ABSTRACT
This chapter gives a rapid overview of the algebraic systems (such as groups, rings, fields, vector spaces, and algebras) that appear later in the book. After giving the axioms defining each of these systems and some basic examples, we describe some constructions (such as subspaces, product spaces, and quotient spaces) for building new systems from old ones. Then we discuss homomorphisms, which are structure-preserving maps between algebraic systems. The chapter concludes with a review of linear independence, spanning, basis, and dimension in the context of vector spaces over a field.
