LU and CR Elimination

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 = WR. The r by r matrix W has full rank r, where B meets C. Then the triple factorization A = CW-1 B treats columns and rows of A (C and B) in the same way.

Automated summary

This paper introduces a new matrix factorization method, which decomposes matrix A into column matrix C and row matrix R, and reimagines the start of a linear algebra course by observing the independent columns, rank, and column space of A.