Here is a bilinear map φ of two vector spaces V₁ & V₂ into another vector space N with(adsbygoogle = window.adsbygoogle || []).push({});

respect to the bases B = {e₁,e₂} & B' = {e₁',e₂'}:

φ : V₁ × V₂ → N | (m₁,m₂) ↦ φ(m₁,m₂) = φ(∑ᵢλᵢeᵢ,∑_{j}μ_{j}e'_{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∑_{j}λᵢμ_{j}φ(eᵢ,e_{j}')

(Hope that's right!)

Now in a discussion of bilinear (quadratic, hermitian) forms the terms φ(eᵢ,e_{j}') in ∑ᵢ∑_{j}λᵢμ_{j}φ(eᵢ,e_{j}')

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,e_{j}') 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 × ... × V_{n}→ N

φ : V₁ × V₂ × ... × V_{n}→ V₁ × V₂ × ... × V_{n}

φ : V₁ × V₂ × ... × V_{n}→ W₁ × W₂ × ... × W_{n}

φ : V₁ × V₂ × ... × V_{n}→ W₁ × W₂ × ... × W_{m}

(all being vector spaces) as in what kind of books there are discussing this, what this would be useful for etc...

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# Multilinear Maps

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