Page 241 - Advanced engineering mathematics

7.6 Nonhomogeneous Systems
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We call AX = B consistent if there is a solution. If there is no solution, the system
is inconsistent.
2. A linear combination of solutions of a nonhomogeneous system AX = B need not
be a solution. Therefore the solutions do not have the vector space structure seen
in the homogeneous case.
Nevertheless, solutions of AX = B do have a property that parallels that for solutions of lin-
ear second order differential equations. We will call AX=O the associated homogeneous system
of the nonhomogeneous system AX = B. Although a sum of solutions of the nonhomogeneous
system need not be a solution, we claim that the difference of any two solutions of the nonhomo-
geneous system is a solution, not of the system, but of the associated homogeneous system. The
reason for this is that, if AU 1 = B and AU 2 = B, then
A(U 1 − U 2 ) = AU 1 − AU 2 = B − B = O.
This is the key to the fundamental theorem for writing the general solution of AX = B.
THEOREM 7.13
Let H be the general solution of the associated homogeneous system. Let U p be any particular
solution of AX=B. Then the expression H+U p contains every solution of the nonhomogeneous
system AX = B.
Proof Suppose H 1 ,··· ,H k form a basis for the solution space of AX = O, where k =
m − number of nonzero rows of (A R ). Then the general solution of the homogeneous system is
H = α 1 H 1 + ··· + α k H k .
If U is any solution of AX=B, then U − U p is a solution of the associated homogeneous system,
and therefore has the form
U − U p = c 1 H 1 + ··· + c k H k
for some constants c 1 ,··· ,c k . But then
U = c 1 H 1 + ··· + c k H k + U p ,
and this solution is contained in the general expression H + U p .
As an immediate consequence, Theorem 7.13 tells us when a nonhomogeneous system can
have only one solution.
COROLLARY 7.6
A consistent nonhomogeneous system AX=B has a unique solution if and only if the associated
homogeneous system has only the trivial solution.
The corollary follows from the fact that the nonhomogeneous system has a unique solution
exactly when H is the zero vector in Theorem 7.13.
Theorem 7.13 suggests a strategy for finding all solutions of AX = B, when the system is
consistent.
October 14, 2010 14:23 THM/NEIL Page-221 27410_07_ch07_p187-246