exterior algebra
The exterior algebra is what you get when you decide that 'a vector times itself is zero' — a rule that captures the geometry of oriented area and volume. Squashing repeated directions to zero forces a sign flip when two factors swap, so the product becomes antisymmetric. It is the algebraic engine behind determinants, differential forms, and the notion of signed k-dimensional volume.
Given an R-module V, the exterior algebra Λ(V) is the quotient of the tensor algebra T(V) by the two-sided ideal generated by all v ⊗ v with v in V. The induced product is written with the wedge ∧, and the relation v ∧ v = 0 forces v ∧ w = −w ∧ v. As a graded algebra Λ(V) = ⊕_{k≥0} Λ^k(V), where Λ^k(V) is spanned by wedges v_1 ∧ ... ∧ v_k; such a wedge is zero exactly when the vectors are linearly dependent.
If V has dimension n, then dim Λ^k(V) = C(n, k) (binomial coefficient), the algebra is finite-dimensional of total dimension 2^n, and Λ^k(V) = 0 for k > n. The top piece Λ^n(V) is one-dimensional, and the action of a linear map on it is multiplication by the determinant. Elements of Λ^k(V) are precisely the alternating k-tensors, the home of antisymmetric multilinear forms.
For V = R^3 with basis e_1, e_2, e_3: Λ^0 = R, Λ^1 = V (dim 3), Λ^2 spanned by e_1∧e_2, e_1∧e_3, e_2∧e_3 (dim 3), Λ^3 spanned by e_1∧e_2∧e_3 (dim 1); total 1 + 3 + 3 + 1 = 8 = 2^3.
The dimensions 1, 3, 3, 1 are row 3 of Pascal's triangle, summing to 2^3.
In characteristic 2 one must define Λ(V) using the ideal generated by v ⊗ v, not by v ⊗ w + w ⊗ v: the relation v ∧ v = 0 is strictly stronger than antisymmetry when 2 is not invertible, and only the former gives the correct exterior algebra.