Say we have a full rank matrix A (i.e., the rows are linearly independent and the same is true for columns). Since using the basic procedures of swapping, scalar multiplication, and addition, we can arrive at a row (or column) reduced echelon form which is the identity matrix (i.e., $rref(A)=I=cref(A)$, for any A that is full rank), it stands to reason that the columns and rows of A are just different basis for the same space (i.e., $span(A)=span(A^T)$).
Is my reasoning correct?