Page 73 - Matrix Analysis & Applied Linear Algebra

66 Chapter 2 Rectangular Systems and Echelon Forms

Furthermore, recall from (2.4.4) that the general solution of the associated
homogeneous system
x 1 +2x 2 +2x 3 +3x 4 =0,
2x 1 +4x 2 + x 3 +3x 4 =0, (2.5.4)
3x 1 +6x 2 + x 3 +4x 4 =0,
is given by
     
−2x 2 − x 4 −2 −1
x 2
 .
   1   0 
  = x 2  0  + x 4  −1
−x 4
x 4 0 1
That is, the general solution of the associated homogeneous system (2.5.4) is a
part of the general solution of the original nonhomogeneous system (2.5.2).
These two observations can be combined by saying that the general solution
of the nonhomogeneous system is given by a particular solution plus the general
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solution of the associated homogeneous system.
To see that the previous statement is always true, suppose [A|b] represents
a general m × n consistent system where rank (A)= r. Consistency guarantees
that b is a nonbasic column in [A|b], and hence the basic columns in [A|b] are
in the same positions as the basic columns in [A|0] so that the nonhomogeneous
system and the associated homogeneous system have exactly the same set of basic
variables as well as free variables. Furthermore, it is not difficult to see that
E [A|0] =[E A |0] and E [A|b] =[E A |c],

 
ξ 1
.
 . 
 . 
 
 . This means that if you solve
 ξ r 
where c is some column of the form c = 
 0 
.
 
.
 . 
0
the i th equation in the reduced homogeneous system for the i th basic variable
to produce
x b i in terms of the free variables x f i ,x f i+1 ,...,x f n−r
, (2.5.5)
x b i = α i x f i + α i+1 x f i+1 + ··· + α n−r x f n−r
then the solution for the i th basic variable in the reduced nonhomogeneous
system must have the form
. (2.5.6)
x b i = ξ i + α i x f i + α i+1 x f i+1 + ··· + α n−r x f n−r
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For those students who have studied differential equations, this statement should have a familiar
ring. Exactly the same situation holds for the general solution to a linear differential equation.
This is no accident—it is due to the inherent linearity in both problems. More will be said
about this issue later in the text.