So $A$ is some $m \times n$ matrix. Now we have $A^T$. We know that both $A$ and $A^T$ have same rank.
How do we then show that the location of columns that have pivots for $A$ and $A^T$ (columns that do not have pivots) may be different?
(By being different, I mean: $A$ may have columns with pivots in the first, third column, while $A^T$ may have columns with pivots in te first and fourth column.)
Addition: What about the case when $A$ is $3 \times 3$ matrix?
$A = [0 \; 1 \; 0]$ and $A^T = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \sim \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$.
$A$'s one and only pivot is in column 2. $A^T$'s one and only pivot is in its one and only column - column 1.