# Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space.

Can one prove or disprove this? This is my continuation of Same row space is equivalent to same column space?

Let $A=\begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}$
and, $B=\begin{pmatrix} 1 & 1\\ 0 & 0 \end{pmatrix}$
If two $n\times n$ matrices have the same column space, then there is a sequence of elementary column operations transforming one matrix to the other. But column operations need not preserve the row space of a matrix.