Reduced Row Echelon Form is a canonical form of a k×n matrix which we take here to represent a sequence of k vectors in an n dimensional vector space over some field which is the reals here when we invoke visual images. The RREF is also a k×n matrix whose vectors span the same subspace as the original. The obvious question is “how is a canonical set chosen?”. The RREF is always relative to some particular basis in the vector space. The basis is a ordered sequence of n independent vectors. We picture them here as arranged from left to right as they are in the matrix representation. Let S be the subspace spanned by the original vectors. Choose the smallest m such that the rightmost m basis elements produce a non-zero linear combination in S. Divide each of the coefficients by left most term so that the left most of the n vector components is 1. That vector is the bottom row of the RREF matrix.

Continue to the left allowing more of the basis vectors until another linear combinations can represent more elements of S but disallow those basis elements that enabled some pre.