Linear algebra MOC

𝕂-tensor product of vector spaces

Let 𝑉1,𝑉2 be vector spaces over 𝕂. The tensor product 𝑉1 ⊗𝕂𝑉2 is a vector space which allows one to treat 𝕂-bilinear maps from 𝑉1 ×𝑉2 as 𝕂-linear maps from 𝑉1 ⊗𝕂𝑉2, as ensured by the Universal property. See Tensor product of modules over a commutative ring for a direct generalization.

Universal property

The tensor product is uniquely characterised by the following universal property:

If 𝑉1,𝑉2,𝑊 are vector spaces over 𝕂 and ℎ :𝑉1 ×𝑉2 →𝑊 is a bilinear map there exists a unique linear map ¯𝑓 :𝑉1 ⊗𝑉2 →𝑊 such that ℎ(𝑢,𝑣) =¯ℎ(𝑢 ⊗𝑣).

Proof

proof Let ℎ :𝑉1 ×𝑉2 →𝑊 be a bilinear map.

Finite dimensional characterization

The tensor product 𝑉 ⊗𝑊 of vector spaces 𝑉,𝑊 over a field 𝕂 is the vector space of bilinear forms 𝑉∗ ×𝑊∗ →𝕂, equipped with a bilinear¹ map ( ⊗) :𝑉 ×𝑊 →𝑉 ⊗𝑊 such that linalg

(𝑣⊗𝑤)(𝑓,𝑔)=𝑓(𝑣)𝑔(𝑤)

for 𝑣 ∈𝑉, 𝑤 ∈𝑊, 𝑓 ∈𝑉∗, 𝑔 ∈𝑊∗.²³ It follows that if {𝑣𝑖}𝑛𝑖=1 and {𝑤𝑗}𝑚𝑗=1 are bases of 𝑉 and 𝑊 respectively, then {𝑣𝑖 ⊗𝑤𝑗}𝑛,𝑚𝑖,𝑗=1 defines a basis for the tensor product space 𝑉 ⊗𝑊. We call

𝑇∈𝑉⊗⋯⊗𝑉⏟__⏟__⏟𝑝⊗𝑉∗⊗⋯⊗𝑉∗⏟__⏟__⏟𝑞=T𝑝𝑞

a type (𝑝,𝑞) Tensor⁴

Warning

This characterization probably requires 𝑉∗∗ ≃𝑉 and hence finite dimensions.

Hilbert spaces

If 𝑉,𝑊 are finite-dimensional Hilbert spaces. then the tensor product 𝑉 ⊗𝑊 is a Hilbert space carrying the unique inner product given by

⟨𝑣⊗𝑤|𝐵⟩=𝐵(⟨𝑣|,⟨𝑤|)

Then if {𝑣𝑖}𝑛𝑖=1 and {𝑤𝑗}𝑚𝑗=1 are orthonormal bases of 𝑉,𝑊 respectively, {𝑣𝑖 ⊗𝑤𝑗}𝑛,𝑚𝑖,𝑗=1 is an orthonormal basis of 𝑉 ⊗𝑊.

Further characterizations

As quotient

One may construct the tensor product 𝑉 ⊗𝑊 as a quotient space of the free module generated by formal products of vectors in 𝑉 and 𝑊.

𝑉∗𝑊=𝕂[{𝑣∗𝑤∣𝑣∈𝑉,𝑤∈𝑊}]𝑈=span⁡{(𝛼𝑣1+𝛽𝑣2)∗(𝛾𝑤1+𝛿𝑤2)−𝛼𝛾𝑣1∗𝑤1−𝛼𝛿𝑣1∗𝑤2−𝛽𝛾𝑣2∗𝑤1−𝛽𝛿𝑣2∗𝑤2∣𝛼,𝛽,𝛾,𝛿∈𝕂,𝑣∈𝑉,𝑤∈𝑊}𝑉⊗𝑊=𝑉∗𝑊/𝑈

Properties

• dim⁡(𝑉 ⊗𝑊) =dim⁡𝑉 ⋅dim⁡𝑊

See also

• Tensor product of linear maps (functor)
• Tensor product of group representations
• Tensor algebra
• Tensor

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Footnotes

1. Simon defines these as “biantilinear” maps 𝑋 ×𝑌 →ℂ, which is of course completely equivalent. ↩

2. 1996, Representations of finite and compact groups, §II.5, p. 29 ↩

3. 2015. An Introduction to Tensors and Group Theory for Physicists, §3.4, p.70 ↩

4. Authors vary on the order of the tensor type, cf. @jeevanjeeIntroductionTensorsGroup2015 with @emamCovariantPhysicsClassical2021 (I use the convention of the latter, also aligns with Wikipedia) ↩