4.7.1 Tensor Product Domain Input

The tensor product domain input is a foundational concept in algebra, defining how tensors combine vectors and scalars across different spaces.

Tensor Product Domain Input is a tuple of vectors, drawn one from each factor of a multilinear domain structure, that serves as the argument fed into the canonical map connecting the Cartesian product domain to the tensor product space. It is the specific object of the form $(v_1, \ldots, v_k)$ that gets sent, via the canonical multilinear map, to the pure or simple tensor $v_1 \otimes \cdots \otimes v_k$ inside the tensor product space, and it is the raw material from which every element of that space is ultimately built.

Formal Definition

The Canonical Map

Given vector spaces $V_1, \ldots, V_k$, the tensor product space $V_1 \otimes \cdots \otimes V_k$ comes equipped with a canonical multilinear map

⌈ : V_{1} \times \dots \times V_{k} \to V_{1} \otimes \dots \otimes V_{k}

sending a domain input $(v_1, \ldots, v_k)$ to the simple (or decomposable) tensor $v_1 \otimes \cdots \otimes v_k$. A tensor product domain input is precisely an element of the source of this map: an ordered tuple with one entry chosen from each factor space, exactly as described by the multilinear domain structure.

Universality With Respect to Domain Input

The defining universal property of the tensor product states that for any multilinear map $T$ on the same domain, there is a unique linear map $\tilde{T}$ on the tensor product space such that $T(v_1, \ldots, v_k) = \tilde{T}(v_1 \otimes \cdots \otimes v_k)$ for every domain input. In this sense, the domain input is the common starting point shared by every multilinear map built on that domain: whatever a particular tensor computes, it computes it as a function of the same underlying domain input.

Structural Properties of Domain Inputs

Not Every Tensor Product Element Is a Single Domain Input's Image

While every domain input maps to a simple tensor, not every element of the tensor product space arises as the image of one domain input. General elements of $V_1 \otimes \cdots \otimes V_k$ are finite sums of simple tensors, and only those special elements expressible as a single simple tensor come directly from one domain input under the canonical map; this distinction is the basis for the notion of tensor rank.

Basis Expansion of a Domain Input

If each factor space $V_i$ has a basis ${e^{(i)}1, \ldots, e^{(i)}{n_i}}$, a domain input's canonical image expands as

v_{1} \otimes \dots \otimes v_{k} = \sum_{i_{1}, \dots, i_{k}} v_{1}^{i_{1}} \dots v_{k}^{i_{k}} (e_{i_{1}}^{(1)} \otimes \dots \otimes e_{i_{k}}^{(k)})

showing that even a single domain input, once expanded, generally maps to a sum of several basis tensors, not a single basis tensor, unless each $v_i$ happens to already be a basis vector.

Role in Constructing and Analyzing Tensors

Testing Multilinear Maps on Domain Inputs

Because every multilinear map $T$ satisfies $T = \tilde{T} \circ \lceil$, evaluating $T$ on a domain input is equivalent to first mapping the input into the tensor product space via the canonical map and then applying the induced linear map $\tilde{T}$. This equivalence is what allows properties of $T$, such as multilinearity itself, to be re-derived from properties of the tensor product space and its associated linear maps.

Domain Input as the Unit of Rank

The minimal number of domain inputs whose canonical images sum to a given tensor element is precisely the tensor's rank; a rank-one tensor is one obtainable from a single domain input, while higher-rank tensors require combining the images of multiple distinct domain inputs.

Summary of Key Points

A tensor product domain input is an ordered tuple of vectors, one from each factor space of the multilinear domain structure.
The canonical multilinear map sends each domain input to a simple tensor inside the tensor product space.
General tensor product elements are sums of simple tensors and need not arise from a single domain input.
Expanding a domain input in a chosen basis for each factor produces a sum of basis tensors, weighted by products of coordinates.
The number of domain inputs needed, in combination, to express a given tensor element defines that element's rank.