](/page/Finite-Dimensional%20Vector%20Space) over $k$, and let $\mathcal{B}=(v_1,\ldots,v_n)$ be an ordered basis of $V$. For each $v \in V$, let $[v]_{\mathcal{B}} \in k^n$ denote the ordered coordinate tuple of $v$ with respect to $\mathcal{B}$. Then the map
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\begin{align*}
\Phi_{\mathcal{B}}: V \to k^n,\qquad v \mapsto [v]_{\mathcal{B}}
\end{align*}
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is a linear isomorphism of vector spaces over $k$.