4.16.2 Tensor Linear Extension per Slot

Tensor Linear Extension per Slot is the elementary fact, applied one slot at a time, that an ordinary linear map on a vector space is completely determined by its values on a basis, used repeatedly as the atomic building block underlying the tensor multilinear extension construction. It isolates the single-slot version of extension, the special case in which every slot but one is already held fixed, reducing the general multilinear extension problem to a sequence of applications of this one elementary linear-algebra fact.

The Elementary Fact in One Slot

Linear Maps Determined by Basis Values

For a vector space V with basis e_1, ..., e_n, any linear map L : V → F into a field, or more generally into any vector space, is uniquely determined by its values L(e_1), ..., L(e_n) on the basis, since any vector v = ∑_j v^j e_j gives

by additivity and homogeneity of L, the ordinary linear-map versions of the corresponding tensor properties restricted to a single argument.

Applying This Fact to a Single Slot of a Tensor

When every slot of a type (p, q) tensor T except one is fixed to specific arguments, the remaining tensor remaining slot map obtained is exactly a linear map on the single remaining space, V or V* as appropriate; the elementary fact above applies directly to this remaining slot map, showing it is determined completely by its values on the basis of that one space.

Building the Multilinear Extension from Repeated Single-Slot Extension

Extending Slot by Slot Using the Elementary Fact

The tensor multilinear extension from basis proceeds by invoking the single-slot linear extension fact once per slot: fixing every other slot to a basis element in turn, extending the current slot linearly using the elementary fact, and then moving on to extend the next slot in the same way, with the previously extended slot now treated as fully general rather than restricted to basis elements.

Why the Order of Slots Does Not Matter

Because the elementary single-slot fact is applied independently in each slot, with the values in the other slots merely held as parameters rather than participating in the extension itself, extending the slots in any order produces the same final multilinear map; this order-independence at the level of the whole tensor is inherited directly from the fact that a single linear extension does not reference or depend on any other slot.

Relation to the General Extension Construction

The Multi-Slot Case as Iterated Single-Slot Cases

The general tensor multilinear extension construction, applied to a full spanning set across every slot simultaneously, can be understood as p + q iterated applications of the single-slot linear extension fact, one for each slot, with each application treating the tensor's dependence on the remaining, not-yet-extended slots as an ambient parameter rather than an obstacle.

Reduction of Compatibility to the Single-Slot Case

The tensor extension compatibility condition, checked across relations among the generators in a given slot, reduces to exactly the ordinary condition needed for the single-slot linear extension fact to apply without contradiction: a linear map is well-defined on a spanning set exactly when its prescribed values respect every linear relation among that spanning set, which is the single-slot instance of the same compatibility requirement stated more generally for full multilinear extension.

Single-Slot Extension and Homogeneous Degree

Degree One in Each Slot Individually

The elementary linear extension fact applies because a linear map has degree exactly one in its single argument; the analogous statement for a multilinear map is that it has degree exactly one in each slot considered separately, which is precisely why tensor multilinear extension construction succeeds slot by slot rather than requiring the joint degree, p + q, of the whole tensor to be handled at once.

Contrast with Extending a Map Nonlinear in Some Slot

If a candidate map failed to be linear in some particular slot, for instance behaving quadratically in that argument, the elementary single-slot extension fact would not apply to that slot, and no amount of correct linear behavior in the other slots could compensate; this is why verifying additivity and homogeneity in every individual slot, not merely in some slots, is a prerequisite before the extension construction can be carried out at all.

Practical Role of the Single-Slot Perspective

Simplifying Verification of Multilinearity

Because multilinearity reduces to linearity checked one slot at a time, verifying that a candidate tensor is well-defined and multilinear can be broken down into p + q separate, simpler verifications, each amounting to nothing more than the elementary fact that a linear map on a finite-dimensional space is determined by, and can be freely extended from, its values on a basis.

A Conceptual Bridge to Ordinary Linear Algebra

The single-slot extension fact is the precise conceptual bridge connecting the ordinary linear algebra fact about linear maps and bases to the more elaborate tensor multilinear extension construction, showing that tensors, despite their higher rank and more elaborate index structure, ultimately rest on nothing more exotic than repeated application of this one familiar linear-algebra principle.

Diagrammatic Summary

The diagram singles out one slot of the tensor, the one currently being extended using the elementary linear-map fact, while the surrounding slots remain fixed, acting only as parameters that do not participate in that particular extension step.