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Math Help - Vector space problem (lin. algebra)

  1. #1
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    Vector space problem (lin. algebra)

    If anyone could explain how the following problem is done, I would really appreciate it!

    Here is a vector candidate. The set is R, and we define scalar multiplication by ax = a * x (usual scalar multiplication) and vector addition by x (insert circle symbol with plus sign in it that means vector addition) y = max(x, y).

    For each of the following three vector space axioms, either verify the axiom or show that it does not hold.

    a) a(x+y) = ax + ay
    b) There exists an element 0 such that for any x in the proposed vector space, x + 0 = x.
    c) x+y = y+x
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  2. #2
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    Quote Originally Posted by buckaroobill View Post
    If anyone could explain how the following problem is done, I would really appreciate it!

    Here is a vector candidate. The set is R, and we define scalar multiplication by ax = a * x (usual scalar multiplication) and vector addition by x (insert circle symbol with plus sign in it that means vector addition) y = max(x, y).

    For each of the following three vector space axioms, either verify the axiom or show that it does not hold.

    a) a(x+y) = ax + ay
    a(x+y) = a max(x,y)

    ax + xy = max(ax,ay)

    let a=-1, x=1, y=2, then a(x+y) = -1 max(1,2) = -2, and ax+ay = max(-1,-2) = -1, so
    in this case a(x+y) != ax + ay, so this axiom is not satisfied.

    b) There exists an element 0 such that for any x in the proposed vector space, x + 0 = x.
    If such an element existed then there is a e in R such that for all x in R e<x, but there is
    no such element in R (it would have to behave like -infty, but this is not an element of R)

    c) x+y = y+x
    Trivialy true as max(x,y)=max(y,x)

    RonL
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