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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 e 1 ;:::;e s Q D f 1 ;:::;f n P 1 A : (8.4.6) According to Lemma 8.4.11 , if we dene f v 1 ;:::;v n g by v 1 ;:::;v n D f 1 ;:::;f n P 1 and f w 1 ;:::;w s g by w 1 ;:::;w s D e 1 ;:::;e s Q; 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 w 1 ;:::;w s D v 1 ;:::;v n A D d 1 v 1 ;:::;d s v s :...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Linear Independence

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