By Spencer, Donald Clayton; Nickerson, Helen Kelsall

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Let 01 T(A) V, = V -> W, T 1 be linear transformations and T'(A) suppose T(A) = for each A e T' (A) for each v. 11), and let V be the subset of functions having continuous derivatives. subspace of W. what is degrees ~ 1m T? n, 8. subspace of Show that If v. T is linear. What is T(Un)? T : V -> W be linear, and let Sha~ ker T, Un is the subspace of polynomials of what is Let V is a linear T : V -> W transform each function in V Let into its derivative. 1) into a parallel of T(U) U in in T(V). ) 9.

2, in the case W = V, Bi = M(TS) Ai, = satisfies M(T)M(S) where the product on the right is the usual matr:Llc multiplication. 8. Show that a product is defined in E(V) by setting 49 [S, Tl = ST - TS, for any pair of elements S, T in E(V). Show S, T, Y e A(V), a e R): that the "bracket product" satisfies (for any [S + T, Y] = [S, Y] + [T, Y] , a[S, Tl [aS, T] = [S, aT] [S, S] = ~ , [S, Tl - [T, S] [[S, T], Y] + [[T, Y], S] + [[Y, SJ, Tl=~. The last result is called the "Jacobi identity" and shows that this product is not associative.

44 Axiom 11. of elements of a(ST) = (aS)T = S(aT) E(V), Axiom 12. for each pair for each real number S, T a. There is a (unique) element I in E(V) such that IT for each TI T T e E(V). Proof. 5. Axiom 12, note that the function linear and therefore in E(V). 6 is The remaining axioms are verified by computation, using the definitions of operations in E(V): Axiom 1o: (S(T 1 + T2 ))(X) S((T 1 + T2 )(X)) = S(T 1 (X) + T2 {X)) S(T 1 {X)) + S(T 2 (X)) = (ST 1 ) (X) + (ST 2 ) (X) (ST 1 ((T 1 + T2 )S)(X) + ST 2 )(X).