Page 356 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 5.2.4

Remark In the preceding examplewe focused on the interval . Had we focused on a closed interval ,
.
then the subspaces corresponding to those defined in the example would be denoted by , , and

Similarly, on an open interval they would be denoted by , , and

.

Solution Spaces of Homogeneous Systems

If is a system of linear equations, then each vector x that satisfies this equation is called a solution vector of the
system. The following theorem shows that the solution vectors of a homogeneous linear system form a vector space, which
we shall call the solution space of the system.

THEOREM 5.2.2

If is a homogeneous linear system of m equations in n unknowns, then the set of solution vectors is a subspace of
.

Proof Let W be the set of solution vectors. There is at least one vector in W, namely . To show that W is closed under

addition and scalar multiplication, we must show that if x and are any solution vectors and k is any scalar, then and

are also solution vectors. But if and are solution vectors, then

from which it follows that

and

which proves that and are solution vectors.

EXAMPLE 7 Solution Spaces That Are Subspaces of