The same naive approach shows that for a finite dimensional affine space there are natural generalised barycentric coordinates for any sequence of points whose affine span is the space. The potential applications of these two results are considerable - whenever there is a natural spanning set for a vector space or affine space, computations can be done directly with it, in an efficient and stable way. This avoids the need to obtain a basis by thinning or applying an analogue of Gram-Schmidt to some ad hoc ordering of the spanning set, which may destroy the inherent geometry. Another typical situation where spanning sequences have natural advantages over bases is when the dimension of the space is difficult to determine, or sensitive to perturbations, as is the case for multivariate spline spaces. A number of indicative examples are given. These include vector spaces over the rationals, such as cyclotomic fields.
Keywords: finite frames, vector spaces over the rationals, least squares (minimum norm) solutions, affine spaces, barycentric coordinates, multivariate splines, cyclotomic fields
Math Review Classification: Primary 15A03, 15A21, 41A45, 42C15; Secondary 12Y05, 15B10, 41A15, 52B11, 65F25
Length: 21 pages
Last Updated: 10 May 2010