4.8.2 Tensor Vector Codomain Case

In tensor algebra, the codomain of a vector defines the space into which the tensor maps, shaping its structure and transformation properties.

Tensor Vector Codomain Case is the situation in which a tensor's multilinear map produces, upon evaluation, an element of a genuine vector space of dimension one or greater, rather than a bare scalar of the base field. It generalizes the scalar codomain case by allowing the output to carry its own internal direction and magnitude structure, and it is the setting required for objects such as vector-valued curvature and torsion operators, cross-product-like multilinear maps, and any tensor field whose result is meant to be combined with other vectors rather than treated as a plain number.

Formal Definition

The Vector-Valued Multilinear Map

A tensor in the vector codomain case is a multilinear map

T : V_{1} \times \dots \times V_{k} \to W

where $W$ is a vector space with $\dim W = m \geq 1$, and $W$ need not equal, or bear any predetermined relationship to, any of the domain factors $V_1, \ldots, V_k$. Slotwise linearity is required exactly as in the scalar case, with additivity and homogeneity holding in every input slot; only the target of the map differs.

Decomposition Into Scalar Components

Fixing a basis ${f_1, \ldots, f_m}$ of $W$, any vector-valued map decomposes uniquely as

T (v_{1}, \dots, v_{k}) = \sum_{j = 1}^{m} T_{j} (v_{1}, \dots, v_{k}) f_{j}

where each $T_j$ is a scalar-valued multilinear map on the same domain. This decomposition reduces the study of the vector codomain case entirely to $m$ instances of the scalar codomain case, one for each basis direction of the target.

Structural Features of the Vector-Valued Case

Independence of Component Maps

Each of the $m$ scalar components $T_j$ is, in principle, an entirely independent multilinear map: no relationship among the $T_j$ is forced by the vector-valued structure alone, though in specific applications, such as curvature tensors, additional geometric constraints often relate the components to one another.

Behavior Under Change of Basis in the Codomain

Choosing a different basis of $W$ mixes the component functions $T_j$ linearly among themselves via the change-of-basis matrix, while leaving the underlying map $T$, as an abstract basis-independent object, unchanged. This is analogous to how changing a basis on the domain side mixes a tensor's ordinary indices, except here the mixing acts on an entirely separate set of indices belonging to the output.

Contraction With the Codomain

When $W$ is equipped with its own dual space $W^{}$, a vector-valued tensor can be paired with an element of $W^{}$ to yield an ordinary scalar-valued tensor, precisely reproducing one of the component maps $T_j$ when the paired covector is chosen from the dual basis. This pairing generalizes the notion of contraction to include the codomain, not merely the domain.

Applications of the Vector-Valued Case

Vector-Valued Curvature and Torsion

The torsion tensor of an affine connection is naturally a vector-valued bilinear map, taking two tangent vectors and producing a third tangent vector, rather than a scalar; it is a canonical example where treating the codomain as a genuine vector space, rather than reducing immediately to scalar components, keeps the geometric meaning of the output transparent.

Operator-Valued Tensors

A tensor with codomain $W = \operatorname{End}(V)$, the space of linear operators on $V$, produces, upon evaluation, a linear map rather than a number or a vector; this is a further extension of the vector codomain case in which the target space itself carries additional algebraic structure, namely composition of operators, beyond simple vector addition and scaling.

Building Higher-Arity Tensors From Vector-Valued Ones

A vector-valued multilinear map with codomain $W$ can itself be viewed as a single slot's worth of input to a still higher-arity multilinear map, provided $W$ is paired with a further domain factor; this layered construction is how tensor-valued differential operators, such as the Riemann curvature operator acting on vector fields, are assembled from simpler vector-valued building blocks.

Summary of Key Points

The vector codomain case is a multilinear map whose output lies in a vector space $W$ of dimension one or greater, rather than in the base field.
Choosing a basis of $W$ decomposes any vector-valued map into a finite family of independent scalar-valued component maps.
Changing the basis of the codomain mixes these component maps linearly without altering the underlying basis-independent map.
Pairing the codomain with its dual generalizes contraction to include the output side of a tensor, not just the input side.
Vector-valued tensors underlie geometric objects such as torsion, and can serve as building blocks for still higher-arity, more structured tensor-valued maps.