Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

may be thought of as ‘transforming’ one geometrical entity (i.e. a vector) into

another.

If we now introduce a basisei,i=1, 2 ,...,N, into our vector space then the

action ofAon each of the basis vectors is to produce a linear combination of

the latter; this may be written as

Aej=

∑N

i=1

Aijei, (8.23)

whereAijis theith component of the vectorAejin this basis; collectively the

numbersAijare called the components of the linear operator in theei-basis.In

this basiswe can express the relationy=Axin component form as

y=

∑N

i=1

yiei=A





∑N

j=1

xjej



=

∑N

j=1

xj

∑N

i=1

Aijei,

and hence, in purely component form, in this basis we have

yi=

∑N

j=1

Aijxj. (8.24)

If we had chosen a different basise′i, in which the components ofx,yandA

arex′i,y′iandA′ijrespectively then the geometrical relationshipy=Axwould be

represented in this new basis by

y′i=

∑N

j=1

A′ijx′j.

We have so far assumed that the vectoryis in the same vector space as

x.If,however,ybelongs to a different vector space, which may in general be

M-dimensional (M=N) then the above analysis needs a slight modification. By

introducing a basis setfi,i=1, 2 ,...,M, into the vector space to whichybelongs

we may generalise (8.23) as

Aej=

∑M

i=1

Aijfi,

where the componentsAijof the linear operatorArelate to both of the basesej

andfi.