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True/False Question on Linear Combos

  1. Oct 1, 2012 #1
    1. The problem statement, all variables and given/known data
    If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.


    2. Relevant equations



    3. The attempt at a solution
    I think it's true. But after that, I don't know.

    I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x1+x2 is a solution?

    I'm worried that that is actually a solution to Ax=2b, though.
     
  2. jcsd
  3. Oct 1, 2012 #2
    Perhaps that is only true for homogeneous equations, now that I think about it.

    Would nonhomogeneous equations require some sort of weighted average linear combination?
     
  4. Oct 1, 2012 #3

    SammyS

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    Is it true that (1/2)Ax = A((1/2)x) ?
     
  5. Oct 1, 2012 #4

    LCKurtz

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    Is it given b is not the zero vector? Think about uniqueness, rank, and free variables.
     
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