Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Vector Addition and Subspaces

  1. Nov 8, 2007 #1
    I don't understand this, can someone help?:

    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. jcsd
  3. Nov 8, 2007 #2


    User Avatar
    Homework Helper
    Gold Member

    Can you think of a subset of R^2 such that it is a subspace?
  4. Nov 8, 2007 #3
    Well you can add things but you can't subtract things. This should be a big enough hint.
  5. Nov 8, 2007 #4
    He mentioned additive inverses and closure under addition (implying 0 being an element), so he has no problem subtracting things. The key lies in the one part of the definition of a vector space that is left out.
  6. Nov 8, 2007 #5
    I misread it. I thought the question asked to find something that fails to be a subspace because it's not closed under additive inverses.
  7. Nov 8, 2007 #6
    To JasonRox: Yes, I can think of a subset, V, of R^2 that is a subspace. V= {(0,0)}. But I still don't understand how to approach this problem.
  8. Nov 9, 2007 #7
    first try to solve a simpler problem.
    are there nontrivial subgroups of the group [tex] (\mathbb{R},+,-,0) [/tex] ? that means is there a set [tex] A [/tex] with [tex] \{0\} \subset A \subset \mathbb{R} [/tex] such that [tex] A [/tex] is closed under addition and substraction?
    once you have found such a subgroup [tex] A [/tex], is [tex] A^2 [/tex] a set with the desired property?
  9. Nov 9, 2007 #8


    User Avatar
    Staff Emeritus
    Science Advisor

    In order that a subset be a subspace, it must be closed under addition, additive inverses, and scalar multiplication. Since your subset is required to be closed under addition and additive inverse, there's only one place left to look!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?