Abstract
The concept of orthogonality lies at the very heart of the method of least squares. The normal equations of least squares in their simplest expression state that the residual vector is orthogonal to all of the basis vectors used in the approximating vector. Suppose that the observed vector F lies in a D dimensional, linear vector space, SD. Now choose a completely arbitrary basis, fi, i = 1, 2, ··· D. (To form a basis, the vectors fi must: (a) lie in SD, (b) be linearly independent, and (c) span SD, so that any vector in SD is expressible as a linear combination of the fi.) Denote ΦN that approximating vector that is a linear combination of the first N of the f iwhere the obvious dependence of the coefficients on the number N has been emphasized by the notation aiN. The equation of condition is which defines the residual at stage N, RN.
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