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Determine whether it's an inner product on R^3

  1. Dec 15, 2011 #1
    1. The problem statement, all variables and given/known data
    Let u = (u1, u2, u3)
    and v = (v1, v2, v3)
    Determine if it's an inner product on R3.
    If it's not, list the axiom that do not hold.



    2. Relevant equations
    the 4 axioms to determine if it's an inner product are
    (all letters representing vectors)

    1. <u,v> = <v,u>
    2. <u+v, w> = <u, w> + <v,w>
    3. <ku, v> = k<u,v>
    4. <v,v> ≥ 0 and < v,v> = 0 if and only if v = 0


    3. The attempt at a solution

    So <u, v> is defined as
    u1v1 + u3v3

    I'll skip the ones that did work and show axiom 4 which did not hold but I'm confused as to why this doesn't hold. I have a guess but have to make sure that I'm thinking correctly.

    Axiom 4 does not hold:
    <v, v > = v1v1 + v3v3
    = v12 + v32 ≥ 0
    and
    <0, 0> = (0)(0) + (0)(0) = 0

    now to check the other way:
    if <v, v > = 0
    implies that since
    v12 = 0 => v1 = 0
    v32 = 0 => v3 = 0

    then it goes to say it's not an inner product on R3. Am I correct to say it's not an inner product on R3 because there are only 2 components for axiom 4? and not 3? (i.e. no v2 showing anywhere)

    Thank you for any help. Will be much appreciated.
     
  2. jcsd
  3. Dec 15, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Okay, I was wondering if you were going to mention that!:tongue:

    Specifically, because there is no "v2", If u= (0, 1, 0) u is not 0 but <u, u>= 0, contradicting that last law.
     
  4. Dec 15, 2011 #3
    Oh okay, that makes sense.
    So just to clarify, because if <v, v > = 0 if and only if v = the zero vector
    but we don't know v2 due to how <u, v > is defined so that means v2 could be a non-zero number.

    Correct me if my logic is wrong.

    Thank you again, HallsofIvy.
     
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