1.5 Homogeneous systems

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A system of homogeneous linear equations is a system of the form

a11x1 + a12x2 + ¢ ¢ ¢ + a1nxn = 0

a21x1 + a22x2 + ¢ ¢ ¢ + a2nxn = 0

...

am1x1 + am2x2 + ¢ ¢ ¢ + amnxn = 0:

Such a system is always consistent as x1 = 0; ¢ ¢ ¢ ; xn = 0 is a solution.

This solution is called the trivial solution. Any other solution is called a

non{trivial solution.

For example the homogeneous system

x ¡ y = 0

x + y = 0

has only the trivial solution, whereas the homogeneous system

x ¡ y + z = 0

x + y + z = 0

has the complete solution x = ¡z; y = 0; z arbitrary. In particular, taking

z = 1 gives the non{trivial solution x = ¡1; y = 0; z = 1.

There is simple but fundamental theorem concerning homogeneous sys-

tems.

THEOREM 1.5.1 A homogeneous system of m linear equations in n un-

knowns always has a non{trivial solution if m < n.

1.6. PROBLEMS 17

Proof. Suppose that m < n and that the coe±cient matrix of the system

is row{equivalent to B, a matrix in reduced row{echelon form. Let r be the

number of non{zero rows in B. Then r · m < n and hence n ¡ r > 0 and

so the number n ¡ r of arbitrary unknowns is in fact positive. Taking one

of these unknowns to be 1 gives a non{trivial solution.

REMARK 1.5.1 Let two systems of homogeneous equations in n un-

knowns have coe±cient matrices A and B, respectively. If each row of B is

a linear combination of the rows of A (i.e. a sum of multiples of the rows

of A) and each row of A is a linear combination of the rows of B, then it is

easy to prove that the two systems have identical solutions. The converse is

true, but is not easy to prove. Similarly if A and B have the same reduced

row{echelon form, apart from possibly zero rows, then the two systems have

identical solutions and conversely.

There is a similar situation in the case of two systems of linear equations

(not necessarily homogeneous), with the proviso that in the statement of

the converse, the extra condition that both the systems are consistent, is

needed.

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