Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESWe reiterate that the vector x (a geometrical entity) is independent of the basis– it is only the components of x that depend on the basis. We note, however,that given a set of vectors u 1 , u 2 ,...,u M ,whereM ≠ N, inanN-dimensionalvector space, then either there exists a vector that cannot be expressed as alinear combination of the u i or, for some vector that can be so expressed, thecomponents are not unique.8.1.2 The inner productWe may usefully add to the description of vectors in a vector space by definingthe inner product of two vectors, denoted in general by 〈a|b〉, which is a scalarfunction of a and b. The scalar or dot product, a · b ≡|a||b| cos θ, ofvectorsin real three-dimensional space (where θ is the angle between the vectors), wasintroduced in the last chapter and is an example of an inner product. In effect thenotion of an inner product 〈a|b〉 is a generalisation of the dot product to moreabstract vector spaces. Alternative notations for 〈a|b〉 are (a, b), or simply a · 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 be orthogonal if 〈a|b〉 =0.Similarly, the norm of a vector a is given by ‖a‖ = 〈a|a〉 1/2 and is clearly ageneralisation of the length or modulus |a| of a vector a in three-dimensionalspace. In a general vector space 〈a|a〉 can be positive or negative; however, weshall be primarily concerned with spaces in which 〈a|a〉 ≥0 and which are thussaid to have a positive semi-definite norm. Insuchaspace〈a|a〉 = 0 implies a = 0.Let us now introduce into our N-dimensional vector space a basis ê 1 , ê 2 ,...,ê Nthat has the desirable property of being orthonormal (the basis vectors are mutuallyorthogonal and each has unit norm), i.e. a basis that has the property〈ê i |ê j 〉 = δ ij . (8.15)Here δ ij is the Kronecker delta symbol (of which we say more in chapter 26) andhas the properties{1 for i = j,δ ij =0 for i ≠ j.244