12.12.5 Tensor Extension Resulting Tensor

Tensor Extension Resulting Tensor is the specific element of the target domain V ⊗_R S produced by applying the tensor domain extension operation to a particular tensor of the source domain V, obtained by pairing that tensor with the multiplicative identity of S under the canonical map v ↦ v ⊗ 1, or more generally by expressing it as a finite sum Σ vₖ ⊗ sₖ once further S-scalar combinations are taken. It is the concrete output of the extension operation applied to a single input, as distinguished from the target domain as a whole, which is the ambient space containing every possible resulting tensor.

Formation of the Resulting Tensor

The Canonical Embedding

For a source tensor v ∈ V, the resulting tensor after extension is v ⊗ 1 ∈ V ⊗_R S, where 1 denotes the multiplicative identity of S. This is the canonical, structure-preserving image of v, and it is the resulting tensor most directly associated with v before any further S-scalar action is applied.

Resulting Tensors Under Further Scalar Action

Once v ⊗ 1 sits inside the S-module V ⊗_R S, it may be acted on by any scalar s ∈ S, producing further resulting tensors v ⊗ s. These are exactly the tensors newly reachable only after extension, since forming v ⊗ s for s outside the image of R → S has no counterpart in the original source domain.

General Resulting Tensors as Finite Sums

Not every element of V ⊗_R S is of the simple form v ⊗ s; a general resulting tensor is a finite sum Σₖ vₖ ⊗ sₖ of such simple tensors, arising for instance when several source tensors are extended and then combined, or when a single source tensor is extended and re-expressed after a change of basis in S.

Well-Definedness of the Resulting Tensor

Independence from Representation of the Input

If a source tensor v is expressed in coordinates relative to a basis {eᵢ} of V as v = Σ aᵢeᵢ, the resulting tensor computed via v ⊗ 1 agrees with the resulting tensor computed via Σ aᵢ(eᵢ ⊗ 1), by bilinearity of ⊗. The resulting tensor is therefore a well-defined function of v alone and does not depend on which particular coordinate expression for v was used to compute it.

Uniqueness Given the Extension Homomorphism

When the extending homomorphism R → S is injective and V is a free R-module, the map v ↦ v ⊗ 1 is itself injective, so distinct source tensors always produce distinct resulting tensors. In this common case the resulting tensor faithfully identifies its source, and the source domain can be regarded as literally sitting inside the target domain via this embedding.

Diagram of a Tensor and Its Resulting Tensor

Properties Preserved and Not Preserved

Linear Operations Are Preserved

Because the extension map v ↦ v ⊗ 1 is R-linear, sums and R-scalar multiples of source tensors extend to the corresponding sums and multiples of resulting tensors: (v + v′) ⊗ 1 = v ⊗ 1 + v′ ⊗ 1 and (r·v) ⊗ 1 = r·(v ⊗ 1) for r ∈ R, where r acts on the right side through its image in S. Any R-linear relation satisfied by source tensors is automatically satisfied by their resulting tensors.

New S-Linear Combinations Are Not Predictable from R Alone

A relation among resulting tensors that uses genuinely S-linear combinations — coefficients from S not arising as images of R — need not correspond to any relation among the original source tensors, since the source domain has no way to express such coefficients. The resulting tensor can therefore participate in S-linear identities that have no meaningful analogue back in the source domain.

Multilinear Structure Carries Over Termwise

If the source tensor v was itself built from a multilinear operation on further source tensors, such as v = v₁ ⊗ v₂, the resulting tensor after extension satisfies the corresponding factorization v ⊗ 1 = (v₁ ⊗ 1) ⊗ (v₂ ⊗ 1) under the natural identification of V ⊗_R S with (V₁ ⊗_R S) ⊗_S (V₂ ⊗_R S) when V = V₁ ⊗_R V₂, so multilinear composition of source tensors is reflected faithfully in the corresponding resulting tensors.

Use of the Resulting Tensor in Computation

As Input to S-Linear Constructions

Once a resulting tensor is formed, it can be used as input to any construction available over S — forming further tensor products over S, applying S-linear maps, or evaluating S-valued multilinear forms — none of which would have been well-defined applied directly to the original source tensor over R.

As the Base Case for Extending a Whole Family

When an entire basis {eᵢ} of the source domain is extended tensor by tensor to {eᵢ ⊗ 1}, the collection of resulting tensors forms the standard basis of the target domain described in the theory of the target domain itself, making the individual resulting tensor the atomic building block from which the extended structure as a whole is assembled.