Math 2.0

Linear and multilinear algebra; matrix theory Definition

Tensor product of vector spaces

Builds on Vector spaceBilinear form

Definition

Let FF be a field, and let VV and WW be vector spaces over FF. Let LL be the vector space over FF with basis the set V×WV \times W; write [v,w][v, w] for the basis vector of (v,w)(v, w). Let NN be the subspace of LL spanned by all elements of these forms, for v,v′∈Vv, v' \in V, w,w′∈Ww, w' \in W and a∈Fa \in F: [v+v′,w]−[v,w]−[v′,w],[v,w+w′]−[v,w]−[v,w′],[v + v', w] - [v, w] - [v', w], \qquad [v, w + w'] - [v, w] - [v, w'], [av,w]−a[v,w],[v,aw]−a[v,w].[av, w] - a[v, w], \qquad [v, aw] - a[v, w]. The tensor product of VV and WW is the quotient space V⊗W=L/NV \otimes W = L / N. For v∈Vv \in V and w∈Ww \in W, the class of [v,w][v, w] is written v⊗wv \otimes w.

Remarks

The map (v,w)↦v⊗w(v, w) \mapsto v \otimes w is bilinear, and the vectors v⊗wv \otimes w span V⊗WV \otimes W. Not every vector of V⊗WV \otimes W is of the form v⊗wv \otimes w; for example, e1⊗e1+e2⊗e2e_1 \otimes e_1 + e_2 \otimes e_2 in F2⊗F2F^2 \otimes F^2 is not. The tensor product is characterised by its universal property. For finite-dimensional spaces, V∗⊗WV^* \otimes W is isomorphic to the space of linear maps from VV to WW.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Tensor_product, As a quotient space
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DefinesTensor product of vector spaces
UsesVector space, Bilinear form

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