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(Solved) to the coefficients ,8 = (,80, . . . ,n)T. Show that this basis transformation matrix is given by the socalled Vandermonde matrix V E Ronn"


Let Pn be the space of functions defined on [-1,1] that can be described by polynomials of degree less of equal to n with coefficients in R.

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to the coefficients ,8 = (,80, . . . ,fin)T. Show that this basis transformation matrix is
given by the socalled Vandermonde matrix V E Ronn“ given by 1 2 71— 'n. 1 $0 1:3 $0 1 m0 _ 1 $1 $1 :5?— is?
V _ :I 2 n—l n 1 SL‘n $1, 33,, (En i.e., the relation between a and {-3 in (1) is given by a = V6. An easy way to see
this is to choose appropriate 3: in (1). (c) Note that since V transforms one basis into another basis, it must be an invertible
matrix. Let us compute the condition number of V numerically.1 Compute the 2-
based condition number 520/) for n = 5, 10,20, 30 with uniformly spaced nodes x,- = —1 + (20/11. 2' = 0,...,n. Based on the condition numbers, can this basis
transformation be performed accurately?

 


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