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College Algebra Exam Review 368

# College Algebra Exam Review 368 - ˚ R as R –modules In...

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378 8. MODULES In particular, d j is nonzero for all j , since f d 1 v 1 ; : : : ; d s v s g is a basis of N . n Exercises 8.4 8.4.1. Let R be a commutative ring with identity element and let M be a module over R . (a) Let A and B be matrices over R of size n –by– s and s –by– t re- spectively. Show that for OEv 1 ; : : : ; v n Ł 2 M n , OEv 1 ; : : : ; v n Ł.AB/ D .OEv 1 ; : : : ; v n ŁA/B: (b) Show that if f v 1 ; : : : ; v n g is linearly independent subset of M , and OEv 1 ; : : : ; v n ŁA D 0 , then A D 0 . 8.4.2. Let R be a PID. Adapt the proof of Lemma 8.4.4 to show that any submodule of a free R –module of rank n is free, with rank no more than n . 8.4.3. Prove Lemma 8.4.7 8.4.4. Let R denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. That is, a matrix is in R if, and only if, each row and each column of the matrix has only finitely many non–zero entries. Show that R is a non-commutative ring with identity, and that R Š
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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 co-ordinate 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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