4.6.4 Tensor Independent Slot Variation

Tensor Independent Slot Variation is the principle that the linearity behavior exhibited when one argument slot of a multilinear map is varied places no constraint on, and is not coupled to, the behavior exhibited when a different slot is varied instead. Each slot's linearity is verified and expressed entirely on its own terms, and this independence across slots is what allows an arity-$k$ multilinear condition to be decomposed into $k$ separate, individually checkable linearity conditions rather than one entangled joint condition.

Formal Statement

Independence of the Linearity Conditions

For a map $T : V_1 \times \cdots \times V_k \to F$, the multilinearity requirement consists of $k$ separate conditions, one for each slot index $i$:

None of these $k$ conditions references or depends on any other; each is a self-contained statement about a single slot, with the other $k-1$ arguments simply held as an arbitrary fixed argument context. Independence here means that verifying, strengthening, or violating the condition at slot $i$ has no logical bearing on the condition at slot $j \neq i$.

No Cross-Slot Coupling in the Definition

The definition of multilinearity does not impose any relationship between how the map behaves as slot $i$ varies and how it behaves as slot $j$ varies; in particular, multilinearity does not require, for instance, that the "rate" of linear response in one slot match or relate in any fixed way to the rate of response in another slot. The only requirement is that each slot's response be linear, considered in isolation.

Consequences of Slot Independence

Verifying Multilinearity Slot by Slot

Because the conditions are independent, a proof or computational check of multilinearity can proceed one slot at a time: fix all other arguments, confirm linearity in the chosen slot, and move on to the next slot with a fresh, unrelated choice of fixed arguments if convenient. No information from checking one slot needs to be carried over to check another.

Independence Does Not Mean Uncoupled Values

Slot independence is a statement about the logical structure of the linearity conditions, not about the numerical values the tensor produces. The actual output $T(v_1, \ldots, v_k)$ generally depends jointly on all $k$ arguments at once, and changing the value in one slot typically does change the output even when other slots are unchanged; what remains independent is only the linearity property itself, not the coupling of numeric outputs across arguments.

Enabling the Component Expansion

Independent slot variation is what justifies expanding a tensor in a basis one slot at a time, since the linear expansion applied to slot 1 does not disturb or require re-verification of the linear expansion already applied to slot 2. Repeating this expansion across all $k$ slots in any order produces the same final sum over components, precisely because no slot's linear behavior depends on the order in which the other slots were expanded.

Relation to Symmetry and Contraction

Symmetry as an Additional, Separate Condition

Because slot linearity is independent per slot, symmetry (equality of the map under permutation of two or more slots) is a genuinely additional condition layered on top of multilinearity, not something multilinearity itself implies or forbids. A multilinear map may be symmetric, antisymmetric, or neither, and independent slot variation is precisely why no symmetry is automatically inherited from the linearity structure alone.

Contraction Acting on One Slot Pair at a Time

Contraction, which pairs and sums over one contravariant and one covariant slot, relies on independent slot variation to be well-defined: the operation only manipulates the two paired slots, and the independence of the remaining slots' linearity conditions guarantees that contraction produces a map that is still multilinear in every slot left untouched.

Summary of Key Points

• Multilinearity decomposes into $k$ separate linearity conditions, one per slot, none of which depends logically on the others.
• Verifying multilinearity can therefore proceed one slot at a time, holding an arbitrary fixed argument context for the remaining slots.
• Independence concerns the logical structure of the linearity conditions, not the numeric coupling of the tensor's actual output values across arguments.
• It is what permits basis expansion of a tensor to proceed slot by slot, in any order, and still yield the same result.
• Symmetry and contraction are additional structures layered on top of, and made tractable by, the independence of the underlying slotwise linearity conditions.