MA554_JAN10 - n such that there exists an n by n matrix A...

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QUALIFYING EXAMINATION January 2010 MA 554 1. (15 points) Let R be a ring (commutative, with identity), M an R -module, and N a submodule of M . Write ι : N , M for the natural inclusion map and - * = Hom R ( - ,R ) for R -duals. (a) Show that if M/N is free, then ι * : M * N * is surjective. (b) Give an example showing that the assumption of freeness is needed in part (a). 2. (17 points) Let M be a finitely generated Z -module (i.e., a finitely generated Abelian group). (a) Let r denote the rank of M (i.e., M Z r T with T a torsion module). Show that r is the maximal number of linearly independent elements in M . (b) Let N be a (necessarily finitely generated) submodule of M . Show that if M/N is a torsion module, then M and N have the same rank. 3. (16 points) Consider the matrix A = 0 1 0 x 3 + x x + 1 - x x 2 - x x 0 0 0 - 1 0 - x 3 - x - 1 x 3 - x 2 + x x 3 - x 2 x 2 0 0 with entries in the polynomial ring R = Q [ x ]. Determine the dimension of the cokernel of A , considered as a vector space over Q . (Recall that A defines an R -linear map R 5 -→ R 4 and that every R -module is a vector space over Q via the inclusion Q R .) 4. (11 points) Determine all positive integers
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Unformatted text preview: n such that there exists an n by n matrix A with coecients in Q satisfying A 3 = 2 I n . (Here I n denotes the n by n identity matrix.) 5. (14 points) Consider the elementary Jordan matrix A = 1 1 0 of size n by n over a eld K . Determine the Jordan canonical form of A 2 . 6. (12 points) Let R be a domain and A an n by n matrix with entries in R , where n 2. Prove that det(adj( A )) = (det( A )) n-1 . (Recall that adj( A ) is the n by n matrix whose ( i,j )-entry is (-1) i + j times the determinant of the matrix obtained from A by deleting row j and column i .) 7. (15 points) Let V be a nite-dimensional vector space over C . Show that f is a symmetric bilinear form of rank at most 2 on V if and only if there exist and in V * such that f ( x,y ) = ( x ) ( y ) + ( y ) ( x ) for every x and y in V ....
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