Tensor products combine vector spaces, creating a new space that captures bilinear relationships. They're a powerful tool for studying multilinear algebra, allowing us to extend linear concepts to higher dimensions.

Understanding tensor products is crucial for grasping advanced topics in multilinear algebra. They provide a framework for working with complex structures and are essential in fields like quantum mechanics and machine learning.

Tensor product of vector spaces

Definition and properties

• Tensor product of vector spaces V and W over field F denoted as V ⊗ W
• V ⊗ W forms a vector space over F
• Equipped with bilinear map ⊗: V × W → V ⊗ W sending (v, w) to v ⊗ w
• Elements of V ⊗ W consist of linear combinations of pure tensors v ⊗ w (v ∈ V, w ∈ W)
• Satisfies distributivity over vector addition $((v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w)$
• Compatible with scalar multiplication $(c(v \otimes w) = (cv) \otimes w = v \otimes (cw))$

Universal property

• Most general bilinear map from V × W
• For any bilinear map f: V × W → U, unique linear map f̃: V ⊗ W → U exists
• Satisfies f = f̃ ∘ ⊗
• Any bilinear map can be factored through tensor product
• Allows reduction of multilinear problems to linear ones (matrix multiplication)

Constructing the tensor product

Free vector space approach

• Start with free vector space F(V × W) generated by V × W
• Define subspace R in F(V × W) generated by elements:
  • $(v_1 + v_2, w) - (v_1, w) - (v_2, w)$
  • $(v, w_1 + w_2) - (v, w_1) - (v, w_2)$
  • $(cv, w) - c(v, w)$ for v, v₁, v₂ ∈ V, w, w₁, w₂ ∈ W, c ∈ F
• Tensor product V ⊗ W defined as quotient space F(V × W) / R
• Canonical bilinear map ⊗: V × W → V ⊗ W defined by (v, w) ↦ [(v, w)]
  • [(v, w)] denotes equivalence class of (v, w) in quotient space

Verification of properties

• Demonstrate constructed tensor product satisfies universal property
• Show any bilinear map f: V × W → U factors uniquely through V ⊗ W
• Verify resulting vector space meets all tensor product requirements
• Prove distributivity and scalar multiplication compatibility
• Confirm bilinearity of canonical map ⊗

Basis for the tensor product

Finite-dimensional case

• Given basis {v₁, ..., vₙ} for V and {w₁, ..., wₘ} for W
• Basis for V ⊗ W formed by {vᵢ ⊗ wⱼ | 1 ≤ i ≤ n, 1 ≤ j ≤ m}
• Dimension of V ⊗ W equals product of dimensions: dim(V ⊗ W) = dim(V) · dim(W)
• Any element in V ⊗ W uniquely expressed as linear combination of vᵢ ⊗ wⱼ
• Coordinates of tensor arrangeable in n × m matrix
• Examples:
  • R² ⊗ R³ has basis {e₁ ⊗ f₁, e₁ ⊗ f₂, e₁ ⊗ f₃, e₂ ⊗ f₁, e₂ ⊗ f₂, e₂ ⊗ f₃}
  • C² ⊗ C² has basis {e₁ ⊗ e₁, e₁ ⊗ e₂, e₂ ⊗ e₁, e₂ ⊗ e₂}

Infinite-dimensional case

• Tensor product basis still formed by tensoring basis elements
• Additional considerations for completeness may be necessary
• Hilbert space tensor products require completion in appropriate topology
• Examples:
  • L²(R) ⊗ L²(R) basis involves infinite tensor products of basis functions
  • Tensor product of function spaces (C[0,1] ⊗ C[0,1])

Uniqueness of the tensor product

Existence proof

• Construct tensor product using universal property
• Verify constructed space satisfies all required tensor product properties
• Show bilinearity of canonical map ⊗: V × W → V ⊗ W
• Demonstrate universal property holds for constructed tensor product
• Example: Construct R² ⊗ R³ and verify its properties

Uniqueness up to isomorphism

• Consider two tensor products V ⊗ W and V ⊗' W with bilinear maps ⊗ and ⊗'
• Use universal property to construct unique linear maps:
  • φ: V ⊗ W → V ⊗' W
  • ψ: V ⊗' W → V ⊗ W
• Prove φ and ψ are inverses establishing isomorphism between V ⊗ W and V ⊗' W
• Show isomorphism preserves bilinear structure φ(v ⊗ w) = v ⊗' w for all v ∈ V, w ∈ W
• Conclude tensor products are isomorphic as vector spaces
• Equivalent as universal objects for bilinear maps from V × W
• Example: Prove uniqueness of R² ⊗ R³ constructed using different methods