12 1. Vecto r Spaces 1.2.6 Tenso r products. Le t V an d % be vector spaces over F. Let Y ® °tt be the set of all the (finite) forma l sum s £a -v }: ® u-, wher e dj G F, Vj G y an d u-e *%. W e define addition formally by Y*ajvj®uj+Hajvj®uj= L a jvj®up jeJx jeJ 2 jeJ x uJ2 and we define multiplication by scalars by a £ a . v . ® W . = £(aa.)v .®M .. With these definitions, Y ® 9/ i s a vector space over F. The tensor product Y ®^ is , by definition, the quotient of ^ ® ty£ by the subspace [^ ® $/]0 spanne d by the elements of the form a. (v j + v2)®w (vj ®w + v2®w), (1.2.9) b . v ® (wj + w2) ~ (v®^ + v®w2), c. a(v®w ) (av)®w, (av)® w —v® (aw), for all v, v. G y,w , UjEfy, an d a G F. In other words, Y ® ^ i s the space of formal sum s Y, a jvj ® M / modulo the equivalence relation generated by: a. (v t + v2) ® w = Vj ® w + v2 ® w, (1.2.10) b . v ® (MJ +U2) = w1 + v®w2, c. a (v ® w) = (av ) ® w = v ® (au). EXAMPLE: lfY = F[x] and $/ = F[y], and we define the mapI of r®T^intoF[ c,) ]by: (1.2.11)D : ^ / / ^ ^ W ^ ^ a ^ / ^ W e F ^ ] , 7 7 then all the elements of [Y ® ^ ]0 are mapped to zero, so that all the elements in an equivalence class modulo [Y ® ^ ]0 ar e mapped to the same polynomial. Fo r example, every formal su m in Y ® °l/ that is
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