Master’s Comprehensive Exam Advanced Linear Algebra January 1996

1. Let A = [ \begin{matrix}1 & 2 & −1 & 1 \\ −1 & −2 & 2 & 1 \\ 2 & 4 & 1 & 9 \end{matrix} ] Find the bases for the following:

(a) The row space of A.

(b) The column space of A.

(c) The nullspace of A.

(d) The left nullspace of A (i.e. the set of all x such that xA = 0.)

2. Let A = [ \begin{matrix}0 & 1 & 1 \\ −1 & 2 & 1 \\ −1 & 1 & 2 \end{matrix} ].

(a) Find the characteristic polynomial c(x) and the polynomial m(x) of A.

(b) Find the eigenvalues of A and a basis for each eigenspace.

(c) Determine whether or not A is diagonalizable; if it is diagonalizable, find an invertible

matrix P and a diagonal matrix D such that P^{−1}AP = D.

3. Let V be a finite dimensional real vector space with an inner product ⟨, ⟩. Let W be a proper subspace of V. Let W^{⊥} = {v ∈ V |⟨v, w⟩ = 0∀w ∈ W }. Prove that V = W ⊕ W^{⊥}.

4. Let V be a vector space over a field F. Let V^{∗} be its dual space.

(a) Suppose T : V → V is a linear transformation. Define the transpose T^{∗} : V^{∗} → V^{∗}.

(b) Now suppose V has a finite basis B = {v_{1}, ..., v_{n}}. Define the dual basis B^{∗} for V^{∗}.

(c) Suppose A is the matrix of T with respect to B. What is the matrix of T^{∗} with respect to B^{∗}? Prove your answer.

5. Suppose A and B are complex n × n matrices having the same characteristic polynomial and the same minimal polynomial.

(a) Prove that if n = 3, then A and B are similar over C.

(b) Is this true for n = 4? Give a proof or a counterexample.

(c) Answer (a) and (b) for real matrices and similarity over R.