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Linear Algebra - Underdetermined Systems

  1. Dec 20, 2008 #1
    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?
     
  2. jcsd
  3. Dec 20, 2008 #2

    HallsofIvy

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    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!
     
  4. Dec 20, 2008 #3
    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.
     
  5. Dec 20, 2008 #4
    Well think about it you can have 2 equations of a plane. How can they intersect?
     
  6. Dec 20, 2008 #5
    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?
     
  7. Dec 20, 2008 #6
    That would be my guess.
     
  8. Dec 20, 2008 #7
    Thanks!
     
  9. Dec 20, 2008 #8

    HallsofIvy

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    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.
     
  10. Dec 20, 2008 #9
    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?
     
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