Intuitive proof of row rank = column rank? Is it possible to give an intuitive/elementary proof of the theorem that says that the row rank of a (finite-dimensional) square matrix matrix equals its column rank?

suppose T be a linear translation such  that $T(x)=Ax$ and A be a m*n matrix.
$T(x)=A_1x_1+A_2x_2+....+A_nx_n$so
rank(T)=rank column space of A
in other hand :
$Rank(T)+null(T)=n$         since (T is a linear translate form $F^n \to F^{m}$)
$null(T)=\{x, Ax=0\}$ so
$dim(null(T))=n-$ rank 's row space of A
with attention to :
$Rank(T)+null(T)=n$
we will have :
$rank column space of A+n -rank row space of A=n$
so : rank column space of A=rank row space of A
