Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.1 VECTOR SPACES

the trivial case in which all the coefficients are zero) then the vectors arelinearly

independent, and no vector in the set can be expressed as a linear sum of the

others.

If, in a given vector space, there exist sets ofNlinearly independent vectors,

but no set ofN+ 1 linearly independent vectors, then the vector space is said to

beN-dimensional. (In this chapter we will limit our discussion to vector spaces of

finite dimensionality; spaces of infinite dimensionality are discussed in chapter 17.)

8.1.1 Basis vectors

IfVis anN-dimensional vector space thenanyset ofNlinearly independent

vectorse 1 ,e 2 ,...,eNforms abasisforV.Ifxis an arbitrary vector lying inVthen

the set ofN+ 1 vectorsx,e 1 ,e 2 ,...,eN, must belinearly dependentand therefore

such that

αe 1 +βe 2 +···+σeN+χx= 0 , (8.9)

where the coefficientsα,β,...,χare not all equal to 0, and in particularχ=0.

Rearranging (8.9) we may writexas a linear sum of the vectorseias follows:

x=x 1 e 1 +x 2 e 2 +···+xNeN=

∑N

i=1

xiei, (8.10)

for some set of coefficientsxithat are simply related to the original coefficients,

e.g.x 1 =−α/χ,x 2 =−β/χ, etc. Since anyxlying in the span ofVcan be

expressed in terms of thebasisorbase vectorsei, the latter are said to form

acompleteset. The coefficientsxiare thecomponentsofxwith respect to the

ei-basis. These components areunique, since if both

x=

∑N

i=1

xiei and x=

∑N

i=1

yiei,

then

∑N

i=1

(xi−yi)ei= 0 , (8.11)

which, since theeiare linearly independent, has only the solutionxi=yifor all

i=1, 2 ,...,N.

From the above discussion we see thatanyset ofNlinearly independent

vectors can form a basis for anN-dimensional space. If we choose a different set

e′i,i=1,...,Nthen we can writexas

x=x′ 1 e′ 1 +x′ 2 e′ 2 +···+x′Ne′N=

∑N

i=1

x′ie′i. (8.12)