College Algebra Exam Review 369

College Algebra Exam Review 369 - . Find a basis f v 1...

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8.5. FINITELY GENERATED MODULES OVER A PID, PART II. 379 8.4.7. Retain the notation of the previous exercise. By Proposition 8.4.6 , there exist invertible matrices P 2 Mat m .R/ and Q 2 Mat n .R/ such that A 0 D PAQ is diagonal, A 0 D PAQ D diag .d 1 ;d 2 ;:::;d s ;0;:::;0/; where s ± min f m;n g . Show that there is a basis f w 0 1 ;:::;w 0 m g of W such that f d 1 w 0 1 ;:::;d s w 0 s g is a basis of range .'/ . 8.4.8. Set A D 2 4 2 5 ² 1 2 ² 2 ² 16 ² 4 4 ² 2 ² 2 0 6 3 5 . Left multiplication by A defines a homomorphism ' of abelian groups from Z 4 to Z 3 . Use the diagonaliza- tion of A to find a basis f w 1 ;w 2 ;w 3 g of Z 3 and integers f d 1 ;:::d s g ( s ± 3 ), such that f d 1 w 1 ;:::;d s w s g is a basis of range .'/ . (Hint: Compute invertible matrices P 2 Mat 3 . Z / and Q 2 Mat 4 . Z / such that A 0 D PAQ is diagonal. Rewrite this as P ± 1 A 0 D AQ .) 8.4.9. Adopt the notation of Exercise 8.4.6 . Observe that the kernel of ' is the set of P j x j v j such that A 2 6 4 x 1 : : : x n 3 7 5 D 0: That is the kernel of ' can be computed by finding the kernel of A (in Z n ). Use the diagonalization A 0 D PAQ to find a description of ker .A/ . Show, in fact, that the kernel of A is the span of the last n ² s columns of Q , where A 0 D diag .d 1 ;d 2 ;:::;d s ;0;:::;0/ . 8.4.10. Set A D ± 2 5 ² 1 2 ² 2 ² 16 ² 4 4 ²
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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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