Linear Algebra - Underdetermined Systems

daveyman

1. The problem statement, all variables and given/known data

Every underdetermined system of linear equations has infinitely many solutions. (True/False)


2. Relevant equations

N/A


3. The attempt at a solution

Every source I have found, including several textbooks, say that underdetermined systems "often" or "usually" have an infinite number of solutions, so I'm assuming the answer is false, but I can't think of an example that shows an underdetermined system that does not have infinitely many solutions.

Any ideas?

HallsofIvy

Well, first, what definition of "underdetermined system"are you using? I've found two different definitions on the internet that give obvious and different answers to your question!

daveyman

Here is the definition straight from my linear algebra book (Moore and Yaqub, 3rd Edition): Systems of linear equations with fewer equations than unknowns are frequently called undetermined systems.

NoMoreExams

Well think about it you can have 2 equations of a plane. How can they intersect?

daveyman

Two planes could intersect on a particular line, thus creating an infinite number of solutions. If the planes are parallel, however, they will never intersect and there will be no solution.

So I guess the answer would be false?

NoMoreExams

That would be my guess.

daveyman

Thanks!

HallsofIvy

One reason for my question, by the way, (besides the absolute importance of precise definitions in mathematics) was that the other reference I found defined "undetermined system" as one having an infinite number of solutions! The definition given here, and the solution to this problem, is the one I would expect.

daveyman

I made an error that becomes extremely important in a discussion about definitions. I wrote undetermined but I meant underdetermined. Sorry about this. I don't think this changes the conclusion, however.

Do you agree?