Artin's "Algebra" book (1st ed, page 114, Proposition 2.9b) claims that given any $m\times n$ matrix A, there're matrices $Q \in GL_m(F)$ and $P \in GL_n(F)$ so that $QAP^{-1}$ has the form:
\begin{equation*} \begin{pmatrix} I_r & O_{n-r} \\\\ O_{m-r} & O_{m-r,n-r} \end{pmatrix} \end{equation*}
where $I_r$ is the $r\times r$ identity matrix, $O_{n-r}$, $O_{m-r}$, $O_{m-r,n-r}$ are matrices of zeros of sizes $(n-r)\times (n-r)$, $(m-r)\times (m-r)$, and $(m-r)\times (n-r)$, respectively, and $r$ is the rank of the associated linear transformation (i.e. the dimension of its column space).
The book claims this can be proved by matrix manipulation and I have attempted it. The one point I believe may be weak or not appropriately developed is claiming that the dimension of the column space of the row-echelon form of a matrix $A$, is the same as the one from $A$. I highlighted it below in bold.
Proof attempt:
There is a series of elementary row operations that take $A$ to its row echelon form, which can be expressed as left multiplication by the corresponding elementary matrices:
$$({E_q} \cdots {E_1})A:= QA$$
and let $s$ be the number of rows left that are not all zero.
Similarly, by a series of elementary column operations, every column that does not contain a pivot, can be zero-ed. Such operations can be expressed as right multiplication by the corresponding elementary matrices:
$$QA({E_p}^\prime \cdots {E_1}^\prime):=QAP^{-1}$$
The number of non-zero columns is the number of l.i. vectors in the columns of $A$, this is, $r$.
What's left to be proved it that $r=s$, but this is so because $r\leq \textrm{dim}(F^s)=s$, and at the same time $s\leq \textrm{dim}(F^r)=r$.