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Vectors axioms

  1. Jun 19, 2009 #1
    1. The problem statement, all variables and given/known data
    show that the collection of all ordered 3-tupples (x1,x2,x3) whose components satisfy 3x1 - x2 + 5x3 = 0 forms a vector space with the respect the usual operation of R3.


    2. Relevant equations
    3x1 - x2 + 5x3


    3. The attempt at a solution
    we tried it by addition and multipication..solutions would be appreciated asap
     
  2. jcsd
  3. Jun 19, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi alexngo! Welcome to PF! :wink:
    ok! … show us what you get! :smile:
     
  4. Jun 19, 2009 #3
    by showing that it respects both addition and multiplication does this proves it to be a vector space or we need to show taht it satisfies all 10 axioms
     
  5. Jun 19, 2009 #4

    CompuChip

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    Technically you need to show all 10 of them. But note that, if your set is a vector space, then it is a subspace of R^3. In simpler terms: the "vectors" from your vector space are just vectors as you know them. So the axioms about distributivity and associativity carry over to the subspace (for example: if the addition of three arbitrary 3-d vectors is associative, then the addition of three special ones of the form (x, y, (y-3x)/5) is definitely associative as well). In fact there is a reduced set of axioms which you can use to show that a subset of a vector space is a vector space.
     
  6. Jun 19, 2009 #5

    tiny-tim

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    You only need the closure axioms …

    you don't need to prove eg a + b = b + a because so long as you've proved closure, ie that b + a is in the collection, then a + b = b + a is automatically satisfied.

    But of course, you do still need to prove closure under multiplication by a scalar. :wink:

    EDIT: oooh, CompuChip :smile: beat me to it! :biggrin:
     
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