The geometry of basic, approximate, and minimum-norm solutions of linear equations Original Research Article

The basic solutions of the linear equations Ax = b are the solutions of subsystems corresponding to maximal nonsingular submatrices of A. The convex hull of the basic solutions is denoted by C = C(A, b). Given 1 ≤ p ≤ ∞, the imagep-approximate solutions of Ax = b, denoted x{p}, are minimizers of short parallelAx − bshort parallelp. Given M set membership, variant Dm, the set of positive diagonal m × m matrices, the solutions of minx short parallelM(Ax − b)short parallelp are called scaledimagep-approximate solutions. For 1 ≤ p1, p2 ≤ ∞, the minimum-imagep2-norm imagep1-approximate solutions are denoted x{p1}{p2}. Main results: 1. (1) If A set membership, variant Rm × nm, then C contains all [some] minimum imagep-norm solutions, for 1 ≤ p < ∞ [p = ∞].2. (2) For general A and any 1 ≤ p1, p2 < ∞ the set C contains all x{p1}{p2}.3. (3) The set of scaled imagep-approximate solutions, with M ranging over Dm, is the same for all 1 < p < ∞.4. (4) The set of scaled least-squares solutions has the same closure as the set of solutions of minx f (Ax − b), where f:Rm+ → R ranges over all strictly isotone functions.