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General solution of linear system

  1. Jul 25, 2010 #1
    I have this question, but don't know how to even start.

    Suppose (M) is a linear system of 2 equations and 3 unknowns, where (2,-3,1) its solution.
    Suppose (O) is a matching homogeneous linear system, where (-1,1,1) and (1,0,1) its solutions.

    How can I find the general solution of (M)?
    I'm totally lost with this one and appreciate any help.
     
    Last edited: Jul 25, 2010
  2. jcsd
  3. Jul 25, 2010 #2

    Mark44

    Staff: Mentor

    Where (2, -3, 1) is what?
    Is there a word missing here?
    I understand what you're trying to say, but I would like you to rephrase things so that you have complete thoughts.
     
  4. Jul 25, 2010 #3
    Edited the first post. Sorry for the missing words (I guess 24+ hours without sleep make me skip words).
     
    Last edited: Jul 26, 2010
  5. Jul 26, 2010 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Still missing words. Also, you cannot say that these are the solutions to the homogeneous system nor that (2, -3, 1) is the solution to the original system. Such a problem has an infinite number of solutions. If you can find a complete set of "independent" solutions then you can write any solution to the homogenous system as a linear combination of them. What is given here is not sufficient to conclude that there isn't a third linearly independent but to get an answer to this, we must assume so. It would have been a lot better if you had simply copied the problem as it was given.

    Anyway, assuming that any solution to the homogeneous system must be of the form A(-1, 1, 1)+ B(1, 0, 1) for some numbers A and B. Notice that L(A(-1, 1, 1)+ B(1, 0, 1))= AL(-1, 1, 1)+ BL((1, 0, 1))= A(0)+ B(0)= 0. Further, if x is a solution to the original system, if L(x)= y where y was the given "right side" of the system, then L(A(-1, 1, 1)+ B(1, 0, 1)+ x)= AL(-1, 1, 1)+ BL(1, 0, 1)+ L(x)= 0+ 0+ L(x)= y.

     
  6. Jul 26, 2010 #5
    Sorry for my grammar, English is not my native tongue.
    Anyway thanks for the hint.
     
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