Tensor Products of Vector Spaces — Mathematics Knowledge System

Overview

The tensor product V ⊗ W of two vector spaces V and W is a new vector space that 'contains all bilinear combinations' of elements from V and W. Tensor products formalize the construction of multilinear objects, underlie the mathematics of quantum mechanics (composite systems), and appear in deep learning (weight tensors, multilinear maps).

Intuition

Multiplication of matrices is not the same as the tensor product. The tensor product is about combining spaces: if V has dimension m and W has dimension n, then V ⊗ W has dimension mn. Each 'pure tensor' u ⊗ v records a pair; but V ⊗ W also includes sums of such pairs (mixed tensors) that can't be written as a single u ⊗ v. The tensor product converts bilinear maps into linear maps — the universal property.

Formal Definition

Definition

The tensor product V ⊗ W is the unique (up to isomorphism) vector space equipped with a bilinear map ⊗: V × W → V ⊗ W satisfying the universal property:

\dim(V \otimes W) = \dim(V) \cdot \dim(W)

Dimension

\text{Basis of } V \otimes W: \{e_i \otimes f_j\}_{i=1,\ldots,m;\, j=1,\ldots,n}

If {e_i} is a basis of V and {f_j} is a basis of W, then {e_i ⊗ f_j} is a basis of V ⊗ W.

Basis

(\alpha u_1 + \beta u_2) \otimes v = \alpha(u_1 \otimes v) + \beta(u_2 \otimes v)

Bilinearity

\text{Universal property: every bilinear map } B: V \times W \to Z \text{ factors uniquely through } \tilde{B}: V \otimes W \to Z

Universal property

Notation

Notation	Meaning
$V \otimes W$	Tensor product of vector spaces V and W
$u \otimes v$	Elementary tensor (pure tensor): element of V ⊗ W

Properties

Commutativity up to isomorphism

V \otimes W \cong W \otimes V \text{ via } u \otimes v \mapsto v \otimes u.

Associativity

(U \otimes V) \otimes W \cong U \otimes (V \otimes W).

Distributivity over direct sum

U \otimes (V \oplus W) \cong (U \otimes V) \oplus (U \otimes W).

Kronecker product

\text{In coordinates, if } A \in M_{m \times n}, B \in M_{p \times q}, \text{ the tensor product of linear maps corresponds to the Kronecker product } A \otimes B \in M_{mp \times nq}.

Applications

Real-world · Physics

Quantum mechanics: the state space of a composite system AB is the tensor product H_A ⊗ H_B. Entangled states are exactly the non-pure-tensor elements.

Worked Examples

1
Basis of V ⊗ W is all e_i ⊗ f_j for i=1,2 and j=1,2,3.
2
The 6 basis elements are: e_1⊗f_1, e_1⊗f_2, e_1⊗f_3, e_2⊗f_1, e_2⊗f_2, e_2⊗f_3.
$\dim(V \otimes W) = 2 \times 3 = 6$

✓ Answer

Basis: {e_1⊗f_1, e_1⊗f_2, e_1⊗f_3, e_2⊗f_1, e_2⊗f_2, e_2⊗f_3}; dimension 6.

Practice Problems

Mediumfree response

If V has dimension 4 and W has dimension 5, what is dim(V ⊗ W)?

Mediumfree response

Show that not every element of V ⊗ W is a pure tensor u ⊗ v by giving an example in R^2 ⊗ R^2.

Common Mistakes

Common Mistake

The tensor product is the same as the direct product V × W.

V × W has dimension dim(V)+dim(W) (direct sum); V ⊗ W has dimension dim(V)·dim(W). They are different constructions.

Quiz

Summary

The tensor product V ⊗ W has dimension dim(V) × dim(W), with basis {e_i ⊗ f_j}.
Pure tensors u ⊗ v span V ⊗ W but not every element is pure (non-pure elements are 'entangled').
Universal property: every bilinear map V×W→Z factors through a unique linear map V⊗W→Z.
In coordinates, tensor product of linear maps is the Kronecker product.
V ⊗ W ≅ W ⊗ V (commutative up to isomorphism); (U⊗V)⊗W ≅ U⊗(V⊗W) (associative).

References

WebsiteWikipedia — Tensor product