This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**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 α 1 ,...,α n of A such that: (i) α 1 e 1 ,...,α n e n form a basis of M ; (ii) α i divides α 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....

View
Full
Document