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 ):
A basis of a vector space is defined as a set, ,,, of vectors for which every can be uniquely written as a linear combination:
To illustrate the power of these general vector space definitions, consider the following example.
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