2.11. Tensor products 31 is given by formal symbols v w, v V , w W , by the subspace spanned by the elements (v1 + v2) w v1 w v2 w, v (w1 + w2) v w1 v w2, av w a(v w), v aw a(v w), where v V, w W, a k. Exercise 2.11.2. Show that V W can be equivalently defined as the quotient of the free abelian group V W generated by v w, v V, w W by the subgroup generated by (v1 + v2) w v1 w v2 w, v (w1 + w2) v w1 v w2, av w v aw, where v V, w W, a k. The elements v w V W , for v V, w W are called pure tensors. Note that in general, there are elements of V W which are not pure tensors. This allows one to define the tensor product of any number of vector spaces, V1 ⊗· · · Vn. Note that this tensor product is associa- tive, in the sense that (V1 V2) V3 can be naturally identified with V1 (V2 V3). In particular, people often consider tensor products of the form V n = V · · · V (n times) for a given vector space V , and, more generally, E := V n (V ) m . This space is called the space of tensors of type (m, n) on V . For instance, tensors of type (0, 1) are vectors, tensors of type (1, 0) linear functionals (covectors), tensors of type (1, 1) linear operators, of type (2, 0) bilinear forms, tensors of type (2, 1) algebra structures, etc. If V is finite dimensional with basis ei, i = 1,...,N, and ei is the dual basis of V , then a basis of E is the set of vectors ei 1 · · · ei n ej1 · · · ejm,
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