Elimination and Factorization

04/05/2023

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by   Gilbert Strang, et al.

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If a matrix A has rank r, then its row echelon form (from elimination) contains the identity matrix in its first r independent columns. How do we interpret the matrix F that appears in the remaining columns of that echelon form ? F multiplies those first r independent columns of A to give its n-r dependent columns. Then F reveals bases for the row space and the nullspace of the original matrix A. And F is the key to the column-row factorization A=CR.