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
∧N
A = 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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