LU and CR Elimination

Abstract The reduced row echelon form rref (A) has traditionally been used for classroom examples : small matrices A with integer entries and low rank r. This paper creates a column-row rank-revealing factorization A = CR, with the first r independent columns of A in C and the r nonzero rows of rref (A) in R. We want to reimagine the start of a linear algebra course, by helping students to see the independent columns of A and the rank and the column space. If B contains the first r independent rows of A, then those rows of A = CR produce B = W R. The r by r matrix W has full rank r, where B meets C. Then the triple factorization A = CW B treats columns and rows of A (C and B) in the same way.