4.2.5 Tensor Bilinear Vector Output Case

The Tensor Bilinear Vector Output Case explores how bilinear maps produce vector outputs in tensor algebra, bridging linear transformations and multilinear structures.

Tensor Bilinear Vector Output Case is the specialization of a bilinear map in which the codomain is a vector space of dimension greater than one rather than the scalar field, so that the map assigns a full vector to every ordered pair of input vectors while remaining linear in each argument separately. Formally this is a map $B : V \times W \to U$ with $dim (U) \geq 1$ treated as an arbitrary vector space rather than the ground field itself, and it is the setting in which most bilinear operations of geometry and physics, including the cross product and matrix multiplication, naturally live.

Structural Description

Component Functions

A vector-valued bilinear map decomposes into a family of scalar-valued bilinear maps, one for each coordinate of the output space. If ${u_{k}}$ is a basis of $U$ , then

B (v, w) = \sum_{k} B^{k} (v, w) u_{k}

where each component $B^{k} : V \times W \to F$ is an ordinary scalar-valued bilinear map. This decomposition shows the vector output case is not a fundamentally different object but a coordinated bundle of scalar output cases indexed by the output dimension.

Three-Index Coefficient Array

In finite dimensions with bases on all three spaces, the map is fully determined by a three-index array of structure constants

c_{i j}^{k} = B^{k} (e_{i}, f_{j})

so that for general inputs

B (v, w) = \sum_{i, j, k} v^{i} w^{j} c_{i j}^{k} u_{k}

This three-index array is precisely a tensor of type $(1, 2)$ : one upper index for the output space and two lower indices for the two input slots.

Relation to Tensor Type

One Contravariant, Two Covariant Indices

Because the output carries a free upper index while each input contributes a lower index, the vector output case corresponds to an element of $U \otimes V^{\otimes W^{}}$ . This is in direct contrast to the scalar output case, which has no free upper index and corresponds instead to an element of $V^{\otimes W^{}}$ alone.

Currying to a Linear Map of Matrices

Fixing the second argument produces a linear map $V \to U$ , and since $U$ has dimension greater than one, this induced map can be represented by a genuine matrix rather than a row vector, distinguishing the computational character of the vector output case from the scalar case where the induced map is a functional.

Canonical Examples

The Cross Product

The three-dimensional cross product $\times : ℝ^{3} \times ℝ^{3} \to ℝ^{3}$ is a vector-valued bilinear map whose structure constants are given by the Levi-Civita symbol, and it is antisymmetric under exchange of its two input slots.

Matrix Multiplication

Multiplication of an $m \times n$ matrix by an $n \times p$ matrix, viewed as a map from pairs of matrices to their product, is bilinear with an output space of dimension $m p$ , exemplifying a vector output case whose output dimension can be large relative to the input dimensions.

Lie Bracket

The Lie bracket $[\cdot, \cdot] : 𝛌 \times 𝛌 \to 𝛌$ on a Lie algebra is a vector-valued bilinear map whose structure constants encode the algebra's entire multiplicative structure.

Consequences for Tensor Construction

Building Higher-Rank Tensors from the Vector Case

Because the vector output case already carries one contravariant index, iterating the construction, by letting the output space itself be a space of linear maps, is the standard route to producing tensors of higher total rank without leaving the bilinear framework, since a vector-valued bilinear map into a space of linear maps is equivalent to a trilinear map with scalar output.

Nondegeneracy Is Not Applicable

Unlike the scalar output case, where nondegeneracy is defined through an induced isomorphism to a dual space, the vector output case has no single canonical notion of nondegeneracy; instead, injectivity or surjectivity must be examined slot by slot or through the rank of the induced linear map for each fixed argument, reflecting the greater structural freedom of mapping into a multidimensional codomain.