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I Vector Space of Alternating Multilinear Functions ...

  1. Mar 17, 2019 at 2:36 AM #1
    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ...

    I need some help in order to fully understand the vector space of alternating multilinear functions ...

    The relevant text from Shifrin reads as follows:



    ?temp_hash=c5a985b43e67ff351bbb3452d56f5618.png




    In the above text from Shifrin we read the following:

    " ... ... In particular, if ##T \in {\bigwedge}^k ( \mathbb{R}^n )^{ \ast }##, then for any increasing ##k##-tuple ##I##, set ##a_I = T( e_{ i_1} , \cdot \cdot \cdot , e_{ i_k} )##. Then we leave it to the reader to check that


    ##T = \sum_{ i \text{ increasing } }a_I \text{dx}_I##

    ... ... ... "



    Can someone please help me to prove/demonstrate that ##T = \sum_{ i \text{ increasing } }a_I \text{dx}_I## ... ...



    Help will be much appreciated ...

    Peter



    ==========================================================================================


    In case someone needs access to the text where Shifrin defines the terms of the above post and explains the notation, I am providing access to the start of Chapter 8, Section 2.1 as follows:



    ?temp_hash=a6ac9c21b636d70cd5c23e172af350a3.png
    ?temp_hash=a6ac9c21b636d70cd5c23e172af350a3.png
    ?temp_hash=a6ac9c21b636d70cd5c23e172af350a3.png



    Hope that helps ...

    Peter
     
    Last edited: Mar 17, 2019 at 3:07 AM
  2. jcsd
  3. Mar 17, 2019 at 3:49 AM #2

    andrewkirk

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    We are told that the set of ##d\mathbf x_I## with ##I## increasing spans the vector space ##\Lambda^k(\mathbb R^n)^*## of alternating multilinear functions from ##(\mathbb R^n)^k## to ##\mathbb R##. So for any ##T## in that space there exists a set of real numbers ##A_T## indexed by the set ##S## of increasing ##k##-tuples in ##\{1, ...,n\}^k##, such that:

    $$T=\sum_{J\in S} a_J d\mathbf x_J$$

    where ##a_J## is the element of ##A_T## indexed by ##k##-tuple ##J##.

    Applying both sides to the ##k##-tuple of vectors ##(\mathbf e_{i_1},...,\mathbf e_{i_k})##, for increasing ##k##-tuple ##I=(i_1,...,i_k)##, we get:

    \begin{align*}
    T(\mathbf e_{i_1},...,\mathbf e_{i_k})
    &=
    \sum_{J\in S} a_J d\mathbf x_J (\mathbf e_{i_1},...,\mathbf e_{i_k})
    \\&=
    \sum_{I\in S} a_J \delta^{i_1,...,i_k}_{j_1,...,j_k}
    \end{align*}

    by the presumed definition of ##d\mathbf x_J## (not supplied in OP). All the Kronecker deltas in terms in that sum are zero except the one where ##J=I##, where the delta is 1. So the sum on the RHS equals ##a_I##, giving us the desired result.
     
  4. Mar 17, 2019 at 4:37 AM #3
    Hi Andrew ...

    Thanks for your help ...

    Shifrin defines ##\text{dx}_I## as follows ... ...

    ... Now if ##I = (i_1, \cdot \cdot \cdot , 1_k )## is an ordered ##k##-tuple, define

    ##\text{dx}_I : \mathbb{R}^n \times \cdot \cdot \cdot \mathbb{R}^n \to \mathbb{R}## by (note ##k## input factors)

    ##\text{dx}_I ( v_1, \cdot \cdot \cdot , v_k ) = \begin{vmatrix} \text{dx}_{ i_1} ( v_1) & \cdot \cdot \cdot & \text{dx}_{ i_1} ( v_k) \\ \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot \cdot \cdot & \cdot \\ \text{dx}_{ i_k} ( v_1) & \cdot \cdot \cdot & \text{dx}_{ i_k} ( v_k) \end{vmatrix}##


    I think that that would give the result you state ...

    Peter
     
  5. Mar 17, 2019 at 4:47 AM #4


    Hi Andrew ...

    Still reflecting on what you have written ...

    Indeed ... you write:

    " ... ... So for any ##T## in that space there exists a set of real numbers ##A_T## indexed by the set ##S## of increasing ##k##-tuples in ##\{1, ...,n\}^k## ... .. "


    Can you explain in simple terms what you mean by this ,,, perhaps also giving a simple example ...

    Also ... what is the exact form and nature of the ##a_J## .... ?

    Peter
     
  6. Mar 17, 2019 at 5:19 AM #5

    andrewkirk

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    Consider where n=5 and k=3 then ##A_T## is the set of all increasing 3-tuples out of the numbers 1,...,5, which is:

    123, 124, 125, 134, 135, 145, 234, 235, 245, 345

    The set ##A_T## has those ten members. The fourth member is the 3-tuple (1,3,4). If we use J to denote that member then we have ##J=(1,3,4)## and ##j_1=1,j_2=3,j_3=4##.

