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Subsets of R^3 as subspaces

  1. Mar 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Which of the following subsets of R3 are subspaces? The set of all vectors of the form (a,b,c) where a, b, and c are...


    2. Relevant equations
    1. integers
    2. rational numbers

    3. The attempt at a solution
    I think neither are subspaces. IIRC, the scalar just needs to be from R3 and not, for example, an integer for 1 or a rational number for 2.

    So for number 1, I can multiply the integers of vector (a,b,c) by some non-integer k, ending up with (ka,kb,kc) outside the subset, and thus not a subspace.

    For number 2, I can multiply the rational numbers of vector (a,b,c) some some irrational number (say, ∏) and end up with (∏a, ∏b, ∏c), all outside the subset and thus not a subspace.

    Or am I totally wrong?
     
  2. jcsd
  3. Mar 9, 2013 #2

    jbunniii

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    No, you are totally correct. The indicated sets are not subspaces of ##\mathbb{R}^3##, for the reasons you stated.
     
  4. Mar 9, 2013 #3

    jbunniii

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    Correction: the scalars are elements of ##\mathbb{R}##, not ##\mathbb{R}^3##.
     
  5. Mar 9, 2013 #4
    If we're dealing with complex space, can scalars be complex?

    Thanks for the help!
     
  6. Mar 9, 2013 #5

    jbunniii

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    They can, but then it wouldn't be ##\mathbb{R}^3## anymore. It would be ##\mathbb{C}^3##.
     
  7. Mar 9, 2013 #6
    True, thanks! Would the correct term be that we're working in the "field" of R^3 or just R^3 space when talking about this?
     
  8. Mar 9, 2013 #7

    jbunniii

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    To be precise, a vector space consists of an abelian group of vectors and a field of scalars, along with some rules governing the multiplication of a vector by a scalar.

    So if we want to be precise, we would say that we are working in the vector space in which the vectors are elements of ##\mathbb{R}^3## and the scalars are elements of ##\mathbb{R}##, with the usual rules of multiplication.

    However, for brevity we typically say that we are working in the vector space ##\mathbb{R}^3##, and unless stated otherwise, it is understood that the scalar field is ##\mathbb{R}##.

    Similarly, we may say that we are working in the vector space ##\mathbb{C}^3##, where the assumption is that unless stated otherwise, the scalar field is ##\mathbb{C}##.
     
  9. Mar 9, 2013 #8
    Thank you, that clears up a lot.
     
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