We will consider the set X = Zn as a discretization of the circle. Let V be the real vector space of functions on X, i.e. V =R*. Let B = (@j , ..., en) be the basis of V defined by e ([k]) = 1 if [k] = [i] and otherwise. Let 8: V V be the map defined by (8f)([k]) = f([k]) f([k 1]). 1. Determine the matrix of 8 in the given basis. In other words, determine [8]B. Now let Y = {1,2, ... n} and W = R', 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 (Df)(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 sk and Dk for k = 2, 3