16.12.3 Tensor Wedge Product Bilinear Behavior

The wedge product's bilinear behavior defines how it combines tensors in a way that preserves alternating properties and ensures linearity in both arguments.

Tensor Wedge Product Bilinear Behavior is the foundational property that the wedge product distributes over vector addition and interacts predictably with scalar multiplication in each of its two argument slots separately, the companion property to anticommutativity without which the wedge product could not be extended from single vectors to arbitrary linear combinations.

Statement of Bilinear Behavior

Additivity in Each Slot

For vectors u, v, w ∈ V, the wedge product distributes over addition in its first slot:

(u + v) \land w = u \land w + v \land w

and, by an entirely symmetric argument (using anticommutativity to move the sum to the other side), also in its second slot:

u \land (v + w) = u \land v + u \land w

Homogeneity Under Scalar Multiplication

The wedge product also respects scalar multiplication in each slot:

(c u) \land v = c (u \land v) = u \land (c v)

for any scalar c, so a scalar can be freely moved into or out of either slot without changing the value of the product.

Why Bilinearity Must Be Established Independently

Not a Consequence of Anticommutativity Alone

Anticommutativity governs how the product responds to reordering its two arguments; it says nothing by itself about how the product interacts with sums or scalar multiples. Bilinear behavior is a logically separate requirement, and a hypothetical "product" satisfying only anticommutativity without bilinearity would not extend consistently to linear combinations of vectors at all.

Both Properties Needed for the Extended Formula

Only once both anticommutativity and bilinearity are established can the wedge product be evaluated on arbitrary vectors expressed in coordinates, since expanding (Σaᵢeᵢ)∧(Σbⱼeⱁ) into a double sum of basis wedges requires bilinearity to justify each distribution step, and requires anticommutativity to simplify the resulting sum down to independent basis terms.

Worked Verification in Coordinates

Expanding the Basic Formula

Applying bilinearity fully to u = a₁e₁+a₂e₂ and v = b₁e₁+b₂e₂, the four cross terms reduce (using e₁∧e₁ = e₂∧e₂ = 0 and e₂∧e₁ = −e₁∧e₂) to a single surviving term:

u \land v = (a_{1} b_{2} - a_{2} b_{1}) (e_{1} \land e_{2})

exactly recovering the original signed-area determinant formula that motivated the wedge product's foundation, now derived from bilinear behavior plus anticommutativity rather than assumed directly.

Bilinear Behavior Extended to Higher Degree

Multilinearity as the Generalized Form

At degree k, bilinear behavior generalizes to full multilinearity: the wedge product v₁∧...∧vₖ is linear in each of its k slots separately, with sums and scalar multiples distributing exactly as in the two-vector case, just applied independently to whichever slot is varied.

Consistency with the Antisymmetrized Tensor Product View

This multilinear behavior is automatically inherited from the fact that the wedge product is built from the antisymmetrized ordinary tensor product, and the ordinary tensor product is itself multilinear by construction; bilinearity at the wedge product level is therefore not an extra assumption layered on top of the antisymmetrization construction, but a direct consequence of starting from an already-multilinear operation before antisymmetrizing it.