4.10.1 Tensor Vector Valued Output

Tensor Vector Valued Output represents outputs as vectors within tensor algebra, enabling multi-dimensional data transformation and analysis.

Tensor Vector Valued Output is the specific element of a vector space $W$ produced by fully evaluating a vector-valued multilinear map on one filled tuple of arguments, carrying, unlike a scalar valued output, its own internal magnitude and directional structure that can itself be added to and combined with other elements of $W$. It is the terminal result of a vector-valued tensor evaluation, and unlike a bare number, it remains a genuine geometric or algebraic object in its own right after the evaluation is complete.

Formal Definition

The Output as an Element of the Codomain

For a vector-valued multilinear map $T : V_1 \times \cdots \times V_k \to W$ and a specific tuple $(v_1, \ldots, v_k)$, the vector valued output is

w = T (v_{1}, \dots, v_{k}) \in W

Unlike a scalar valued output, $w$ is not a terminal, structureless quantity: it can be added to other elements of $W$, scaled, and, if a basis of $W$ is chosen, expanded into its own set of coordinates. It is a full-fledged vector, retaining all the algebraic structure of $W$.

Determination by Basis Expansion

If ${f_1, \ldots, f_m}$ is a basis of $W$, the vector valued output expands as

w = \sum_{a = 1}^{m} w^{a} f_{a}

where each coefficient $w^{a}$ is the scalar valued output of the $a$-th component map $T_a$ evaluated on the same input tuple, showing that a single vector valued output is, once a basis is fixed, simply an ordered collection of $m$ ordinary scalar valued outputs assembled together.

Properties of the Vector Valued Output

Basis-Independence as an Abstract Object

The vector valued output $w$ itself, considered as an abstract element of $W$, does not depend on any choice of basis, even though its coordinate expansion $w^{a}$ does; changing the basis of $W$ changes the numeric coefficients used to describe $w$, but not the underlying vector, exactly as with any ordinary vector in linear algebra.

Additivity of the Output Under Additivity in Any Slot

Because $T$ is multilinear, splitting one argument as a sum $u + v$ splits the resulting vector valued output correspondingly as a sum of two vector valued outputs, $T(\ldots, u, \ldots) + T(\ldots, v, \ldots)$, computed and combined within $W$ using $W$'s own vector addition; this is a direct carryover of slotwise additivity, now expressed at the level of genuine vector sums in the codomain rather than scalar sums.

Applications of Vector Valued Outputs

Torsion as a Directly Usable Vector

The vector valued output of the torsion tensor, evaluated on two tangent vectors, is itself a tangent vector describing the failure of a small parallelogram of flows to close; because this output remains a vector rather than reducing to a number, it can be directly compared, added, and further manipulated using the ambient vector space's own geometric operations.

Operator-Valued Output as an Applicable Transformation

When the codomain is a space of linear operators, the vector valued output, itself an operator, can be applied to a further vector to produce yet another vector, chaining the tensor's output into subsequent computations in a way that a bare scalar output could never support.

Aggregating Outputs Across a Domain

A vector valued tensor field, assigning a vector valued output to every point of a manifold, produces at each point a genuine tangent (or otherwise structured) vector, which can then be integrated, differentiated, or compared with neighboring outputs using the vector space operations native to the codomain fiber.

Summary of Key Points

A vector valued output is a genuine element of the codomain space $W$, retaining $W$'s own vector addition and scalar multiplication after evaluation.
Expanding the output in a basis of $W$ decomposes it into $m$ scalar coordinates, each itself the output of a separate scalar-valued component map.
The abstract vector valued output does not depend on the choice of basis of $W$, only its coordinate expansion does.
Slotwise additivity of the tensor carries over directly to additivity of the vector valued output, computed using the codomain's own vector addition.
Vector valued outputs support further vector-space operations, such as addition, comparison, and, when operator-valued, direct application to other vectors.