This chapter is the most abstract of the book. You may skip it at first reading, and jump directly to Chapter 4. But make sure you return to it later. Matrix theory can be viewed from an algebraic viewpoint or from a geometric viewpoint — both are equally important. The theory of vector spaces is essential in understanding the geometric viewpoint.

Associated with every vector space is a set of scalars, used to define scalar multiplication on the space. In the most abstract setting these scalars are required only to be elements of an algebraic field. We shall, however, always take the scalars to be the set of complex numbers (complex vector space) or, as an important special case, the set of real numbers (real vector space).

A vector space (or linear space) V is a nonempty set of elements (called vectors) together with two operations and a set of axioms. The first operation is addition, which associates with any two vectors x, y ∈ V a vector x + y ∈ V (the sum of x and y). The second operation is scalar multiplication, which associates with any vector x ∈ V and any real (or complex) scalar α, a vector αx ∈ V. It is the scalar (rather than the vectors) that determines whether the space is real or complex.