Short proof that rows-rank=column-rank?

Assuming that if $\displaystyle A$ is an $\displaystyle m\times n$ matrix whose rank is $\displaystyle k$, then $\displaystyle A^{T}$ (the transpose of $\displaystyle A$) is an $\displaystyle n\times m$ matrix whose rank is also $\displaystyle k$.

From this I conclude that if the rank of the rows of $\displaystyle A$ is not equal to the rank of the column of $\displaystyle A$ then what I first assumed is not true, thus it's a contradiction.

If what I wrote is right then this proof is quite simpler than the one my professor gave us! And also way simpler than the one given in the Hoffman's book.

Hope to hear your comments soon. (I'm not confident in myself enough not to post this here...)