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**Unformatted text preview: **LINEAR ALGEBRA COMPREHENSIVE EXAM JAN, 2011 Attempt all questions. Time 2 hrs (1) Let C be a commutative ring with identity, E be a finitely generated projective C-module, and u End C ( E ). (a) (2 pts) Define Tr( u ) (trace of u ). (b) (8 pts) Let F be another finitely generated projective mod- ule and v : E F , and w : F E be two linear maps. Prove that Tr( v w ) = Tr( w v ) . (2) Let L be a free module over a principal ideal domain A , and let M be a submodule of finite rank n . (a) (2 pts) Given x L define the content, c L ( x ), of x . (b) (10 pts) Prove that there exists a basis B of L , and n ele- ments e 1 ,...,e n of B , and corresponding elements 1 ,..., n of A such that: (i) 1 e 1 ,..., n e n form a basis of M ; (ii) i divides i +1 for 1 i n- 1. (c) (8 pts) Prove that every finitely generated module E over a principal ideal domain A is a direct sum of a finite number of cyclic modules....

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