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Unformatted text preview: Serge Ballif MATH 535 Homework 6 October 12, 2007 1. Define the linear transformation U on C n by v 7→ Fv , where √ nF is the Vandermonde matrix for the values ω j n := exp ((2 πij ) /n ) , i = 0 ,...,n- 1 . a) Show that the sum S j = n X k =1 ( ω j n ) k = ( n if n | j otherwise by multiplying the expression by ω i n and simplifying. b) Conclude that U is unitary. c) Show that, for u = ( z 1 ,...,z n ) t ∈ C n , U- 1 u = Uv , where v = ( z n ,...,z 1 ) t . Remark: This mapping has revolutionized mathematical modelling and is the basis for the fastest known method for multiplication of extremely large integers. (a) If n | j , then ω i n = e 2 πij/n = 1, so each of the summands of S j is 1. In this case S j = n . If n- j , then we can examine S j /ω j n = ∑ n- 1 k =0 ( ω j n ) k , a geometric series with sum S j ω j n = ( ω j n ) n- 1 ω j n- 1 = 1- 1 ω j n- 1 = 0. Thus, S j = 0 in the case n- j . (b) We know that the columns of a vandermonde matrix with distinct values is a basis for the space V . U is unitary iff the column vectors form an orthonormal basis of V . To see that the basis is indeed orthonormal we note that the inner product of the i-th and j-th columns is δ ij as determined by the formula in part (a)....
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