ABSTRACT
RREF, leading coefficient, pivot column, matrix equivalence, modified RREF, rank normal form, row equivalence, column equivalence
Thoughwehave avoided it so far, one of themost useful reductions of amatrix is to bring it into row reduced echelon form (RREF). This is the fundamental result used in elementary linear algebra to accomplish so many tasks. Even so, it often goes unproved. We have seen that a matrix A can be reduced to many matrices in row echelon form. To get uniqueness, we need to add some requirements. First, a little language. Given a matrix A, the leading coefficient of a row of A is the first nonzero entry in that row (if there is one). Evidently, every row not consisting entirely of zeros has a unique leading coefficient. A column of A that contains the leading coefficient of at least one row is called a pivot column.
