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!

Vector Spaces, Subsets, and Subspaces

  1. Nov 8, 2007 #1
    1. The problem statement, all variables and given/known data


    What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2?

    R, in this question, is the real numbers.


    2. Relevant equations

    I know that, for example, V={(0,0)} is a subset for R^2 that is also a subspace, but I can't figure out how something can be a subset and not a subspace.



    3. The attempt at a solution

    Does this have anything to do with scalar multiplication being closed on the vector space?
     
  2. jcsd
  3. Nov 9, 2007 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Think about the set of all (x,y) where x and y are both integers.
     
  4. Nov 9, 2007 #3
    So for example, if we let the subset = (a,b) s.t. a,b are elements of Z. Then it is closed under addition but not under scalar multiplication. i.e. Let (a,b) = (1,3) and multiply by 1/2 for example (which is the example we used to figure it out). Then you get (1/2, 3/2), neither of which are in Z.
     
  5. Nov 9, 2007 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure, but it does have additive inverses.
     
  6. Nov 9, 2007 #5
    or both rational numbers
     
  7. Nov 9, 2007 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    It's very easy to be a subset without being a subspace! Just look at any subset that does not satisfy the requirements for a subpace- what about { (1, 0)}?



    In order for a subset to be a subspace, it must be closed under addition, have additive inverses, and be closed under scalar multiplication. Since you are asked about a subset that IS closed under addition and has additive inverses, looks like there is only one thing left!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Vector Spaces, Subsets, and Subspaces
Loading...