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 productWe 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 ,...,ˆeNthat 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.