Page 405 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

DEFINITION

If A is an matrix, then the subspace of spanned by the row vectors of A is called the rowspace of A, and the subspace

of spanned by the column vectors of A is called the column space of A. The solution space of the homogeneous system of

equations , which is a subspace of , is called the nullspace of A.

In this section and the next we shall be concerned with the following two general questions:

What relationships exist between the solutions of a linear system and the row space, column space, and nullspace of
the coefficient matrix A?

What relationships exist among the row space, column space, and nullspace of a matrix?

To investigate the first of these questions, suppose that

It follows from Formula 10 of Section 1.3 that if denote the column vectors of A, then the product can be

expressed as a linear combination of these column vectors with coefficients from x; that is,

(1)

Thus a linear system, , of m equations in n unknowns can be written as

(2)

from which we conclude that is consistent if and only if b is expressible as a linear combination of the column vectors of A

or, equivalently, if and only if b is in the column space of A. This yields the following theorem.

THEOREM 5.5.1

A system of linear equations is consistent if and only if b is in the column space of A.

EXAMPLE 2 A Vector b in the Column Space of A
Let be the linear system

Show that b is in the column space of A, and express b as a linear combination of the column vectors of A.