We begin our investigation of linear algebra by studying vectors, the basic components of much of the theory and applications of linear algebra. Historically, vectors were not the ﬁrst objects in linear algebra to be studied intensively. The solution of systems of linear equations provided much of the early motivation for the subject, with determinants arising as an early topic of interest. Anyone reading this book has probably worked with linear systems and may even have encountered determinants in this context. We will investigate these topics fully in Chapters 2 and 4.