According to Wikipedia on matrix equivalence:

"Equivalent matrices represent the same linear transformation $V \to W$ under two different choices of a pair of bases of $V$ and $W$, with $P$ and $Q$ being the change of basis matrices in $V$ and $W$ respectively." source

"Two matrices are equivalent if and only if they have the same rank." source

"2x2 matrices only have three possible ranks: zero, one, or two. This means all $2 \times 2$ matrices fit into one of three matrix equivalent classes." source

However, there are infinitely many distinct linear transformations from $\mathbb{R}^2$ to $\mathbb{R}^2$.

How can there be only three classes of $2 \times 2$ matrices (based on rank) if there are infinitely many linear transformations from $\mathbb{R}^2$ to $\mathbb{R}^2$?

What I’m trying to understand is if there is a relation between two $m \times n$ matrices that represent the same linear transformation $T: F^n \to F^m$. (I know about similarity, but it applies only to square matrices.)

Sorry if I’ve misunderstood something. Thanks!