16.9.4 Tensor Alternating Multilinear Exterior Relation
Tensor Alternating Multilinear Exterior Relation explains how antisymmetric multilinear forms build exterior algebra via wedge products and alternating properties.
Tensor Alternating Multilinear Exterior Relation is the precise dictionary connecting alternating multilinear forms, viewed as functions on tuples of vectors, to elements of the exterior algebra, viewed as formal wedge products of covectors, establishing the isomorphism that lets the two languages be used interchangeably without loss of information.
The Two Languages Being Related
The Functional Language
In the functional language, an alternating multilinear form of rank k is a map T: V^k → 𝔽 satisfying the sign-reversal law on argument swaps. This language emphasizes evaluation: given specific vectors, what number does T produce.
The Formal Wedge Language
In the formal wedge language, the same object is represented as a linear combination of formal symbols e*_{i₁} ∧ ... ∧ e*_{iₖ} for increasing index tuples. This language emphasizes algebraic structure: how the object combines with others under ∧.
The Relation Connecting the Two
From Wedge Expression to Evaluation Formula
The exterior relation states that a decomposable wedge of covectors evaluates on a tuple of vectors via a determinant of pairings:
This is the precise formula translating any formal wedge expression into a concrete evaluation rule, valid for any k.
Well-Definedness of the Relation
The relation is consistent with the alternating property expected from a rank-k multilinear form, since swapping any two rows of the determinant on the right side (corresponding to swapping two arguments vᵢ, vⱼ) negates the determinant, exactly matching the sign-reversal law required in the functional language.
The Isomorphism Established by the Relation
Structure-Preserving Correspondence
The exterior relation defines a linear isomorphism between the space Λᵏ(V*) of formal wedge expressions and the space of alternating multilinear forms of rank k on V:
Both spaces have the same dimension C(n,k), consistent with this being a genuine isomorphism rather than a mere embedding of one structure into a larger one.
Preservation of the Wedge Operation
Under this relation, formal wedge multiplication on one side corresponds exactly to the antisymmetrized tensor product operation on the other, so computations can be carried out in whichever language is more convenient and translated back without any discrepancy.
Using the Relation in Practice
Translating a Concrete Form into Wedge Notation
Given an explicit alternating form defined by its action on basis vectors, the exterior relation allows it to be rewritten as a sum of basis wedge elements weighted by its independent component values:
giving a canonical wedge decomposition directly from the independent component values already established for the tensor.
Translating Wedge Algebra Back into Component Computations
Conversely, any formal manipulation performed using ∧ — associativity, graded commutativity, distributivity — translates directly into corresponding index manipulations on components, allowing symbolic wedge computations to be checked or implemented at the component level whenever needed.