(a) To each vector v = [20, ...,2N_]* e C~ we associate a second vector v = [ao, ....av_i]t e CN. The vector 7 is known as the Discrete Fourier Transform of v. The entries of the Discrete Fourier Transform are given by the formula N-1 an = U(n) := 2 ki -kin for n E ZN. *=0 (b) The building blocks of the Discrete Fourier Transform are the N discrete trigonometric functions en : Zx - C given by en ( k ) := 1 when KEZN, for ne {0, 1, ..., N -1}. Thus the nth Fourier coefficient of v can be written in terms of the C~ inner product as N-1 an = 1 (n) = > zken (k) =: (v, en). *=0 (c) The original vector v can be exactly determined from its Discrete Fourier Transform. The &th entry of v is given, for ke ZN, by 1 N-1 N-1 v(k) := 2 = Ev (n)whin = [ (v, en)en (k). 1=0 n=0 (d) The set {eo, ..., ex-1} forms an orthonormal basis for the vector space C~ with respect to the standard inner product, or dot product, in CN, and the Pythagorean theorem holds: N-1 1 7 1 12= (ZN) = 1/012- = 10(12)12. n=0Exercise 6.33. Show that the Discrete Fourier Transform in CN of the Fourier basis vector e; is given by the standard basis vector s;, that is, e; = s;, for 0