 # 5. Matrix Norms, Eigenvalues, Eigenvectors, &amp; Condition Num...

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5. Matrix Norms, Eigenvalues, Eigenvectors, & Condition Numbers (Part 2) The problem is that (speaking a bit loosely) each successive x' increasingly lies in the space spanned by the lower powers of x and thus incrementally spans only a rapidly shrinking portion of the y(x) function space. The solution is to construct an orthonormal basis using the Gram-Schmidt process. Let: = x° = 1 and proceed successively: x¹ - x1 (x¹, x¹) - (x¹, Po(x)) 2 - - - p1 p1 and then (a) Construct explicitly the first three elements of this orthonormal polynomial basis: Po(x), P1 (x), and P2(x). (b) What is the moment form of the least squares problem in the p basis and, in particular, what are the corresponding coefficients Aij? If possible, interpret your results as an explicit formula for the c:- (c) Given the form taken by the Gram matrix in (b), discuss (with as few calculations as possible) how your answers to 4(c)-(f) would differ in the p basis VS. xi basis.

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