Page 429 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

parameters.

In earlier sections we obtained a wide range of conditions under which a homogeneous linear system of n equations in n

unknowns is guaranteed to have only the trivial solution. (See Theorem 4.3.4.) The following theorem obtains some

corresponding results for systems of m equations in n unknowns, where m and n may differ.

THEOREM 5.6.8

If A is an matrix, then the following are equivalent.

(a) has only the trivial solution.

(b) The column vectors of A are linearly independent.

(c) has at most one solution (none or one) for every matrix b.

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

If , , …, are the column vectors of A, then the linear system can be written as

(6)

If , , …, are linearly independent vectors, then this equation is satisfied only by , which means that

has only the trivial solution. Conversely, if has only the trivial solution, then Equation 6 is satisfied only by

, which means that , , … , are linearly independent.

Assume that has only the trivial solution. Either is consistent or it is not. If it is not consistent, then

there are no solutions of , and we are done. If is consistent, let be any solution. From the discussion following

Theorem 5.5.2 and the fact that has only the trivial solution, we conclude that the general solution of is

. Thus the only solution of is .

Assume that has at most one solution for every matrix b. Then, in particular, has at most one

solution. Thus has only the trivial solution.

A linear system with more unknowns than equations is called an underdetermined linear system. If is a consistent

underdetermined linear system of m equations in n unknowns (so that ), then it follows from Theorem 5.6.7 that the

general solution has at least one parameter (why?); hence a consistent underdetermined linear system must have infinitely many

solutions. In particular, an underdetermined homogeneous linear system has infinitely many solutions, though this was already

proved in Chapter 1 (Theorem 1.2.1).

EXAMPLE 7 An Underdetermined System