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Multilinear Maps

  1. Oct 9, 2011 #1
    Here is a bilinear map φ of two vector spaces V₁ & V₂ into another vector space N with
    respect to the bases B = {e₁,e₂} & B' = {e₁',e₂'}:

    φ : V₁ × V₂ → N | (m₁,m₂) ↦ φ(m₁,m₂) = φ(∑ᵢλᵢeᵢ,∑jμje'j)
    _______________________________= φ(λ₁e₁ + λ₂e₂,μ₁e₁' + μ₂e₂') = φ(λ₁e₁ + λ₂e₂,μ₁e₁') + φ(λ₁e₁ + λe₂,μ₂e₂')
    _______________________________= φ(λ₁e₁,μ₁e₁') + φ(λ₂e₂,μ₁e₁') + φ(λ₁e₁,μ₂e₂') + φ(λe₂,μ₂e₂')
    _______________________________= λ₁μ₁φ(e₁,e₁') + λ₂μ₁φ(e₂,e₁') + λ₁μ₂φ(e₁,e₂') + λ₂μ₂φ(e₂,e₂')
    _______________________________= ∑ᵢ∑jλᵢμjφ(eᵢ,ej')
    (Hope that's right!)

    Now in a discussion of bilinear (quadratic, hermitian) forms the terms φ(eᵢ,ej') in ∑ᵢ∑jλᵢμjφ(eᵢ,ej')
    are elements of a field, & there seems to be quite a lot of theory built up for the special
    case of N being a field, but thus far I can't really find any discussion of what happens
    when the φ(eᵢ,ej') elements are elements of a general vector space.
    Furthermore, the general case of multilinear (n-linear) maps expressed in this way
    seem to follow the same pattern of focusing on fields (or rings), e.g. determinants.
    The limited discussions of tensors & exterior algebra I've seen also follow this format of
    focusing on maps into a field.

    Just wondering what there is on maps of the form
    φ : V₁ × V₂ × ... × Vn→ N
    φ : V₁ × V₂ × ... × Vn→ V₁ × V₂ × ... × Vn
    φ : V₁ × V₂ × ... × Vn→ W₁ × W₂ × ... × Wn
    φ : V₁ × V₂ × ... × Vn→ W₁ × W₂ × ... × Wm

    (all being vector spaces) as in what kind of books there are discussing this, what this would be useful for etc...
     
    Last edited: Oct 9, 2011
  2. jcsd
  3. Oct 9, 2011 #2

    Deveno

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    i believe that what you are looking for is "the tensor product" which is "the most general bilinear map" (with two vector spaces) or "multilinear map" (with more than two).
     
  4. Oct 10, 2011 #3
    Thanks for the response, so I take it that φ : V₁ × V₂ → N is the "most general bilinear map"
    that can be formed because of the "universal property" [that φ is the most general bilinear map because
    for any other bilinear map h : V₁ × V₂ → M you have a unique linear map ω : N → M such that h = ω o φ]

    which if I understand it correctly says that:
    ω o φ : V₁ × V₂ → M | (m₁,m₂) ↦ (ω o φ)(m₁,m₂) = ω[φ(∑ᵢλᵢeᵢ,∑jμje'j)]
    __________________________ = ω[λ₁μ₁φ(e₁,e₁') + λ₂μ₁φ(e₂,e₁') + λ₁μ₂φ(e₁,e₂') + λ₂μ₂φ(e₂,e₂')]
    __________________________ = ω[λ₁μ₁φ(e₁,e₁')] + ω[λ₂μ₁φ(e₂,e₁')] + ω[λ₁μ₂φ(e₁,e₂')] + ω[λ₂μ₂φ(e₂,e₂')]
    __________________________ = λ₁μ₁ω[φ(e₁,e₁')] + λ₂μ₁ω[φ(e₂,e₁')] + λ₁μ₂ω[φ(e₁,e₂')] + λ₂μ₂ω[φ(e₂,e₂')]
    __________________________ = ∑ᵢ∑jλᵢμjω[φ(eᵢ,ej')]
    is linear, which itself is just a fancy way of ensuring that image elements of φ are linear
    because they can be mapped to image elements of a separate bilinear map h.

    So I take it that when you generalize this to an indexed product of vector spaces the
    tensor product that results is also the the most general multilinear map φ : ΠᵢVᵢ→ N?
     
  5. Oct 10, 2011 #4

    lavinia

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    A vector space by definition has scalars in a field.

    Linearity makes sense when vector space is replaced by R-module where R is a commutative ring.
     
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