5.12.4 Tensor Product Bilinear Extension

Tensor Product Bilinear Extension is the process by which values specified only on a spanning set of pairs — most commonly pairs of basis vectors from two factor spaces — are propagated, using bilinearity alone, to a fully determined bilinear map defined on every pair in the whole Cartesian product, and the accompanying guarantee that this propagation is always both possible and unique.

Statement of the Extension Procedure

Let V and W be vector spaces over F with bases {e1, …, ep} and {f1, …, fq}. Given any assignment of target values

for every pair of basis indices (i, j), there exists a unique bilinear map B : V × W → U extending this assignment, obtained by expanding an arbitrary pair (v, w) in the chosen bases and setting

where vi and wj are the coordinates of v and w in the chosen bases. This is bilinear extension: a full bilinear map manufactured entirely from a finite table of values on basis pairs.

Why the Extension Is Well Defined and Unique

Bilinear extension relies on the fact that a bilinear map's behavior on a spanning set of pairs completely determines its behavior everywhere, with no ambiguity and no possibility of inconsistency.

Existence via Direct Construction

The formula above manifestly defines a map on all of V × W, and checking bilinearity directly — additivity and scalar compatibility in each argument — follows immediately from the formula's structure as a sum of scalar multiples of the fixed values uij, since expanding a sum in either v or w simply produces the corresponding additional terms in the sum defining B.

Uniqueness from Bilinearity Alone

Any bilinear map agreeing with the prescribed values on basis pairs must, by bilinearity, agree with the formula above on every pair (v, w), since bilinearity itself forces the expansion of B(v, w) in terms of B(ei, fj) once v and w are written in the chosen bases; there is therefore no second, different bilinear map consistent with the same table of basis values.

Bilinear Extension as the Concrete Face of the Universal Property

Bilinear extension is the down-to-earth, coordinate-based instance of the same guarantee that the universal property of the tensor product states abstractly.

Matching the Two Descriptions

The universal property guarantees that a multilinear (here, bilinear) map corresponds to a unique linear map on the tensor product; bilinear extension supplies the concrete recipe for constructing the bilinear map itself once only its values on a spanning family of pairs are known. The linear map on V ⊗ W corresponding to the extended B, via the universal property, is precisely the linear map sending each basis tensor ei ⊗ fj to the same prescribed value uij — so bilinear extension and the linear extension guaranteed by the induced basis are, under this correspondence, two descriptions of a single underlying construction.

Reduction to a Finite Specification Problem

Because bilinear extension shows that specifying a bilinear map is equivalent to filling in a finite table of pq values (one per pair of basis vectors), the seemingly infinite-dimensional problem of describing all bilinear maps on V × W reduces to the finite combinatorial problem of choosing an arbitrary element of U for each of the pq basis pairs — matching, via the universal property, the dimension count for Hom(V ⊗ W, U) obtained independently from the tensor product's dimension formula.

Extension Beyond Bases: Spanning Sets in General

Although bases are the most convenient spanning family to use, bilinear extension works with any spanning set, provided that the resulting relations remain consistent.

Consistency Requirement for Non-Basis Spanning Sets

If a spanning set used to prescribe values is not linearly independent, the prescribed values must satisfy compatibility conditions inherited from any linear relations among the spanning vectors, or the proposed extension will be inconsistent (no bilinear map can satisfy contradictory prescribed values simultaneously); using an actual basis sidesteps this issue entirely, since a basis has no linear relations to violate.

Practical Preference for Bases

For this reason, bilinear extension in practice is almost always carried out relative to a genuine basis of each factor space, both to guarantee automatic consistency of the extension and to obtain the cleanest possible correspondence with the coordinate array representation of the resulting bilinear map or its associated tensor.

Illustrative Diagram

A finite table of prescribed values on basis pairs, once fixed, extends uniquely to a fully defined bilinear map covering every pair in the ambient Cartesian product.