12.14.4 Tensor Inclusion Component Placement

Tensor Inclusion Component Placement is the specific rule by which each coordinate of a tensor in the source subspace U is assigned to a corresponding coordinate slot within the larger coordinate system of the target space T, once bases for both spaces have been fixed. Where structure preservation guarantees that relations among elements of U survive intact inside ι(U), component placement addresses a narrower, more mechanical question: given the numerical coordinates of a tensor in U, exactly where and how do those numbers appear among the (typically larger) list of coordinates used to describe the same tensor as an element of T.

The Placement Rule Relative to Fixed Bases

Basis Correspondence

Given a basis {fₖ} of U and a basis {gᵢ} of T, the placement of U's basis vectors inside T's coordinate system is recorded by the matrix Q, whose column k lists the coordinates of ι(fₖ) relative to {gᵢ}. This matrix is the complete description of how U's components are placed within T.

Placement of an Arbitrary Tensor

For a tensor u with coordinates cₖ relative to U's basis, its placement inside T's coordinate system is given by the matrix-vector product Qc, distributing and combining U's original coordinates according to Q's entries. This is the direct component-level analogue of the abstract inclusion map, made fully explicit once bases are chosen.

Typical Placement Patterns

Block Placement into a Direct Sum

When T = U ⊕ W and ι(u) = (u, 0), the component placement is the simplest possible: U's coordinates occupy a contiguous block of T's coordinate list unchanged, while every coordinate corresponding to the complementary block W is placed as exactly zero. The placement matrix Q in this case consists of an identity block stacked above (or beside) a zero block.

Scaled Placement into a Tensor Product Factor

For the inclusion v₁ ↦ v₁ ⊗ v₂ of V₁ into V₁ ⊗ V₂ along a fixed v₂ = Σⱼ bⱼfⱼ, the placement of a coordinate cᵢ of v₁ distributes across every (i,j) coordinate slot of the tensor product as cᵢ·bⱼ, so a single coordinate of the source subspace is placed into an entire row (or column) of coordinates in the target space, scaled by the fixed coefficients bⱼ.

Interleaved Placement

In some inclusions — for instance, embedding a space of even-indexed sequence terms into a space of all sequence terms — the placement rule assigns each source coordinate to a non-contiguous, interleaved position in the target coordinate list, illustrating that placement need not respect any block or contiguous structure at all, so long as the resulting map remains linear and injective.

Diagram of Component Placement

Effect of Basis Choice on Placement

Placement Is Not Intrinsic to the Inclusion Alone

The placement rule described by Q depends on the specific bases chosen for both U and T; changing either basis changes the entries of Q, even though the underlying abstract map ι and its image ι(U) remain exactly the same. Two different placement matrices can therefore describe the identical inclusion operation, differing only by the change-of-basis transformations applied on each side.

Adapted Bases Simplify Placement

If the basis {gᵢ} of T is chosen to extend the placed basis vectors {ι(fₖ)} directly — that is, gᵢ = ι(fᵢ) for i up to dim(U), with additional basis vectors chosen for a complement — the placement matrix Q reduces to the simple identity-and-zero block form seen in the direct-sum case, regardless of how complicated the original inclusion formula appeared before this adapted basis was chosen.

Recovering the Abstract Map from Any Valid Placement

Regardless of which bases are used, the placement matrix Q, together with the two chosen bases, completely determines the abstract inclusion map ι, and any two placement matrices arising from the same ι under different basis choices are related by Q′ = A·Q·B⁻¹ for the appropriate change-of-basis matrices A on T and B on U.