Page 427 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

reduced row-echelon form. For example, the augmented matrix for the system

which has the following reduced row-echelon form (verify):

We see from the third row in this matrix that the system is inconsistent. However, it is also because of this row that the reduced
row-echelon form of the augmented matrix has fewer zero rows than the reduced row-echelon form of the coefficient matrix.
This forces the coefficient matrix and the augmented matrix for the system to have different ranks.

The Consistency Theorem is concerned with conditions under which a linear system is consistent for a specific vector b.

The following theorem is concerned with conditions under which a linear system is consistent for all possible choices of b.

THEOREM 5.6.6

If is a linear system of m equations in n unknowns, then the following are equivalent.

(a) is consistent for every matrix b.

(b) The column vectors of A span .

(c) .

Proof It suffices to prove the two equivalences and , since it will then follow as a matter of logic that
. can be expressed as

From Formula 2 of Section 5.5, the system

from which we can conclude that is consistent for every matrix b if and only if every such b is expressible as a

linear combination of the column vectors , ,…, , or, equivalently, if and only if these column vectors span .

From the assumption that is consistent for every matrix b, and from parts (a) and (b) of the Consistency

Theorem (Theorem 5.6.5), it follows that every vector b in lies in the column space of A; that is, the column space of A is

all of . Thus .

From the assumption that , it follows that the column space of A is a subspace of of dimension m and

hence must be all of by Theorem 5.4.7. It now follows from parts (a) and (b) of the Consistency Theorem (Theorem 5.6.5)

that is consistent for every vector b in , since every such b is in the column space of A.

A linear system with more equations than unknowns is called an overdetermined linear system. If is an overdetermined