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Math Help - Bases and order in vector spaces.

  1. #1
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    Bases and order in vector spaces.

    Hi:

    Let V be a vector space over some field F. {V, F, +, .}
    would be more formal but let us call V also to the abelian
    group over which + acts. In wikipedia.org, article
    "Vector space", I read: "A basis is defined as a set B =
    {v_i} indexed by some set I", etc. If V is 3-dimentional,
    v_1, v_2, v_3 belong to V and I = {1,2,3}, then, according
    to wikipedia, (v_1, v_2, v_3} is not the same basis of V as
    {v_2, v_1, v_3}. But according to some authors, which
    define a basis merely as a set, they are. Who am I to believe?
    Any suggestion will be welcome. Thanks for reading.
    Last edited by ENRIQUESTEFANINI; July 28th 2010 at 12:23 PM.
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  2. #2
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:

    Let V be a vector space over some field F. {V, F, +, .}
    would be more formal but let us call V also to the abelian
    group over which + acts. In wikipedia.org, article
    "Vector space", I read: "A basis is defined as a set B =
    {v_i} indexed by some set I", etc. If V is 3-dimentional,
    v_1, v_2, v_3 belong to V and I = {1,2,3}, then, according
    to wikipedia, (v_1, v_2, v_3} is not the same basis of V as
    {v_2, v_1, v_3}. But according to some authors, which
    define a basis merely as a set, they are. Who am I to believe?
    Any suggestion will be welcome. Thanks for reading.

    A basis is an ordered maximal linearly independent set. Some authors certainly don't stress this, but I know none that only require "basis" to be a non-ordered set.

    Tonio
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  3. #3
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    Tonio. I think I'll make a hard copy of your post, for the sake of keeping it in a safe place. Asking the set to be ordered is (implicitly) in accordance with what I have learned in physics. E.g., the cross product of A and B would not be well defined. However, my doubt aroused in the context of a theorem about vector spaces with a denumerable basis. I do not know how to thank you. All the best,

    Enrique.
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  4. #4
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    Of course, the only situation that I can think of when the order of a basis is needed is when you are writing a linear transformation as a matrix.

    Changing the order of the basis vectors in the domain space will change the order of the columns in the matrix and changing the order of the basis vectors in the range space will change the order of the rows.
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  5. #5
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    Thanks for your kind reply. So you are speaking about finite dimentional vector spaces. And what about vector spaces with infinite bases? Hum... I think the answer to this question is implied by your post. I.e., order is needed when treating finite-dimentional vec. spaces and only when the relationship between linear transf. and matrices is of interest. All the best.
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