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Math Help - Tensor

  1. #1
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    Tensor

    Prove that every element  v\in (\mathbb{C}^{2})^{\otimes 3} is the sum of two pure tensors  u_{1}\otimes u_{2}\otimes u_{3} ., where  u_{1},u_{2},u_{3}\in\mathbb{C}^{2} .

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by smith09 View Post
    Prove that every element  v\in (\mathbb{C}^{2})^{\otimes 3} is the sum of two pure tensors  u_{1}\otimes u_{2}\otimes u_{3} ., where  u_{1},u_{2},u_{3}\in\mathbb{C}^{2} .

    Thanks in advance.
    you're sure that two is not three then? the reason for asking this is that, in general, for any 2-dimensional vector space V over a field F, the "maximum rank" of V^{\otimes 3} is 3 and not 2, i.e. every

    element of V^{\otimes 3} is a sum of at most 3 simple tensors.
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  3. #3
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    Thks, but the question requires 2, not 3. They give some hints that consider v as a linear map from  \mathbb{C}^2 to  M_2 (group of 2x2 matrices) (WHY? and HOW?P) and consider two possibilities of the dimension of the image (1 or 2). But I really dont get it.
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  4. #4
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    Oh, sorry,  v here must be in a dense open subset.
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  5. #5
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    Is there any ideal then? I think the first case (1 dim) is not so hard. But the second case is ...not easy.

    Thks in advance.
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