1. The problem statement, all variables and given/known data Let T: C∞(R)→C∞(R) be given by T(f) = f'''' where T sends a function to the fourth derivative. a) Find a basis for the 0-eigenspace. b) Find a basis for the 1-eigenspace. 3. The attempt at a solution I just want to verify my thought process for this problem. For a), finding the basis for the 0-eigenspace, essentially I needed to find a basis for the vectors v in V such that T(v) = 0v . So, would the basis for this 0-eigenspace be all polynomials in P3? If you solve the fourth derivative of any polynomial in P3, you will get 0. As for b), when finding the basis for the 1-eigenspace, we need to find a basis for the vectors v in V such that T(v) = 1v, or that after solving the fourth derivative, you get a function that is equal to 1? Is this the correct logic? So would the basis for the 1-eigenspace be any polynomial in P4? Thanks much for your help.