The Column-Row Factorization of a Matrix

The active ideas in linear algebra are often expressed by matrix factorizations : $S=Q\Lambda Q^{\mathrm{T}}$ for symmetric matrices (the spectral theorem) and $A=U\Sigma V^{\mathrm{T}}$ for all matrices (singular value decomposition). Far back near the beginning comes $A=LU$ for successful elimination : Lower triangular times upper triangular. This paper is one step earlier, with bases in $A=CR$ for the column space and row space of any matrix – and a proof that column rank = row rank. The echelon form of $A$ and the pseudoinverse $A^+$ appear naturally. The ``proofs'' are mostly ``observations''.