1. Suppose T is a linear transformation such that Find the matrix of T. In other words, find the matrix A such that T() = AT for each a in R3. 2. Any cubic polynomial can be written as ao p(x) = do + aIT + a212 + a3x3 = [ do a1 42 a3 ] for scalars a3 do, a1, a2, a3. Observe that differentiating p(x) corresponds to a linear map on the coefficients, in other words do P'(x) = Tdiff al a2 a3 for an appropriate linear map Tdiff. (a) Find the matrix Adiff that represents Tdiff (b) Similarly, integrating an arbitrary quadratic function q(x) = bo + bix + 62x2 - bi|2 back- (2 ([ 2 ])) : corresponds to an appropriate linear map Tint. Find the matrix Aint that represents Tint. (c) Compute the products Aint Adiff and Adiff Aint. Optional: how can these products be interpreted in the context of "differentiating then integrating" us. "integrating then differentiating"? 3. Let A = 12. Find the eigenvalues and eigenvectors of the following matrices. (a) A (b) A2 (c) A-1 (d) A + 41 where I is the 2 x 2 identity matrix. 4. Let v = and A = vv. Find the eigenvalues and eigenvectors of A. Hint: one of the eigenvalues for A has two linearly independent eigenvectors