18. (Dual Spaces) Let V and W be finite dimensional vector spaces over a field F. (a) Show that the set of all linear transformations from V into W, denoted by Hom(V, W), is a vector space over F, where we define vector addition as follows: (S+T)(v) = S(v) +T(v) (aS)(v) = aS(v), where S, TЄ Hom(V, W), a Є F, and v € V. (b) Let V be an F-vector space. Define the dual space of V to be V* = Hom(V, F). Elements in the dual space of V are called linear functionals. Let v₁,..., un be an ordered basis for V. If v = a₁₁ann is any vector in V, define a linear functional V → F by oi(v) = a. Show that the oi's form a basis for V*. This basis is called the dual basis of v₁,..., Un (or simply the dual basis if the context makes the meaning clear). (c) Consider the basis {(3, 1), (2,-2)} for R2. What is the dual basis for (R²)*? (d) Let V be a vector space of dimension n over a field F and let V** be the dual space V*. Show that each element vЄ V gives rise to an element A, in V** and that the map is an isomorphism of V with V**