1. Given vector spaces U, V, W and linear transformations T : U - V and S : V - W we can define a new linear transformation R : U -> W such that R(u) = S(T(u)) for uEU. (a) Let V = C"(R) be the vector space of infinitely differentiable functions. Let T : V - V be the linear transformation defined by d' f df T()= "dx2 dx and let S : V - R be defined by s ( 1) = J's(2 ) de . Define the new linear transformation R = SoT (the o symbol means "compose"). This composition is defined so that R(f) = S(T(f)). Compute R(cos(x)), R(2e-) and R(x3). (b) Is it possible to compose S and T the other way around? In other words, is To S defined? Why or why not? (c) Let A E Mnxm (R) and let BE Mmxx(R). If T(x) = Ax and S(y) = By. Which of the two compositions SoT or To S is defined? Find a formula for the one that is defined in terms of matrix multiplication (HW 2 problem 2 is related). (d) Let T : V - W and S : W - V be linear transformations. If (So T) (x) = x and (To S)(y) = y for all a E V and all y E W, then we say that S is the inverse of T and write S = T-1. If V = W = " and T(x) = Ax for invertible A, then what is a formula for the linear transformation S(y) = T-(y)