TABLE OF CONTENTS After matrix theory became established toward the end of the nineteenth century, it was realized that many mathematical entities that were considered to be quite different from matrices were in fact quite similar. For example, objects such as points in the plane R 2, points in 3-space R 3, polynomials, continuous functions, and differentiable functions (to name only a few) were recognized to satisfy the same additive properties and scalar multiplication properties given in 3.2 for matrices. Rather than studying each topic separately, it was reasoned that it is more efficient and productive to study many topics at one time by studying the common properties that they satisfy. This eventually led to the axiomatic definition of a vector space.
A vector space involves four things-two sets V and F, and two algebraic operations called vector addition and scalar multiplication.
V is a nonempty set of objects called vectors. Although V can be quite general, we will usually consider V to be a set of n-tuples or a set of matrices.
F is a scalar field-for us F is either the field R of real numbers or the field C of complex numbers.
Vector addition (denoted by x + y ) is an operation between elements of V.
Scalar multiplication (denoted by ? x) is an operation between elements of F and V.
The formal definition of a vector space stipulates how these four things relate to each other. In essence,...
