1.2.49 Type One Zero Tensor Definition

A Type One Zero Tensor is a rank-zero tensor with specific algebraic properties in tensor algebra.

Type One Zero Tensor Definition is the characterization of a tensor with exactly one contravariant factor and no covariant factors, the case in which the type $(r, s)$ reduces to $(1, 0)$ . A type $(1, 0)$ tensor is, by definition, simply an ordinary vector, so that vectors themselves occupy the most elementary nonzero-order slot of the type-graded tensor algebra, standing as the first genuinely contravariant object beyond the scalars of type $(0, 0)$ .

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . Setting $r = 1$ and $s = 0$ in the general tensor product space

\underset{r}{⏨} V \otimes \dots \otimes V \otimes \underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

leaves a single copy of $V$ and no copies of $V^{*}$ , giving

T_{0}^{1} (V) = V

so that a type $(1, 0)$ tensor is precisely an element $v \in V$ : an ordinary vector, carrying a single contravariant index $v^{i}$ and no covariant index.

The Unary Multilinear Map Interpretation

Under the correspondence between tensors of type $(r, s)$ and multilinear maps taking $r$ covector and $s$ vector arguments, a type $(1, 0)$ tensor corresponds to a linear map

v : V^{*} \to F

defined by $v (ω) = ω (v)$ for a covector $ω$ , using the natural pairing between $V$ and its dual. This is exactly the canonical embedding of $V$ into its double dual $V^{**}$ , which is an isomorphism in finite dimensions, and it is what justifies treating a vector interchangeably as an element of $V$ and as a linear functional on $V^{*}$ .

Transformation Law

The single index of a type $(1, 0)$ tensor transforms contravariantly: under a change of basis with matrix $A$ , the components transform using the inverse matrix,

{\tilde{v}}^{i} = \sum_{j} {(A^{-1})}_{j}^{i} v^{j}

This inverse transformation is exactly the ordinary rule by which the coordinates of a vector change when the basis is rescaled, and it is this transformation behavior, applied to a single index, that gives the type $(1, 0)$ case its name as the prototypical contravariant tensor.

Order, Degree, and Arity

A type $(1, 0)$ tensor has total order one, occupies the degree-one summand $V^{\otimes} 1 = V$ of the tensor algebra $T (V)$ , and has arity one when regarded as a multilinear map on a single covector argument. Because contravariant order and covariant order coincide with the two type entries themselves in this simplest nontrivial case, a type $(1, 0)$ tensor has contravariant order exactly one and covariant order exactly zero, with no ambiguity between the various invariants used elsewhere to classify tensors of more general type.

Distinction from Type (0, 1)

A type $(1, 0)$ tensor is not the same kind of object as a type $(0, 1)$ tensor, a covector, even though both have total order one. The two differ in their transformation law — contravariant for type $(1, 0)$ , covariant for type $(0, 1)$ — and cannot be canonically identified with one another without extra structure on $V$ , such as a metric tensor, which provides the additional data needed to convert a type $(1, 0)$ tensor into a type $(0, 1)$ tensor through the operation of lowering an index.

Role Within Tensor Algebra

Type $(1, 0)$ tensors are the building blocks from which every higher-order purely contravariant tensor is assembled by repeated tensor product, since a tensor of type $(r, 0)$ for $r \geq 1$ is, when decomposable, a product of $r$ individual type $(1, 0)$ tensors, and every element of the full contravariant tensor algebra is a finite sum of such products. Recognizing vectors as type $(1, 0)$ tensors situates them within the same unified definitional framework used for tensors of every other type, confirming that the vector space $V$ one begins with is itself already the simplest nonzero example of a tensor.