MA554_JAN11

MA554_JAN11 - LINEAR ALGEBRA COMPREHENSIVE EXAM JAN, 2011...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 2

MA554_JAN11 - LINEAR ALGEBRA COMPREHENSIVE EXAM JAN, 2011...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online