Math 322

3rd Midterm

Spring 2001

Dr. Wilson

 

1. Let R be a commutative ring with 1. Let a, b e R. Show that

is an ideal of R.

2. a) Find the GCD(f(x), g(x)) where

f(x) = x3 + 3x2 - 4x - 12 and

g(x) = x3 + 2x2 - x - 2

b) Find polynomials s(x) and t(x) such that

GCD(f(x), g(x)) = s(x)f(x) + t(x)g(x)

3. Compute det (x In - A) where

 

4. Let V be a vector space over F, a field of scalars. Let T be a linear operator on V , let W be a T-invariant subspace of V, and let

Show that the conductor of alpha into W by T,

is an ideal of F[x].

5. Let

Find all the powers of A.

6. Let T be the transformation whose matrix representation with respect to the standard basis is

Find a basis for which the matrix representation of T is in Jordan normal form, and find the Jordan normal form.

7. Let A be a matrix with entries from a field F, and suppose that the characteristic polynomial of A has all its roots in F. Prove that if the characteristic polynomial is square free, that the matrix is diagonalizable and that the minimal polynomial is the characteristic polynomial. Are the converses of these two assertions true? If not give a counterexample.

8. A matrix A is called idempotent if A2 = A. Find the Jordan normal forms of all idempotent matrices.

9. Let

a) Find the Jordan normal form for A.

b) Find a matrix X such that X-1AX is equal to the Jordan normal form of A.