1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Staff Emeritus
    Science Advisor

    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].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?