Master’s Comprehensive Exam Advanced Linear Algebra January 1997

Do only four of the five problems.

1. Consider the vector space V of all polynomials of degree less than or equal to 5. For the

linear map T = \frac{d^{2}}{dx^{2}} + 1 find the Jordan form AND give a basis of V with respect to which the matrix is the Jordan form.

2. Let A = ( \begin{matrix}0 & 0 & 1 & 1 \\ 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 2 & 0 \end{matrix} )

Find a polynomial P (x) so that A^{−1} = P (A). (Hint: Think about characteristic and/or the minimal polynomial.)

3. Let V be a finite dimensional vector space over an arbirtary field. Let T : V → V be an arbitrary linear transformation.

(a) Give the definition of the matrix of T with respect to a basis.

(b) Prove that the matrices of T with respect to two different bases are similar (i.e. conjugate).

4. Suppose a 7 × 7 matrix A has minimal polynomial m(x) = (x^{3} − 1)^{2}.

(a) Give the possible rational canonical forms which A can have when the base field is Q.

(b) Give the possible rational canonical forms which A can have when the base field is C.

5. Let V be an inner product space. For any vector w ≠ 0 recall that the reflection in the

hyperplane orthogonal to w is the linear transformation T_{w}(x) = −\frac{2(x, w)}{(w, w)}w.

(a) Show that the inverse of a reflection is a reflection.

(b) Show that a reflection is an orthogonal linear transformation.

(c) Determine the minimal polynomial, the determinant and the Jordan form of an arbirtary reflection.

(d) Is the composition of two reflections a reflection?