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.

Answer

You can apply elementary row operations and elementary column operations to bring a matrix A to a matrix that is in both row reduced echelon form and column reduced echelon form. In other words, there exist invertible matrices P and Q (which are products of elementary matrices) such that
PAQ=E:=(Ik0(nk)×(nk)).
As P and Q are invertible, the maximum number of linearly independent rows in A is equal to the maximum number of linearly independent rows in E. That is, the row rank of A is equal to the row rank of E. Similarly for the column ranks. Now it is evident that the row rank and column rank of E are identical (to k). Hence the same holds for A.

Attribution
Source : Link , Question Author : hari_sree , Answer Author : darij grinberg

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