16.8.1 Tensor Alternating Bilinear Argument Pair
A tensor alternating bilinear argument pair combines bilinearity with antisymmetry, key in algebra and geometry.
Tensor Alternating Bilinear Argument Pair is the ordered pair of vectors (u, v) fed into an alternating bilinear form, treated here as the fundamental unit of input in its own right — how such pairs are structured, when two different pairs yield the same output, and what geometric object a pair naturally represents.
The Argument Pair as an Ordered Object
Ordered, Not Unordered
An argument pair (u, v) for a bilinear form B is fundamentally ordered: (u, v) and (v, u) are different inputs, even though they involve the same two vectors, because B is defined on V × V, the set of ordered pairs, not on unordered two-element subsets of V:
The alternating property links the values B(u,v) and B(v,u) by a sign, but it does not identify the two pairs as the same input.
The Degenerate Pair
When u = v, the ordered pair (v, v) is its own reversal, (v, v) = (v, v), and this self-identity is exactly what forces B(v,v) = 0 via the sign cancellation argument; the degenerate pair is the unique case where an argument pair coincides with its own reversal.
Classifying Argument Pairs
Independent Pairs
An argument pair (u, v) is called independent if u and v are linearly independent vectors; for these pairs, B(u,v) may take any nonzero value consistent with the alternating structure, and the pair spans a genuine two-dimensional subspace of V.
Dependent Pairs
If v = cu for some scalar c, the pair (u, cu) is dependent, and bilinearity combined with the vanishing-on-repeated-argument fact forces B(u, cu) = cB(u,u) = 0 regardless of the value of c; dependent pairs always evaluate to zero under an alternating bilinear form.
Geometric Reading of the Argument Pair
The Pair as a Spanned Parallelogram
An independent argument pair (u, v) naturally represents the parallelogram spanned by u and v, traced first along u then along v; the value B(u,v) is interpreted, in the canonical two-dimensional case, as the signed area of that parallelogram, with the ordering of the pair fixing the sign convention (traversal direction).
Reversal as Traversal Reversal
Swapping the pair to (v, u) corresponds to tracing the same parallelogram's boundary in the opposite rotational direction, and the sign flip B(v,u) = −B(u,v) captures exactly this reversal of traversal sense without altering the parallelogram's shape or size.
Argument Pairs Under Linear Transformation
Transformed Pairs and Their Values
If a linear map f: V → V is applied to both members of an argument pair, the resulting value transforms according to B(f(u), f(v)), which for the canonical two-dimensional area form equals det(f) · B(u,v); the argument pair's transformed value thus tracks exactly how the spanned parallelogram's area scales and possibly reflects under f.
Equivalence Classes of Pairs with the Same Value
Two argument pairs (u,v) and (u',v') can yield the same value under B without being related by any obvious transformation; the set of all pairs sharing a fixed value B(u,v) = c forms a level set of B on V × V, generally a codimension-one subset once degenerate pairs are excluded.