\begin{definition}[Definition:Matrix Equivalence/Definition 2]
Let $R$ be a ring with unity.
Let $\mathbf A, \mathbf B$ be $m \times n$ matrices over $R$.
$\mathbf A$ and $\mathbf B$ are '''equivalent''' {{iff}} they are the relative matrices, to (possibly) different ordered bases, of the same linear transformation.
We write:
:$\mathbf A \equiv \mathbf B$
\end{definition}