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Exercise on vector space

  1. Aug 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine if the following set is a vector space under the given operations. List all the axioms that fail to hold.

    The set of all pairs of real numbers of the form (x,y), where x >= 0, with the standard operations on R^2

    2. Relevant equations

    3. The attempt at a solution

    By the axioms of a vector space the set fail on hold this axiom:

    for each u in V, there is an object -u in V, called negative of u, such that u + (-u) = (-u) + u = 0.

    If the x term in the pair (x,y) is positive (or zero) then -u = (-x, -y) can not exists. Thus, the negative of u does not exist, and V is not a vector space.

    Is this correct?
     
  2. jcsd
  3. Aug 13, 2012 #2

    Mark44

    Staff: Mentor

    It's OK as far as you went, but you have some more work to do. You need to check all the axioms. There is at least one more that isn't satisfied.
     
  4. Aug 13, 2012 #3
    It also fails in:

    K is any scalar, u is in V, ku is in V.

    u = (x,y)

    If I choose k < 0, then ku = k(x,y) = (kx,ky) and kx < 0 and ku is not in V.
     
  5. Aug 14, 2012 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Note that if the problem had asked only if this was a vector space, you could have stopped after showing one axiom did not hold. But this problem specifically asks you to list all axioms that do not hold.
     
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