Master’s Comprehensive Exam Advanced Linear Algebra May 1995

1. Suppose T : V → W is a linear map between finite dimensional vector spaces over any field. (a) What is the matrix A_{T} of T with respect to bases of V and W (give definition).

(b) Prove that T is invertible in A_{T} is invertible.

(c) Prove the converse of (b).

(d) If V = W = \{ real polynomials of degree ≤ n\} and T = \frac{d}{dx} then five the matrix of T

with respect to your favorite basis.

2. For (a) and (b) consider the 2 × 2 matrix

B = ( \begin{matrix}1 & 5 \\ 0 & −1 \end{matrix} ) .

(a) Find the Jordan form B.

(b) Find B^{100}.

(c) How many similarity (i.e. conjugacy) classes of nilpotent 5 × 5 matrices are there?

3. Suppose p(x) is a polynomial with complex coefficients and A is an n × n matrix. (a) Prove that if λ is an eigenvalue of A then p(λ) is an eigenvalue of P (A).

(b) For which of the following polynomials is p_{k}(C) invertible for

C = ( \begin{matrix}1 & 1 & 0 \\ 0 & −1 & 3 \\ 0 & 0 & 1 \end{matrix} )?

p_{1}(x) = x^{8} + x^{6} + x^{4} + x^{2} + 1, p_{2}(x) = x^{5} + x^{4} + x^{3} + x^{2} + x + 1, p_{3}(x) = x^{20} + x^{10} + 1. Explain your answer.

4. Suppose S, T : V → V are linear transformations. Prove: If ST = T S then the eigenspaces for S are invariant under T (i.e. If V_{λ} is an eigenspace for S, then T (V_{λ}) ⊆ V_{λ}).