## 3rd Midterm

## Spring 2001

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) = x^{3} + 3x^{2} - 4x - 12 and
g(x) = x^{3} + 2x^{2} - 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 I_{n} - 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 A^{2} = 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^{-1}AX is equal to the
Jordan normal form of A.