Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I’ve read the proof on Wikipedia and I understand the proof, but I don’t “get it”. Can someone help me out with this ?

I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level.

You can apply elementary row operations and elementary column operations to bring a matrix $$AA$$ to a matrix that is in both row reduced echelon form and column reduced echelon form. In other words, there exist invertible matrices $$PP$$ and $$QQ$$ (which are products of elementary matrices) such that
$$PAQ=E:=(Ik0(n−k)×(n−k)).PAQ=E:=\begin{pmatrix}I_k\\&0_{(n-k)\times(n-k)}\end{pmatrix}.$$
As $$PP$$ and $$QQ$$ are invertible, the maximum number of linearly independent rows in $$AA$$ is equal to the maximum number of linearly independent rows in $$EE$$. That is, the row rank of $$AA$$ is equal to the row rank of $$EE$$. Similarly for the column ranks. Now it is evident that the row rank and column rank of $$EE$$ are identical (to $$kk$$). Hence the same holds for $$AA$$.