MASTER’S COMPREHENSIVE EXAM ADVANCED LINEAR ALGEBRA AUGUST 1996

1.a) Find a basis for the nullspace V ⊂ R^{4} of the matrix (one row!)

A = [ 1 1 1 1 ] .

b) Find an orthogonal basis for the same nullspace V ⊂ R^{4}.

2. Given are two n × 1-column vectors u, v ∈ R^{n}. Then A = I + u v^{T} is an n × n-matrix, provided that the transpose v^{T} is regarded as a 1 × n-matrix. Show that A is invertible if and only if v^{T} u ≠ −1. Can you find an expression for A^{−1} similar to that for A?

3. Consider the eigenvalues of a real 2 × 2 matrix A = [ \begin{matrix}a & b \\ c & d \end{matrix} ] . Show that a sufficient condition for A to have real eigenvalues is that the entries b, c have the same signs (the result can be strengthened slightly).

4. Suppose a linear transformation T : V → V of an n-dimensional complex vectorspace V has n distinct eigenvalues λ_{1}, …, λ_{n}, λ_{i} ≠ λ_{j} if i ≠ j. Find an expression, in terms of eigenvectors of T, for a cyclic vector v ∈ V for T, i.e., such that {v, T v, T^{2}v, …, T^{n−1}v} is a basis of V.

5. Given is an n × n real symmetric matrix A with the special condition that its characteristic polynomial can be factored in the form p(λ) = det(λI − A) = λ^{k}q(λ), where q(λ) is a polynomial of degree n − k such that q(0) ≠ 0. Prove that the rank of A is n − k. Your proof should not depend on the Principal Axes Theorem. HINT: Consider an orthonormal basis B of R^{n} that extends an orthonormal basis of the nullspace of A. Then compute the characteristic polynomial p(λ) from the matrix representation of the linear transformation corresponding to A in the basis B.