4.6.3 Tensor Fixed Argument Context

Tensor Fixed Argument Context is the specific collection of values assigned to all argument slots of a multilinear map except one, treated as a background condition against which the slotwise linearity property of the remaining, varying slot is stated and verified. It is the "all else held constant" backdrop that gives meaning to the phrase "linear in each slot separately," since linearity in one slot can only be evaluated relative to some particular choice of values occupying every other slot.

Formal Definition

The Context as a Parameter Tuple

For a multilinear map of arity $k$,

a fixed argument context relative to slot $i$ is a choice of elements

drawn from the remaining $k-1$ slots. Given such a context, the slotwise linearity property for slot $i$ is the statement that the map $x \mapsto T(v_1, \ldots, x, \ldots, v_k)$, with $x$ inserted at position $i$, is linear in $x$ for this particular fixed context $C$.

Quantification Over All Contexts

Full slotwise linearity in slot $i$ requires the linearity statement to hold for every possible fixed argument context, not just one. The universal quantifier over contexts is essential: a map could accidentally behave linearly in slot $i$ for one specific context while failing to do so for another, and such a map would not qualify as multilinear.

Role in the Structure of Multilinear Maps

Separating the Slots

The notion of a fixed argument context is what makes it possible to speak of "slot $i$" as an independent object at all. Without holding a context fixed, an argument tuple is an indivisible whole, and there is no meaningful way to isolate the contribution of a single position; the fixed context is the device that carves the joint tuple into "the slot under study" and "everything else."

Context-Dependence of the Induced Linear Functional

For a fixed context $C$, the induced map $\phi_i^{C}(x) = T(\ldots, x, \ldots)$ is a linear functional on $V_i$, but a different context $C'$ generally produces a different linear functional $\phi_i^{C'}$. The tensor's full behavior in slot $i$ is therefore not captured by a single linear functional but by an entire family of them, indexed by the context, and this family itself varies linearly in the remaining slots.

Use in Verifying and Applying Multilinearity

Verification Strategy

To confirm a candidate map is multilinear, one fixes an arbitrary context for every slot but one, checks the linearity condition in the remaining slot, and repeats this for every slot in turn, allowing the fixed context in each case to range over all possibilities. Because context is arbitrary but must be held fixed during each individual check, this strategy correctly reduces a $k$-argument condition into $k$ separate, tractable single-variable linearity checks.

Context in Basis Expansion

When a tensor is expanded in a chosen basis, the fixed argument context for a given slot is typically taken to range over all combinations of basis vectors in the remaining slots; the resulting family of linear functionals, one per combination of basis context, is exactly what generates the tensor's full array of components once linearity in the studied slot is also expanded over its own basis.

Context Sensitivity in Non-Symmetric Tensors

For a tensor that lacks symmetry among some of its slots, changing which values sit in the fixed context, even by only permuting them among the non-varying slots, can produce a genuinely different induced linear functional in the slot under study. This context sensitivity is precisely what a symmetric or antisymmetric tensor is defined to lack across a specified set of slots.

Summary of Key Points

• A fixed argument context is a choice of values for every slot except the one whose linearity is being examined.
• Slotwise linearity in a given slot must hold for every possible fixed context, not merely for one particular choice.
• The context is what allows a single slot to be isolated and treated as an independent linear functional's domain.
• Different contexts generally induce different linear functionals in the varying slot, unless the tensor possesses symmetry relating them.
• Ranging the context over all basis combinations in the remaining slots is the standard technique used to generate a tensor's full set of components.