Master’s Comprehensive Exam Advanced Linear Algebra August 1995

1. (20 pts.) Let the nonzero linear transformation T : R^{2} → R^{2} be given by T (x⃗) = Ax⃗, for some 2 × 2 matrix A. For each of the conditions (a) and (b) below, give an example of a nonzero matrix A so that T will satisfy the given condition. Give all possible values of rk(T) in each case. Justify your answers.

(a) kerT ⊆ imT

(b) imT ⊆ kerT

2. (25 pts.) Let A and B be two n × n real matrices satisfying that AB = −BA (In this case, we say that A and B anticommute.)

(a) If n is odd, show that one of A or B is not invertible.

(b) Show that this can be false if n is even.

3. (25 pts.) Let V be finite-dimensional real or complex inner product space with inner product ⟨·, ·⟩. Let T : V → V be a self-adjoint linear transformation and let u and v be eigenvectors corresponding to different eigenvalues λ and μ. Prove that u and v are orthogonal.

4. (30 pts.) Let A be the matrix

A = [ \begin{matrix}−2 & −1 & 2 \\ −5 & −6 & 16 \\ −3 & −3 & 8 \end{matrix} ] .

The characteristic polynomial of A is h(x) =det(xI_{3} − A) = (x + 1)^{2}(x − 2).

(a) Find an invertible matrix P and a matrix J in Jordan canonical form so that J = P^{−1}AP.

(b) Find the minimal polynomial of A. Explain your reasoning.