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Tensor product

  1. Jan 12, 2009 #1
    If V1 and V2 are both subspaces of a vector space V, then in order for their tensor product to be defined, does the intersection of V1 and V2 have to be 0?
    Last edited: Jan 12, 2009
  2. jcsd
  3. Jan 13, 2009 #2


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    No, infact an important example of tensor product is that between a vector space and itself.

    One possible definition of tensor product is that it maps two vector spaces onto ordered pairs.
    UxV contains ordered pairs (u,v) with u from U and v from V.
    Edited to add. UxV also contains linear combinations of ordered pairs.
    Last edited: Jan 13, 2009
  4. Jan 13, 2009 #3
    I see. But I have to say, I'm puzzled about this: according to one definition I have, the operation on pairs (u,v), u belonging to U and v belonging to V, that is used to construct the tensor product has to be bilinear. But that certainly isn't a sufficient condition, is it? One must also have that the operation in question applied to the whole of the cartesian product of a basis of U and a basis of V forms a basis for the tensor product, right? Does the tensor product always exist (i.e. is there always such a bilinear operation and vector space for any ordered pair of vector spaces)?
  5. Jan 13, 2009 #4


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    The tensor product always exists.
    Existence is not a problem because tensor product is more a construction than an operation. That is the operation always endows the product with the required properties.
    lets assume U and V are vector spaces over the same field F.
    we want
    so in declaring the tensor product has this property we determin the tensor product uniquely.
    Since the tensor product is bilinear we can obtain a basis from the bases of the spaces used to construct it.
    {u(i)} is a basis for U
    {v(j)} is a basis for V
    {u(i),v(j)} is a basis for UxV

    it is important to notice that
    X is an element of UxV does not mean X is of the form (u,v)
    UxV contains all elements of the form (u,v), but also all linear combinations of such terms.
    By a counting argument if
    dim(elements of the form (u,v))=m+n
    I see I was unclear above I said
    UxV contains ordered pairs (u,v) with u from U and v from V
    which is true but it contains more ie sums of such.
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