Before defining a vector field, it is helpful to be precise about what
is meant by a *vector*. A *vector space* (or *linear
space*) is defined as a set, , that is closed under two algebraic
operations called *vector addition* and *scalar
multiplication* and satisfies several axioms, which will be given
shortly. The vector space used in this section is
, in which
the scalars are real numbers, and a vector is represented as a
sequence of real numbers. Scalar multiplication multiplies each
component of the vector by the scalar value. Vector addition forms a
new vector by adding each component of two vectors.

A *vector space* can be defined over any field
(recall
the definition from Section 4.4.1). The field
represents the *scalars*, and represents the *vectors*.
The concepts presented below generalize the familiar case of the
vector space
. In this case,
and
. In
the definitions that follow, you may make these substitutions, if
desired. We will not develop vector spaces that are more general than
this; the definitions are nevertheless given in terms of and
to clearly separate scalars from vectors. The *vector
addition* is denoted by , and the *scalar multiplication* is
denoted by . These operations must satisfy the following
axioms (a good exercise is to verify these for the case of
treated as a vector space over the field
):

- (
**Commutative Group Under Vector Addition**) The set is a commutative group with respect to vector addition, . - (
**Associativity of Scalar Multiplication**) For any and any , . - (
**Distributivity of Scalar Sums**) For any and any , . - (
**Distributivity of Vector Sums**) For any and any , . - (
**Scalar Multiplication Identity**) For any , for the multiplicative identity .

A *basis* of a vector space is defined as a set,
,,, of vectors for which every can be
uniquely written as a *linear combination*:

(8.7) |

for some . This means that every vector has a unique representation as a linear combination of basis elements. In the case of , a familiar basis is , , and . All vectors can be expressed as a linear combination of these three. Remember that a basis is not necessarily unique. From linear algebra, recall that any three linearly independent vectors can be used as a basis for . In general, the basis must only include linearly independent vectors. Even though a basis is not necessarily unique, the number of vectors in a basis is the same for any possible basis over the same vector space. This number, , is called the

To illustrate the power of these general vector space definitions, consider the following example.

is finite, forms a vector space over . It is straightforward to verify that the vector space axioms are satisfied. For example, if two functions and are added, the integral remains finite. Furthermore, , and all of the group axioms are satisfied with respect to addition. Any function that satisfies (8.8) can be multiplied by a scalar in , and the integral remains finite. The axioms that involve scalar multiplication can also be verified.

It turns out that this vector space is infinite-dimensional. One way
to see this is to restrict the functions to the set of all those for
which the Taylor series exists and converges to the function (these
are called *analytic functions*). Each
function can be expressed via a Taylor series as a polynomial that may
have an infinite number of terms. The set of all monomials, ,
, , and so on, represents a basis. Every continuous
function can be considered as an infinite vector of coefficients; each
coefficient is multiplied by one of the monomials to produce the
function. This provides a simple example of a *function space*;
with some additional definitions, this leads to a *Hilbert space*,
which is crucial in functional analysis, a subject that characterizes
spaces of functions [836,838].

The remainder of this chapter considers only finite-dimensional vector spaces over . It is important, however, to keep in mind the basic properties of vector spaces that have been provided.

Steven M LaValle 2012-04-20