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Math Help - linear algebra proof help

  1. #1
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    linear algebra proof help

    The real numbers form a vector space over the rationals(i.e. with Q as the scalar field).

    a) prove this

    b) prove that this vector space has uncountable dimension
    you may assume the Hamel Basis Theorem, which states that this
    space has a basis.

    Any help is appreciated
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  2. #2
    Super Member girdav's Avatar
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    Re: linear algebra proof help

    a) What have you tried?

    b) It's a cardinality argument.
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  3. #3
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    Re: linear algebra proof help

    If a countable basis exists,every real number is a linear combination over Q of finite subset of basis element.Now use cardinals consideretions to explain why this is impossible.
    Last edited by hedi; November 4th 2012 at 03:46 PM.
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  4. #4
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    Re: linear algebra proof help

    I haven't tried anything yet, im not sure where to start. Would i have to prove that the cardinality of the Real numbers is greater than the cardinality of the rational numbers?
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  5. #5
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    Re: linear algebra proof help

    I suppose it belongs to set theory or descrete mathematics.Just use that countable union of finite sets is countable.
    Thanks from darts3
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  6. #6
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    Re: linear algebra proof help

    for part (a) just verify that R satisfies the vector space axioms.

    you can "take a short-cut" by noting that R is an abelian group under addition. then use the fact that R is a field (and thus a ring) to show that:

    a(r + s) = ar + as, for any a in Q, and r,s in R.

    (a+b)r = ar + br, for any a,b in Q and r in R.

    a(br) = (ab)r, for any a,b in Q and r in R.

    1r = r, for any real number r.
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