Problems: Linear Algebra

In the following R denotes the field of real numbers while C denotes the field of complex numbers. In general, U, V , and W denote vector spaces. The set of all linear transformations from V into W is denoted by L(V, W) , while L(V) denotes the set of linear operators on V. For a linear transformation T , the null space of T (also known as the kernel of T) is denoted by null T , while the range space of T (also known as the image of T), is denoted by range T. Problem 0. Let V be a finite-dimensional vector space and let T be a linear operator on V. Suppose that T commutes with every diagonalizable linear operator on V. Prove that T is a scalar multiple of the identity operator. Problem 1. Let V and W be vector spaces and let T be a linear transformation from V into W. Suppose that V is finite-dimensional. Prove rank(T) + nullity(T) = dim V. Problem 2. Let A and B be n × n matrices over a field F (1) Prove that if A or B is nonsingular, then AB is similar to BA. (2) Show that there exist matrices A and B so that AB is not similar to BA. (3) What can you deduce about the eigenvalues of AB and BA. Prove your answer.