What is a "cohomological type" automorphic representation?

Linear Algebra L(p(x)) = p'(0) + 2p'(x) maps P3 into P2. Find the Matrix representation of L w/respect to ...?

• L(p(x)) = p'(0) + 2p'(x) maps P3 into P2. Find the Matrix representation of L w/respect to the ordered bases [x^2, x, 1] and [1+x, 1-x] A = Use A to find the coordinate vector L(p(x)) with respect to the ordered basis [1+x, 1-x]. p(x) = -x^2 + 2x +7. Coordinate vector L(p(x)) is: While the answers would be nice, I would much rather someone take the time to explain(in detail) how to go about solving this type of problem. This is for myself and a group of study partners and at the moment we're completely lost.

• Answer:

  a) Substitute x^2, x, 1 into L (in this order): L(x^2) = 2x {at x = 0} + 2 * 2x = 4x L(x) = 1 {at x = 0} + 2 * 1 = 3 L(1) = 0 {at x = 0} + 2 * 0 = 0. Now, write these in terms of the basis {1 + x, 1 - x}: L(x^2) = 4x = 2(1 + x) + (-2)(1 - x) L(x) = 3 = (3/2)(1 + x) + (3/2)(1 - x) L(1) = 0 = 0(1 + x) + 0(1 - x). Hence, the matrix A equals [2 3/2 0] [-2 3/2 0]. ----------- b) Since -x^2 + 2x + 7 has coordinate vector (-1, 2, 7)^t with respect to {x^2, x, 1}, L(-x^2 + 2x + 7) has coordinate vector with respect to {1 + x, 1 - x}: [2 3/2 0][-1]...[1] [-2 3/2 0][2].=.[5]. ...............[7] (That is, L(-x^2 + 2x + 7) = 1(1 + x) + 5(1 - x).) Double check: L(-x^2 + 2x + 7) = (-2x + 2) {for x = 0} + 2(-2x + 2) = -4x + 6. Indeed, 1(1 + x) + 5(1 - x) = -4x + 6. I hope this helps!