2.11. Tensor products 33 called the nth symmetric power, respectively exterior power of V . If {vi} is a basis of V , can you construct a basis of SnV, ∧nV ? If dimV = m, what are their dimensions? (e) If k has characteristic zero, find a natural identification of SnV with the space of T ∈ V n such that T = sT for all transpositions s, and find a natural identification of ∧nV with the space of T ∈ V n such that T = −sT for all transpositions s. (f) Let A : V → W be a linear operator. Then we have an op- erator A n : V n → W n and its symmetric and exterior powers SnA : SnV → SnW , ∧nA : ∧nV → ∧nW which are defined in an obvious way. Suppose that V = W and that dim V = N, and that the eigenvalues of A are λ1,...,λN. Find Tr(SnA) and Tr(∧nA). (g) Show that ∧NA = det(A)Id, and use this equality to give a one-line proof of the fact that det(AB) = det(A) det(B). Remark 2.11.4. Note that a similar definition to the above can be used to define the tensor product V ⊗A W , where A is any ring, V is a right A-module, and W is a left A-module. Namely, V ⊗A W is the abelian group which is the quotient of the group V • W freely generated by formal symbols v ⊗ w, v ∈ V , w ∈ W , modulo the relations (v1 + v2) ⊗ w − v1 ⊗ w − v2 ⊗ w, v ⊗ (w1 + w2) − v ⊗ w1 − v ⊗ w2, va ⊗ w − v ⊗ aw, a ∈ A. Exercise 2.11.5. Let K be a field, and let L be an extension of K. If A is an algebra over K, show that A ⊗K L is naturally an algebra over L. Show that if V is an A-module, then V ⊗K L has a natural structure of a module over the algebra A ⊗K L. Problem 2.11.6. Throughout this problem, we let k be an arbi- trary field (not necessarily of characteristic zero and not necessarily algebraically closed). If A and B are two k-algebras, then an (A, B)-bimodule will mean a k-vector space V with both a left A-module structure and a right B-module structure which satisfy (av) b = a (vb) for any v ∈ V , a ∈ A, and b ∈ B. Note that both the notions of “left A-module”

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