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Solution set

  1. Jan 26, 2005 #1
    Hi everyone,
    general question: is a solution set for a particular system a vector space? I know it can be if there is a unique solution, but is it generally true?
    Could someone explain, please?

    Thanks.
     
  2. jcsd
  3. Jan 26, 2005 #2

    AKG

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    No. Vector spaces are closed under scalar multiplication. If b is a scalar not equal to 1, Y is non-zero, and X is a solution of AX = Y, then:

    A(bX) = b(AX) = bY is not equal to Y, so (bX) is not a solution, so the set of solutions is not closed under scalar multiplication, so the set of solutions is not a vector space. Perhaps I've misinterpreted your question. If there is a unique solution, then there would only be that 1 element of the vector space. The only vector space that has only one element is the degenerate vector space {0}.
     
  4. Jan 26, 2005 #3

    HallsofIvy

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    For a particular system? Do you mean a system of linear equations?

    The solution set of a system of homogenous equations is a subspace.

    If the system consists of n independent equations in n unknowns, then it is just the 0 vector but if the rank is lower than the number of unknowns, then it is a non-trivial subspace of Rn[/sub].
     
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