SOME DUAL SPACE QUESTIONS 1. Let T : V ! W be an onto map of finite dimensional vector spaces. Prove that T : V ! W is one to one. 2. Let F be a field and T : M22 (F ) ! F be the trace map. For any matrix A 2 M22 (F ), we have the map A (B) = T (A B) a) Prove that this map is linear; i.e., that A 2 M22 (F ) . b) Let {E ij } be the standard basis of M22 (F ). Compute all four linear maps { E ij } in terms of the dual basis of M22 (F ); i.e., evaluate E ij (E kl ) for all k, l. c) Consider the linear map : M22 (F ) ! M22 (F ) defined (on the standard basis) by (E ij ) = E ij . Prove that this map is an isomorphism. 3. Let V be a finite dimensional vector space over a field F . Suppose T : V ! V is a linear map, and suppose W V is an invariant subspace for T ; i.e., T (W ) W . Any element of V gives rise to an element of W by restriction: if : V ! F then |W : W ! F is also linear; so we have a map : V ! W a) Prove that this map is linear and onto. b) Since T : W ! W , we have a dual map TW : W ! W , in addition to the usual dual map, TV : V ! V . Prove that TV = TW . 1