    The set ##A_T## doesn't depend much on what T is. All it uses from T is the value of ##k##. Given a ##k##-tensor T, the set ##A_T## is the set of all increasing k-tuples out of the numbers 1...n. In this example, the tensor T has order 3, and ##A_T## is the set of all 3-tuples (triples) from 1,...,5.
     
  7. Mar 17, 2019 at 5:31 AM #6
    Thanks Andrew ... but ... I'm lost ... I'm probably trying you patience ,,,

    But can you explain from the definition of the space ##\Lambda^k(\mathbb R^n)^*## how we end up considering a set a set of real numbers ##A_T## in the first place ...

    My apologies for being slow to catch on ....

    Peter
     
  8. Mar 17, 2019 at 1:58 PM #7

    Hi Andrew ... I should be more specific ...

    ... indeed ... I should ask a specific question ...

    ... so ...

    Now ... ##\Lambda^k(\mathbb R^n)^*## is the vector space of alternating tensors or alternating multilinear mappings of degree or rank ##k## ... and I am assuming that it is a vector space over the field ##\mathbb{R}## ... is that assumption correct?

    The basis of ##\Lambda^k(\mathbb R^n)^*## is the set of ##k##-linear covectors or mappings ...

    ##\text{dx}_{I_1}, \text{dx}_{I_2}, \text{dx}_{I_3}, \cdot \cdot \cdot , \text{dx}_{I_l}## where ##l = \begin{pmatrix} n \\ k \end{pmatrix}##

    ... so an element ##T## of ##\Lambda^k(\mathbb R^n)^*## can be expressed as

    ##T = a_1 \text{dx}_{I_1} + a_2 \text{dx}_{I_2} + \cdot \cdot \cdot + a_l \text{dx}_{I_l}## ... where ##a_i \in \mathbb{R}## for all ##i## such that ##1 \le i \le l ## ... ...

    ... ... is that correct?

    So I am wondering how the ##a_i## above relate to your ##a_J## ... indeed are the ##a_J## just real numbers ...


    Hope you can help ...

    Peter
     
  9. Mar 17, 2019 at 4:16 PM #8

    andrewkirk

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    That is a spanning set, not a basis, because it includes ##d\mathbf x_I## for non-increasing ##I##. So for ##k=3## it would include ##d\mathbf x_{1,3,5}## and ##d\mathbf x_{3,1,5}##, which are linearly dependent since ##d\mathbf x_{1,3,5}=-d\mathbf x_{3,1,5}##.

    To make it a basis, we need to restrict to ##d\mathbf x_I## for increasing ##I##.

    That is correct if you use the numbers ##1,...,m## to index ##A_T##, where ##m## is the number of distinct, increasing ##k##-tuples that can be drawn from ##1,...,n##. In my example above with k=3,n=5, we have m=10. But you also need to specify an indexing of ##A_T##, ie label its elements with the numbers 1 to m.

    My approach avoids having to do that, as well as simplifying the notation, by noting that indexes don't have to be integers. We can use the k-tuples themselves to index the k-forms. That is what the author has done above when he writes things like ##d\mathbf x_{2,4,5,1}##. That is for the case k=4 and##n\ge 5##, and the example k-tuple is (2,4,5,1). The corresponding coefficient would be ##a_{(2,4,5,1)}##.

    A longer, but more explanatory presentation of the formula [EDIT: fixed an error in that, which used ##I## instead of ##J## for the index]:

    $$T = \sum_{J\in A_T} a_J d\mathbf x_J$$

    is

    $$T = \sum_{\substack{(i_1,...,i_k)\in \{1,...,n\}^k\\i_1<i_2<....<i_k}} a_{(i_1,...,i_k)} d\mathbf x_{(i_1,...,i_k)}$$

    or, even more lengthily:

    $$T = \sum_{i_1=1}^n \sum_{i_2=i_1+1}^n...\sum_{i_k=i_{k-1}+1}^n
    a_{(i_1,...,i_k)} d\mathbf x_{(i_1,...,i_k)}$$

    It may help to read the short wiki article on index sets. We are first introduced to indexing in cases where the index set is always a contiguous set of natural numbers starting with 1 or 0. But as we move into more advanced maths, it becomes useful to realise that an indexing of a set is just a surjection from another set (the index set) onto the first one, and the index set doesn't have to be integers. It can for instance be k-tuples. Doing this allows simplification of notation and avoids begging questions such as 'how do we order and number the set ##A_T## of increasing k-tuples?'.
     
    Last edited: Mar 17, 2019 at 5:45 PM
  10. Mar 17, 2019 at 5:03 PM #9
    Andrew ... thank you for a most helpful post ...

    Have gotten the main ideas now ...

    Still reflecting on what you have written ...

    Thanks again...

    Peter
     
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