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Math Help - Prove this vector space is not finite-dimensional.

  1. #1
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    Prove this vector space is not finite-dimensional.

    Let V be the set of real numbers.Regard V as a vector space over the field of rational numbers,with the usual operations.Prove that this vector space is not finite-dimensional.
    It is easy to give countable real numbers which are linearly independent intuitively.But I always fail to prove this.
    Can you help me?Thanks.
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  2. #2
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    Quote Originally Posted by tianye View Post
    Let V be the set of real numbers.Regard V as a vector space over the field of rational numbers,with the usual operations.Prove that this vector space is not finite-dimensional.
    It is easy to give countable real numbers which are linearly independent intuitively.But I always fail to prove this.
    Can you help me?Thanks.


    Using the fact that \pi is a transcendental number , show that the set \{1,\pi,\pi^2,\ldots,\pi^n\} is linearly independent over \mathbb{Q}\,\,\,\forall\,n\in\mathbb{N}

    Tonio
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