Consider the vector space

Question 5. Consider the vector space V = span ({cos(x)e", sin(x)e , xcos(x)el, rsin(x)e }) C D() (RR) and the map T : V - D() (R) defined by T(f) = f + f' for all fe V. You can assume that T is a well-defined linear map. a) What is the dimension of V? Justify your answer. (5 points) b) Show that Im(T) C V. (3 points) c) Since Im(T) C V, the linear map S : V - V defined by S(f) = f + f' for all f E V is well- defined. Show that S is invertible and compute S-(a cos(x)er + bsin(x)e# + crcos(r)ed + dr sin(x)e) for all a, b, c, de R. (5 points)