Unlike the column space, the row space is not affected by performing row reduction.
So just do the row reduction, and the non-zero rows in the reduced form give you your basis.
I know to work out the column space of a matrix you should put it into row echelon form and then look at which columns have a pivot- a basis for the column space is then the corresponding columns of the original matrix. Is it also valid to transpose the original matrix work out a basis for the row space by performing row reduction and then viewing these rows as columns?
Thanks!