Page 71 - Matrix Analysis & Applied Linear Algebra

64 Chapter 2 Rectangular Systems and Echelon Forms
2.5 NONHOMOGENEOUS SYSTEMS

Recall that a system of m linear equations in n unknowns

a 11 x 1 + a 12 x 2 + ··· + a 1n x n = b 1 ,
a 21 x 1 + a 22 x 2 + ··· + a 2n x n = b 2 ,
. . .

a m1 x 1 + a m2 x 2 + ··· + a mn x n = b m ,

is said to be nonhomogeneous whenever b i
= 0 for at least one i. Unlike
homogeneous systems, a nonhomogeneous system may be inconsistent and the
techniques of §2.3 must be applied in order to determine if solutions do indeed
exist. Unless otherwise stated, it is assumed that all systems in this section are
consistent.
To describe the set of all possible solutions of a consistent nonhomogeneous
system, construct a general solution by exactly the same method used for homo-
geneous systems as follows.

• Use Gaussian elimination to reduce the associated augmented matrix [A|b]
to a row echelon form [E|c].

• Identify the basic variables and the free variables in the same manner de-
scribed in §2.4.

• Apply back substitution to [E|c] and solve for the basic variables in terms
of the free variables.

• Write the result in the form
h n−r , (2.5.1)
x = p + x f 1 h 1 + x f 2 h 2 + ··· + x f n−r
are the free variables and p, h 1 , h 2 ,..., h n−r are
where x f 1 ,x f 2 ,...,x f n−r
n × 1 columns. This is the general solution of the nonhomogeneous system.

range over all possible values, the general solu-
As the free variables x f i
tion (2.5.1) generates all possible solutions of the system [A|b]. Just as in the
homogeneous case, the columns h i and p are independent of which row eche-
lon form [E|c] is used. Therefore, [A|b] may be completely reduced to E [A|b]
by using the Gauss–Jordan method thereby avoiding the need to perform back
substitution. We will use this approach whenever it is convenient.
The difference between the general solution of a nonhomogeneous system
and the general solution of a homogeneous system is the column p that appears