Let $V$ and $W$ be finite-dimensional vector spaces over a field $k$. Let $T:V\to W$ be a [linear map](/page/Linear%20Map). Let $\mathcal B$ and $\mathcal C$ be ordered bases of $V$, and let $\mathcal D$ and $\mathcal E$ be ordered bases of $W$. Then the two representing matrices of $T$ have the same rank:
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\begin{align*}
\operatorname{rank}([T]_{\mathcal D \leftarrow \mathcal B}) = \operatorname{rank}([T]_{\mathcal E \leftarrow \mathcal C}).
\end{align*}