Unformatted text preview: ˚ R as R –modules. In the remaining exercises, R denotes a principal ideal domain. 8.4.5. Let M be a a free module of rank n over R . Let N be a submodule of M . Suppose we know that N is ﬁnitely generated (but not that N is free). Adapt the proof of Theorem 8.4.12 to show that N is free. 8.4.6. Let V and W be free modules over R with ordered bases .v 1 ;v 2 ;:::;v n / and .w 1 ;w 2 ;:::;w m / . Let ' W V ! W be a module homomorphism. Let A D .a i;j / be the m –by– n matrix whose j th column is the coordinate vector of '.v j / with respect to the ordered basis .w 1 ;w 2 ;:::;w m / , '.v j / D X i a i;j w j : Show that for any element P j x j v j of M , '. X j x j v j / D Œw 1 ;:::;w m ŁA 2 6 4 x 1 : : : x n 3 7 5 :...
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 Fall '08
 EVERAGE
 Algebra, Matrices, Vector Space, Ring, Let, Principal ideal domain, R. Let

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