5.10.1 Tensor Product Dimension Multiplication

Tensor Product Dimension Multiplication is the specific algebraic law stating that the dimension of a tensor product of finite-dimensional vector spaces equals the product of the dimensions of the individual factors, together with the several independent arguments — basis counting, bilinear form counting, and inductive extension from two factors to many — that establish this law rigorously.

Statement of the Law

For finite-dimensional vector spaces V and W over a field F, with dim(V) = p and dim(W) = q,

extending to any finite number of factors as dim(V1 ⊗ ⋯ ⊗ Vn) = d1 × d2 × ⋯ × dn. The law is exact, holding regardless of the field F, and it is the source of the multiplicative (rather than additive) growth that distinguishes the tensor product from the direct sum.

Proof by Basis Counting

The most direct argument for dimension multiplication proceeds by explicitly constructing a basis of the tensor product and counting its size.

Constructing the Induced Basis

Choosing bases {e1, …, ep} of V and {f1, …, fq} of W, the tensors ei ⊗ fj, ranging over all pairs (i, j), form a basis of V ⊗ W: they span the space because every simple tensor v ⊗ w expands, by multilinearity, into a linear combination of these pairwise products once v and w are expressed in their respective bases; and they are linearly independent by the dual-functional argument used to establish the general induced basis construction.

Counting the Basis

Because this basis has exactly pq elements, dim(V ⊗ W) = pq follows immediately once the basis property is established; the dimension multiplication law is, in this sense, nothing more than a restatement of the size of a Cartesian product of two finite index sets.

Proof via Bilinear Form Counting

A second, independent argument derives the same law from the universal property of the tensor product, without constructing a basis explicitly.

Tensor Product as Classifying Space for Bilinear Maps

By the universal property, linear maps out of V ⊗ W correspond bijectively to bilinear maps out of V × W, so in particular dim(V ⊗ W) = dim of the space of bilinear forms on V × W (taking the target space to be F). The space of bilinear forms on V × W is well known, by direct construction using dual bases, to have dimension exactly pq, since a bilinear form is determined by, and freely specifiable at, its pq values on pairs of basis vectors.

Consistency of the Two Proofs

That both the basis-counting argument and the bilinear-form-counting argument arrive at the identical value pq is not a coincidence but a consequence of the universal property being precisely the statement that ties these two descriptions of the tensor product together; the agreement serves as an internal consistency check on the construction of the tensor product itself.

Extension to Many Factors

The two-factor law extends to n factors by an inductive argument that leans on the associativity of the tensor product.

Inductive Step

Assuming dim(V1 ⊗ ⋯ ⊗ Vk) = d1 d2 ⋯ dk has been established for k factors, the (k+1)-factor case follows by applying the two-factor law to the pair of spaces (V1 ⊗ ⋯ ⊗ Vk) and Vk+1, using the canonical associativity isomorphism (V1 ⊗ ⋯ ⊗ Vk) ⊗ Vk+1 ≅ V1 ⊗ ⋯ ⊗ Vk ⊗ Vk+1 to identify the two-factor result with the (k+1)-factor tensor product:

completing the induction with a base case (n = 1, or the trivially verified n = 2 case) already established.

Order-Independence

Because the associativity isomorphism used at each inductive step is canonical and the multiplication of positive integers is associative and commutative, the resulting product d1 d2 ⋯ dn does not depend on the order in which the factors are grouped or multiplied together, matching the fact that the abstract n-factor tensor product itself is, up to canonical isomorphism, independent of any particular grouping of its factors.

Illustrative Diagram

The grid's fifteen cells, five columns by three rows, correspond exactly to the fifteen basis tensors of a tensor product of a five-dimensional and a three-dimensional space, giving a direct visual confirmation of the dimension multiplication law.