Fall 2016 Problem 3: Show that every linear map A : ℝ^{3} → ℝ^{3} has both a 1-dimensional invariant subspace and a 2-dimensional invariant subspace.

Fall 2015 Problem 3: What is the smallest possible n for which there is an n by n real matrix M which has both:

(1) the rank of M^{2} is smaller than the rank of M,

(2) M leaves infinitely many length one vectors fixed.

Spring 2016 Problem 5 Find all possible Jordan canonical forms for a matrix A = T ((123)) if T is a two dimensional complex representation of the symmetric group S_{3}.

Spring 2015 Problem 1: Show that if M is a nondiagonalizable complex matrix and M^{n} is diagonalizable then det(M)=0.

Fall 2017 Problem 3: Let M_{n}(ℝ) denote the ring of n × n matrices over ℝ, and consider a (possibly non-unital) ring homomorphism f : M_{n+1}(ℝ) → M_{n}(ℝ). Can f be nonzero?