Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

If a vectorylies in the null space ofAthenAy= 0 ,whichwemaywriteas

y 1 v 1 +y 2 v 2 +···+yNvN= 0. (8.121)

As just shown above, however, onlyr(≤N) of these vectors are linearly independent. By
renumbering, if necessary, we may assume thatv 1 ,v 2 ,...,vrform a linearly independent
set; the remaining vectors,vr+1,vr+2,...,vN, can then be written as a linear superposition
ofv 1 ,v 2 ,...,vr. We are therefore free to choose theN−rcoefficientsyr+1,yr+2,...,yN
arbitrarily and (8.121) will still be satisfied for some set ofrcoefficientsy 1 ,y 2 ,...,yr(which
are not all zero). The dimension of the null space is thereforeN−r, and this completes
the proof of (8.119).

Equation (8.119) has far-reaching consequences for the existence of solutions

to sets of simultaneous linear equations such as (8.118). As mentioned previously,

these equations may haveno solution,aunique solutionorinfinitely many solutions.

We now discuss these three cases in turn.

No solution

The system of equations possesses no solution unlessblies in the range ofA;in

this case (8.120) will be satisfied for somex 1 ,x 2 ,...,xN. This in turn requires the

setofvectorsb,v 1 ,v 2 ,...,vNto have the same span (see (8.8)) asv 1 ,v 2 ,...,vN.In

terms of matrices, this is equivalent to the requirement that the matrixAand the

augmented matrix

M=











A 11 A 12 ... A 1 N b 1

A 21 A 22 ... A 2 N b 1

..

.

..

.

..

.

AM 1 AM 2 ... AMN bM











have thesamerankr. If this condition is satisfied thenbdoes lie in the range of

A, and the set of equations (8.118) will have either a unique solution or infinitely

many solutions. If, however,AandMhave different ranks then there will be no

solution.

A unique solution

Ifblies in the range ofAand ifr=Nthen all the vectorsv 1 ,v 2 ,...,vNin (8.120)

are linearly independent and the equation has aunique solutionx 1 ,x 2 ,...,xN.

Infinitely many solutions

Ifblies in the range ofAand ifr<Nthen onlyrof the vectorsv 1 ,v 2 ,...,vN

in (8.120) are linearly independent. We may therefore choose the coefficients of

n−rvectors in an arbitrary way, while still satisfying (8.120) for some set of

coefficientsx 1 ,x 2 ,...,xN. There are thereforeinfinitely many solutions,whichspan

an (n−r)-dimensional vector space. We may also consider this space of solutions

in terms of the null space ofA:ifxis some vector satisfyingAx=bandyis