We will consider the set X" = Z, as a discretization of the circle. Let | be the real vector space of functions on X, i.e. V = R. Let B = (e1, ..., en) be the basis of / defined by e;([k]) = 1 if [k] = and 0 otherwise. Let 6 : V - V be the map defined by (8f)([k]) = f([k]) - f([k - 1]). 1. Determine the matrix of & in the given basis. In other words, determine B B. Now let Y = {1, 2, ... n} and W = RY , with a similar basis (e1, ... en), defined by e; (k) = 1 if i = k and 0 otherwise. Also, let D : W - W be the map defined by (DO)(1) = f(1) and (Df)(k) = f(k) - f(k - 1) for k > 1. 2. Determine the matrix of D in the given basis. 3. If either of the two maps above are invertible, compute the inverse. 4. Determine an explicit formula for the composite maps 8* and D* for k = 2, 3