Page 428 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

linear system of m equations in n unknowns (so that ), then the column vectors of A cannot span ; it follows from the

last theorem that for a fixed matrix A with , the overdetermined linear system cannot be consistent for every

possible b.

EXAMPLE 5 An Overdetermined System
The linear system

is overdetermined, so it cannot be consistent for all possible values of , , , , and . Exact conditions under which the
system is consistent can be obtained by solving the linear system by Gauss–Jordan elimination. We leave it for the reader to
show that the augmented matrix is row equivalent to

Thus, the system is consistent if and only if , , , , and satisfy the conditions

or, on solving this homogeneous linear system,
where r and s are arbitrary.

In Formula 3 of Theorem 5.5.2, the scalars , , … , are the arbitrary parameters in the general solutions of both
and . Thus these two systems have the same number of parameters in their general solutions. Moreover, it follows from
part (b) of Theorem 5.6.4 that the number of such parameters is nullity(A). This fact and the Dimension Theorem for Matrices
(Theorem 5.6.3) yield the following theorem.

THEOREM 5.6.7

If is a consistent linear system of m equations in n unknowns, and if A has rank r, then the general solution of the
system contains parameters.

EXAMPLE 6 Number of Parameters in a General Solution

If A is a matrix with rank 4, and if is a consistent linear system, then the general solution of the system contains