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Thread: Product Basis?

  1. #1
    Super Member Showcase_22's Avatar
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    Product Basis?

    I have this theorem:

    Theorem 33.2: Let $\displaystyle \{e_i \}$ and $\displaystyle \{ e^i \}$ be dual bases for $\displaystyle V$ and $\displaystyle V^*$. Then the set of tensor products:

    $\displaystyle \{ e_{i_1} \otimes \ldots \otimes e_{i_p} \otimes e^{j_1} \otimes \ldots \otimes e^{j_q}, \ i_1, \ldots , i_p,j_1, \ldots , j_q=1, \ldots , N \}$

    forms a basis for $\displaystyle \vartheta_q^p(V)$ (the set of all tensors of order $\displaystyle (p,q)$ on a vector space $\displaystyle V$).
    The proof starts by proving that the set is linearly independent. It does this as follows:

    "We shall prove that the set of tensor products is a linearly independent generating set for $\displaystyle \vartheta_q^p(V)$. To prove that the set is linearly independent, let

    $\displaystyle A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q} e_{i_1} \otimes \ldots \otimes e_{i_p} \otimes e^{j_1} \otimes \ldots \otimes e^{j_q}=0$

    where the RHS is the 0 tensor".

    Firstly, what is $\displaystyle A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q} e_{i_1}$? Is it a linear map going from $\displaystyle V$ to $\displaystyle V^*$ with the bases outlined in the question?

    Secondly, how will this prove that the set is linearly independent?
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    I have this theorem:



    The proof starts by proving that the set is linearly independent. It does this as follows:

    "We shall prove that the set of tensor products is a linearly independent generating set for $\displaystyle \vartheta_q^p(V)$. To prove that the set is linearly independent, let

    $\displaystyle A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q} e_{i_1} \otimes \ldots \otimes e_{i_p} \otimes e^{j_1} \otimes \ldots \otimes e^{j_q}=0$

    where the RHS is the 0 tensor".

    Firstly, what is $\displaystyle A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q} e_{i_1}$? Is it a linear map going from $\displaystyle V$ to $\displaystyle V^*$ with the bases outlined in the question?

    Secondly, how will this prove that the set is linearly independent?
    $\displaystyle A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q}$ is a scalar, for each set of indices $\displaystyle \{i_1, \ldots, i_p,j_1, \ldots , j_q\}$. You should read this expression as $\displaystyle \sum A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q}(e_{i_1} \otimes \ldots \otimes e_{i_p} \otimes e^{j_1} \otimes \ldots \otimes e^{j_q})$. The summation convention is being used, which means that you are meant to sum (from 1 to N) over each index that appears both as a superscript and as a subscript. So that expression simply denotes a linear combination of the given tensors. To prove that they are linearly independent, you put a linear combination of them equal to 0, and then you are aiming to show that each coefficient $\displaystyle A^{i_1, \ldots, i_p}_{j_1, \ldots , j_q}$ is zero.
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