Linear Algebra - Underdetermined Systems

[daveyman]

Homework Statement

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

Homework Equations

N/A

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]

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.

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]

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.

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?