Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

MATRICES AND VECTOR SPACES


We reiterate that the vectorx(a geometrical entity) is independent of the basis



  • it is only the components ofxthat depend on the basis. We note, however,


that given a set of vectorsu 1 ,u 2 ,...,uM,whereM=N,inanN-dimensional


vector space, theneitherthere exists a vector that cannot be expressed as a


linear combination of theuior, for some vector that can be so expressed, the


components are not unique.


8.1.2 The inner product

We may usefully add to the description of vectors in a vector space by defining


theinner productof two vectors, denoted in general by〈a|b〉, which is a scalar


function ofaandb. The scalar or dot product,a·b≡|a||b|cosθ,ofvectors


in real three-dimensional space (whereθis the angle between the vectors), was


introduced in the last chapter and is an example of an inner product. In effect the


notion of an inner product〈a|b〉is a generalisation of the dot product to more


abstract vector spaces. Alternative notations for〈a|b〉are (a,b), or simplya·b.


The inner product has the following properties:

(i)〈a|b〉=〈b|a〉∗,
(ii)〈a|λb+μc〉=λ〈a|b〉+μ〈a|c〉.

We note that in general, for a complex vector space, (i) and (ii) imply that


〈λa+μb|c〉=λ∗〈a|c〉+μ∗〈b|c〉, (8.13)

〈λa|μb〉=λ∗μ〈a|b〉. (8.14)

Following the analogy with the dot product in three-dimensional real space,

two vectors in a general vector space are defined to beorthogonalif〈a|b〉=0.


Similarly, thenormof a vectorais given by‖a‖=〈a|a〉^1 /^2 and is clearly a


generalisation of the length or modulus|a|of a vectorain three-dimensional


space. In a general vector space〈a|a〉can be positive or negative; however, we


shall be primarily concerned with spaces in which〈a|a〉≥0 and which are thus


said to have apositive semi-definite norm.Insuchaspace〈a|a〉= 0 impliesa= 0.


Let us now introduce into ourN-dimensional vector space a basisˆe 1 ,ˆe 2 ,...,ˆeN

that has the desirable property of beingorthonormal(the basis vectors are mutually


orthogonal and each has unit norm), i.e. a basis that has the property


〈eˆi|ˆej〉=δij. (8.15)

Hereδijis theKronecker deltasymbol (of which we say more in chapter 26) and


has the properties


δij=

{
1fori=j,
0fori=j.
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