Every m by n matrix A leads to two orthogonal subspaces of n-dimensional space (the row space and the nullspace of A) and two orthogonal subspaces of m- dimensional space (the row space and the nullspace of A). Then the five great factorizations of A create better and better bases for these four subspaces. This is pure linear algebra but it has very valuable applications. The winners are orthonormal bases of v’s and u’s such that each Avᵢ = sᵢuᵢ.
Four Subspaces, Five Factorizations
