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
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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 ) ⊗ w = v 1 ⊗ w + v 2 ⊗ w ) ((v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w) (( v 1 + v 2 ) ⊗ w = v 1 ⊗ w + v 2 ⊗ w )
Compatible with scalar multiplication ( c ( v ⊗ w ) = ( c v ) ⊗ w = v ⊗ ( c w ) ) (c(v \otimes w) = (cv) \otimes w = v \otimes (cw)) ( c ( v ⊗ w ) = ( c v ) ⊗ w = v ⊗ ( c w ))
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_1 + v_2, w) - (v_1, w) - (v_2, w) ( v 1 + v 2 , w ) − ( v 1 , w ) − ( v 2 , w )
( v , w 1 + w 2 ) − ( v , w 1 ) − ( v , w 2 ) (v, w_1 + w_2) - (v, w_1) - (v, w_2) ( v , w 1 + w 2 ) − ( v , w 1 ) − ( v , w 2 )
( c v , w ) − c ( v , w ) (cv, w) - c(v, w) ( c v , 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