Explore BrainMass

# Linear Algebra : Modules, Linear Operators, Characteristic and Minimal Polynomials, Generators, Abelian Groups and Annihilators

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

See the attachments.
Let F be a field and . Then is an n - dimensional vector space
over F. Define a function by .

(a) Show that T is a linear operator.

(b) Find the characteristic and minimal polynomials for T, with explanation. (For the characteristic polynomial, recall that you will need to choose a basis for , find the matrix of T relative to that basis, and find the characteristic polynomial of the matrix.)

(c) By Example 47, we can use T to make into a module over the polynomial ring . Show that is cyclic by giving a generator for M, with explanation. Find the (as defined in Exercise 38).

Example 47) Let F be a field and Let be the polynomial ring over F. What does it mean for an Abelian group M to be a module over ?
First, note that the "constant polynomials" in form a copy of the
field F. Thus scalar multiplication of on M gives scalar multiplication of F on M, which makes M a vector space over F.

Second, consider the function defined by . (That is, equals the result of multiplying the polynomial x by v, using scalar multiplication of on M.) For any and , the requirements (M, where M are the properties of a module-see below) on the scalar multiplication imply and . That is, T is a linear operator on the F-vector space M.

Then given any and , the properties (M) imply

Conversely, suppose that M is any F-vector space and is a linear operator on M. We can define a scalar multiplication on on M by . This satisfies the requirements, making M an -module.

Exercise 38) Let M be an R-module. The annihilator of M in R is defined by
. Show that is an ideal of R.

Definition: Let R be a commutative ring with identity 1. A module over R (also called an
R - module) is an Abelian group M (operation written +) together with a
scalar multiplication which associates to each and an element rm
of M; and , we require: