Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

a discussion of how to use these properties to solve systems of linear equations.

The application of matrices to the study of oscillations in physical systems is

takenupinchapter9.

8.1 Vector spaces

A set of objects (vectors)a,b,c,...is said to form alinear vector spaceVif:

(i) the set is closed under commutative and associative addition, so that

a+b=b+a, (8.2)

(a+b)+c=a+(b+c); (8.3)

(ii) the set is closed under multiplication by a scalar (any complex number) to

form a new vectorλa, the operation being both distributive and associative

so that

λ(a+b)=λa+λb, (8.4)

(λ+μ)a=λa+μa, (8.5)

λ(μa)=(λμ)a, (8.6)

whereλandμare arbitrary scalars;

(iii) there exists anull vector 0 such thata+ 0 =afor alla;

(iv) multiplication by unity leaves any vector unchanged, i.e. 1×a=a;

(v) all vectors have a correspondingnegative vector−asuch thata+(−a)= 0.

It follows from (8.5) withλ= 1 andμ=−1that−ais the same vector as

(−1)×a.

We note that if we restrict all scalars to be real then we obtain areal vector

space(an example of which is our familiar three-dimensional space); otherwise,

in general, we obtain acomplex vector space. We note that it is common to use the

terms ‘vector space’ and ‘space’, instead of the more formal ‘linear vector space’.

Thespanof a set of vectorsa,b,...,sis defined as the set of all vectors that

may be written as a linear sum of the original set, i.e. all vectors

x=αa+βb+···+σs (8.7)

that result from the infinite number of possible values of the (in general complex)

scalarsα,β,...,σ.Ifxin (8.7) is equal to 0 for some choice ofα,β,...,σ(notall

zero), i.e. if

αa+βb+···+σs= 0 , (8.8)

then the set of vectorsa,b,...,s,issaidtobelinearly dependent.Insuchaset

at least one vector is redundant, since it can be expressed as a linear sum of

the others. If, however, (8.8) is not satisfied byanyset of coefficients (other than