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Unformatted text preview: . Find a basis f v 1 ;:::;v 4 g of Z 4 and integers f a 1 ;:::;a r g such that f a 1 v 1 ;:::;a r v r g is a basis of ker .A/ . (Hint: If s is the rank of the range of A , then r D 4 s . Moreover, if A D PAQ is the Smith normal form of A , then ker .A/ is the span of the last r columns of Q , that is the range of the matrix Q consisting of the last r columns of Q . Now we have a new problem of the same sort as in Exercise 8.4.8 .) 8.5. Finitely generated Modules over a PID, part II. The Invariant Factor Decomposition Consider a nitely generated module M over a principal ideal domain R . Let x 1 ;:::;x n be a set of generators of minimal cardinality. Then M is the homomorphic image of a free R module of rank n . Namely consider a free R module F with basis f f 1 ;:::;f n g . Dene an R module...
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 Fall '08
 EVERAGE
 Algebra, Matrices

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