Mathematical Methods for Physicists: A concise introduction - Site Map

5
Linear vector spaces
Linear vector space is to quantum mechanics what calculus is to classical
mechanics. In this chapter the essential ideas of linear vector spaces will be discussed.
The reader is already familiar with vector calculus in three-dimensional
Euclidean space E 3 (Chapter 1). We therefore present our discussion as a generalization
of elementary vector calculus. The presentation will be, however, slightly
abstract and more formal than the discussion of vectors in Chapter 1. Any reader
who is not already familiar with this sort of discussion should be patient with the
®rst few sections. You will then be amply repaid by ®nding the rest of this chapter
relatively easy reading.
Euclidean n-space E n
In the study of vector analysis in E 3 , an ordered triple of numbers (a 1 , a 2 , a 3 ) has
two di€erent geometric interpretations. It represents a point in space, with a 1 , a 2 ,
a 3 being its coordinates; it also represents a vector, with a 1 , a 2 , and a 3 being its
components along the three coordinate axes (Fig. 5.1). This idea of using triples of
numbers to locate points in three-dimensional space was ®rst introduced in the
mid-seventeenth century. By the latter part of the nineteenth century physicists
and mathematicians began to use the quadruples of numbers (a 1 , a 2 , a 3 , a 4 )as
points in four-dimensional space, quintuples (a 1 , a 2 , a 3 , a 4 , a 5 ) as points in ®vedimensional
space etc. We now extend this to n-dimensional space E n , where n is a
positive integer. Although our geometric visualization doesn't extend beyond
three-dimensional space, we can extend many familiar ideas beyond three-dimensional
space by working with analytic or numerical properties of points and
vectors rather than their geometric properties.
For two- or three-dimensional space, we use the terms `ordered pair' and
`ordered triple.' When n > 3, we use the term `ordered-n-tuplet' for a sequence
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