Page 354 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Subspaces of Subspaces of

{} {}
Lines through the origin Lines through the origin
Planes through the origin

Later, we will show that these are the only subspaces of and .

EXAMPLE 4 Subspaces of
From Theorem 1.7.2, the sum of two symmetric matrices is symmetric, and a scalar multiple of a symmetric matrix is
symmetric. Thus the set of symmetric matrices is a subspace of the vector space of all matrices. Similarly,
the set of upper triangular matrices, the set of lower triangular matrices, and the set of diagonal matrices all
form subspaces of , since each of these sets is closed under addition and scalar multiplication.

EXAMPLE 5 A Subspace of Polynomials of Degree

Let n be a nonnegative integer, and let W consist of all functions expressible in the form

(1)

where are real numbers. Thus W consists of all real polynomials of degree n or less. The set W is a subspace of the

vector space of all real-valued functions discussed in Example 4 of the preceding section. To see this, let p and q be the

polynomials

Then

and

These functions have the form given in 1, so and lie in W. As in Section 4.4, we shall denote the vector space W in
this example by the symbol .