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Quick vector space question

  1. Mar 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Determine whether this set equipped with the given operations is a vector space. For those that are not vector spaces identify the axiom that fails.

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


    3. The attempt at a solution

    This set is not a vector space because it is not closed under scalar multiplication I.e. -1*(1,1)=(-1,-1) which is not in V as x<0 and because there is not always a vector in V such that u+(-u)=(-u)+u=0 I.e. when u=(1,1) then -u=(-1,-1) which is not in V as again x<0.

    My question is why does axiom 8 hold which states:

    (K+m)u=ku+km

    I.e. if k=-1, m=-1, u=(1,1) ----> (-1+-1)u=(-1,-1)+(-1,-1)=(-2,-2) which is not in V as x<0.

    Does axiom 8 not require the solution to be in the set V?
     
  2. jcsd
  3. Mar 18, 2013 #2

    micromass

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    You showed that the space is not closed under scalar multiplication, so it's not a vector space. But also, since it's not closed under scalar multiplication, all the axioms which use scalar multiplication make no sense. It makes no sense to asl ##(\alpha + \beta) v = \alpha v + \beta v##, since scalar multiplication is not well-defined.

    so I wouldn't say that Axiom 8 holds in this case. I would rather say that it makes no sense.
     
  4. Mar 18, 2013 #3
    Oh I see, but it could be the case that a set is closed under scalar multiplication but one of the axioms that depend on that, such as axiom 8, do not hold. Just not vice versa.

    Thanks!
     
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