Unformatted text preview: A D PAQ D diag .d 1 ;d 2 ;:::;d s /: We will see below that all the d j are necessarily nonzero. Again, according to Proposition 8.4.6 , P and Q can be chosen so that d i divides d j whenever i ± j . We rewrite ( 8.4.5 ) as Œe 1 ;:::;e s ŁQ D Œf 1 ;:::;f n ŁP ± 1 A : (8.4.6) According to Lemma 8.4.11 , if we deﬁne f v 1 ;:::;v n g by Œv 1 ;:::;v n Ł D Œf 1 ;:::;f n ŁP ± 1 and f w 1 ;:::;w s g by Œw 1 ;:::;w s Ł D Œe 1 ;:::;e s ŁQ; then f v 1 ;:::;v n g is a basis of F and f w 1 ;:::;w s g is a basis of N . By Equation ( 8.4.6 ), we have Œw 1 ;:::;w s Ł D Œv 1 ;:::;v n ŁA D Œd 1 v 1 ;:::;d s v s Ł:...
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Linear Independence, Lemma, Free module, Finitely generated module, FINITELY GENERATED MODULES

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