Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

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