Page 406 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution

Solving the system by Gaussian elimination yields (verify)

Since the system is consistent, b is in the column space of A. Moreover, from 2 and the solution obtained, it follows that

The next theorem establishes a fundamental relationship between the solutions of a nonhomogeneous linear system and

those of the corresponding homogeneous linear system with the same coefficient matrix.

THEOREM 5.5.2

If denotes any single solution of a consistent linear system , and if , , …, form a basis for the nullspace of

A—that is, the solution space of the homogeneous system —then every solution of can be expressed in the form

(3)

and, conversely, for all choices of scalars , , … , , the vector x in this formula is a solution of
.

Proof Assume that is any fixed solution of and that x is an arbitrary solution.
Then

Subtracting these equations yields

which shows that is a solution of the homogeneous system . Since , , … is a basis for the solution space of

this system, we can express as a linear combination of these vectors, say

Thus,

which proves the first part of the theorem. Conversely, for all choices of the scalars , ,…, in 3, we have

or

But is a solution of the nonhomogeneous system, and , ,…, are solutions of the homogeneous system, so the last
equation implies that

which shows that x is a solution of .

General and Particular Solutions . The expression

There is some terminology associated with Formula 3. The vector is called a particular solution of