The SVD-Fundamental Theorem of Linear Algebra

Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and A T ; a 1 through a n and h 1 through h m are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N (A) is the orthogonal complement of R(A T). These four subspaces tell the whole story of the Linear System Ax = y. So, for example, the absence of N (A T) indicates that a solution always exists, whereas the absence of N (A) indicates that this solution 123 is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In " Classical " Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl's excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the " SVD-Fundamental Theorem " of Linear Algebra, and apply it in solving systems of linear equations.