Say a subspace S of R^3 is spanned by a basis = <(-1,2,5),(3,0,3),(5,1,8)> By putting these vectors into a matrix and reducing it to rref, a basis for the row space can be found as <(1,-2,-5),(0,1,3)>. Furthermore, the book goes on to say that this basis spans the subspace S. Cool, not suprising. My question then is if the basis for the column space also spans S. If so, that means span(basis of column space) = span(basis of row space)? Why doesn't my book say this straight up!?