Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.1 VECTOR SPACES

In the above basis we may express any two vectorsaandbas

a=

∑N

i=1

aiˆei and b=

∑N

i=1

bieˆi.

Furthermore,in such an orthonormal basiswe have, for anya,

〈ˆej|a〉=

∑N

i=1

〈ˆej|aieˆi〉=

∑N

i=1

ai〈ˆej|eˆi〉=aj. (8.16)

Thus the components ofaare given byai=〈ˆei|a〉. Note that this isnottrue

unless the basis is orthonormal. We can write the inner product ofaandbin

terms of their components in an orthonormal basis as

〈a|b〉=〈a 1 ˆe 1 +a 2 eˆ 2 +···+aNˆeN|b 1 eˆ 1 +b 2 ˆe 2 +···+bNeˆN〉

=

∑N

i=1

a∗ibi〈ˆei|eˆi〉+

∑N

i=1

∑N

j=i

a∗ibj〈eˆi|ˆej〉

=

∑N

i=1

a∗ibi,

where the second equality follows from (8.14) and the third from (8.15). This is

clearly a generalisation of the expression (7.21) for the dot product of vectors in

three-dimensional space.

We may generalise the above to the case where the base vectorse 1 ,e 2 ,...,eN

arenotorthonormal (or orthogonal). In general we can define theN^2 numbers

Gij=〈ei|ej〉. (8.17)

Then, ifa=

∑N

i=1aieiandb=

∑N

i=1biei, the inner product ofaandbis given by

〈a|b〉=

〈N

∑

i=1

aiei

∣

∣

∣

∣

∣

∣

∑N

j=1

bjej

〉

=

∑N

i=1

∑N

j=1

a∗ibj〈ei|ej〉

=

∑N

i=1

∑N

j=1

a∗iGijbj. (8.18)

We further note that from (8.17) and the properties of the inner product we

requireGij=G∗ji. This in turn ensures that‖a‖=〈a|a〉is real, since then

〈a|a〉∗=

∑N

i=1

∑N

j=1

aiG∗ija∗j=

∑N

j=1

∑N

i=1

a∗jGjiai=〈a|a〉.