I know that
(i) row operations do not change the row space
(ii) column operations do not change the column space
and (iii) row rank = column rank (but this is sort of unrelated, I think).
But, is it true that row operations do not change both the row space and the column space of a matrix?
EDIT: I am guessing that it's most likely true, since in Guassian elimination, solving Ax=b involves only row operations -- there's something about column operations that makes the algorithm not work, I think (according to the book by Friedberg, Insel and Spence.)