This article looks at some of the conceptual difficulties that students have in a linear algebra course. An overview of previous research in this area is given, and the various theories that have been espoused regarding the reasons that students find linear algebra so difficult are discussed. Student responses to two questions testing the ability to prove that a set is a subspace of a vector space are examined in some detail. Some conjectures are made as to why students found the task so difficult. (Contains 1 note.)