Page 407 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

is called the general solution of , and the expression is called the
is the sum of any particular
general solution of . With this terminology, Formula 3 states that the general solution of

solution of and the general solution of .

For linear systems with two or three unknowns, Theorem 5.5.2 has a nice geometric interpretation in and . For example,

consider the case where and are linear systems with two unknowns. The solutions of form a subspace of

and hence constitute a line through the origin, the origin only, or all of . From Theorem 5.5.2, the solutions of can be

obtained by adding any particular solution of , say , to the solutions of . Assuming that is positioned with its

initial point at the origin, this has the geometric effect of translating the solution space of , so that the point at the origin is

moved to the tip of (Figure 5.5.1). This means that the solution vectors of form a line through the tip of , the point at

the tip of , or all of . (Can you visualize the last case?) Similarly, for linear systems with three unknowns, the solutions of

constitute a plane through the tip of any particular solution , a line through the tip of , the point at the tip of , or all of

.

Figure 5.5.1 translates the solution space.
Adding to each vector x in the solution space of

EXAMPLE 3 General Solution of a Linear System (4)
In Example 4 of Section 1.2 we solved the nonhomogeneous linear system

and obtained
This result can be written in vector form as

(5)
which is the general solution of 4. The vector in 5 is a particular solution of 4; the linear combination x in 5 is the general