From this question link, I got to know that row operation(row subtraction and row permutation) do change column space.
But still it seems that it does not change the column rank.
I am trying to prove that row rank == columns rank, but for that I need confirm the statement above. I am referring to this note
Have any intuitive explanation or proof for that row operation does not change the column rank?
Best Answer
If a collection of columns are linearly independent (respectively linearly dependent) then they remain so under elementary row operations. Therefore elementary row operations do not change the largest set of linearly independent columns.
To see this, notice that a linear dependence relation between some columns of a matrix $A$ is given by a nonzero column vector $v$ with $Av=0$. If $E$ is an elementary matrix and $B=EA$ then $Bv=EAv=0$; conversely if $Bv=0$ then $Av=E^{-1}Bv=0$. Therefore if a set of columns of $A$ is linearly dependent then the corresponding set of columns of $B$ is also linearly dependent and vice versa. This means $A$ and $B$ have the same column rank.