(a) Show that the set of all functions of the form f(x) = (ax2 + bx + c) ex for a, b, c, ∈ R forms a vector space. What is its dimension?
(b) Show that the derivative D[f(x)] = f′(x) defines an invertible linear transformation on this vector space, and determine its inverse.
(c) Generalize your result in part (b) to the infinite-dimensional vector space consisting of all functions of the form p(x)ex, where p(x) is an arbitrary polynomial.