4.10.5 Tensor Vector Valued Tensor Role

Tensor Vector Valued Tensor Role describes how tensors act on vector spaces, mapping vectors to tensors through structured linear operations.

Tensor Vector Valued Tensor Role is the identification of a vector-valued multilinear map with a single tensor that carries an extra output slot, so that the map itself is represented as one algebraic object rather than as a family of scalar-valued maps. Where an ordinary tensor is built to accept vectors and covectors and return a scalar in the underlying field, a vector-valued multilinear map instead returns an element of some target vector space W. The tensor role captures how this map can still be encoded tensorially, by treating the output space W as an additional tensor factor rather than as the codomain of a scalar pairing, which places vector-valued multilinear maps and ordinary tensors inside the same unifying framework.

From Scalar-Valued Maps to Vector-Valued Maps

The Scalar Case as a Starting Point

A type (p, q) tensor on a vector space V is ordinarily understood as a multilinear map

T : \times_{i = 1}^{p} V^{*} \times \times_{j = 1}^{q} V \to F

whose value always lands in the base field F. This scalar output is what allows the map to be identified with an element of a tensor product of copies of V and V*.

Replacing the Field with a Vector Space

A vector-valued multilinear map generalizes this by replacing the field F with an arbitrary finite-dimensional vector space W:

M : V_{1} \times V_{2} \times \dots \times V_{k} \to W

Here each V_i may be a vector space or its dual, and M is linear in each argument separately. Because the codomain is no longer the field itself, M cannot immediately be read off as an element of a tensor product built purely from the V_i; the tensor role addresses exactly this gap.

Constructing the Tensor Role

Absorbing the Output Space

The key step is to absorb W into the tensor product as an additional factor, using the dual pairing between W and its dual W*. Any vector-valued multilinear map M corresponds to a unique element

\tilde{M} \in W \otimes V_{1}^{*} \otimes V_{2}^{*} \otimes \dots \otimes V_{k}^{*}

so that W occupies the role formerly played by the base field, and the remaining factors record the dual spaces paired against each argument. This element is what is meant by the tensor role of the vector-valued map: it is a genuine tensor, only one of whose factors happens to be the target space W rather than V or V*.

Recovering the Map from the Tensor

Given the tensor M~, the original map M is recovered by contracting each argument against its corresponding dual factor and leaving the W factor untouched, so that feeding in vectors v_1, ..., v_k produces an element of W:

M (v_{1}, \dots, v_{k}) = (id \otimes_{1} v_{1} ⋮ v_{k}) (\tilde{M})

using the natural evaluation pairing between each V_i* factor and the argument v_i supplied to it, while the identity map on W leaves the output factor as an element of W.

Component Description of the Role

Indexing the Output Factor

In component form, choosing a basis w_1, ..., w_m of W and dual bases for each V_i*, the tensor M~ acquires one extra index ranging over the basis of W, alongside the indices already carried by the argument spaces:

{\tilde{M}}_{i_{1} i_{2} \dots i_{k}}^{a}

where the lower indices i_1, ..., i_k correspond to the argument slots and the upper index a ranges over the basis of W. Because a is not contracted against any argument, it survives as the free index of the output vector once all argument slots are filled.

Interpreting the Free Index

This free index is the component-level signature of the tensor role: it marks the slot of the tensor that is not paired away during evaluation, and its presence is what distinguishes a vector-valued tensor from a scalar-valued one. Every other index behaves exactly as it would in an ordinary tensor, transforming contravariantly or covariantly according to whether it is paired with a vector or a covector argument.

Special Cases of the Role

Linear Maps as Vector-Valued Tensors of Rank One

When k = 1 and the single argument space is V, the tensor role reduces to the familiar identification of a linear map V → W with an element of W ⊗ V*, which is the ordinary tensor description of a linear transformation between two different spaces.

Bilinear Vector-Valued Products

When k = 2, the tensor role describes bilinear operations that output vectors rather than scalars, such as a cross product or a Lie bracket, each of which corresponds to an element of W ⊗ V_1* ⊗ V_2* once the tensor role is applied.

Scalar-Valued Maps as a Degenerate Case

Setting W equal to the base field F recovers the ordinary scalar-valued tensor, since F is one-dimensional and its dual is naturally identified with F itself, showing that the tensor role for scalar-valued maps is simply the special case where the extra output factor carries no independent information.

Diagrammatic Summary

The diagram shows the tensor M~ with k argument slots drawn from the dual spaces V_1*, ..., V_k*, contracted away when vectors are supplied, and a single distinguished output slot corresponding to W, which remains after every argument is paired off and carries the resulting vector value of the map.