bladbasis

Bladbasis, in the context of exterior and geometric algebra, refers to the canonical basis of the exterior algebra generated by a vector space. It is a set of blades formed by wedge products of the space’s basis vectors, organized by grade (dimension of the subspace they represent). For a vector space V of dimension n, the bladbasis consists of 2^n elements, including blades of all grades from 0 to n.

Construction and examples: Let {e1, e2, ..., en} be a basis for V. The bladbasis comprises all wedge products e_{i1} ∧ e_{i2} ∧ ... ∧ e_{ik} for all 1 ≤ i1 < i2 < ... < ik ≤ n and for k = 0, where e_{i1} ∧ ... ∧ e_{ik} is understood with k = 0 as the scalar 1. The elements of grade k are called k-blades. In particular, the 0-blade is the scalar 1, and the n-blade e1 ∧ e2 ∧ ... ∧ en is the pseudoscalar. The blades form a basis for the exterior algebra Λ(V), which decomposes as a direct sum of the graded subspaces Λ^k(V).

Properties: The exterior (wedge) product is antisymmetric, so e_i ∧ e_j = - e_j ∧ e_i and e_i ∧ e_i = 0. Each k-blade represents a oriented k-dimensional subspace (a simple blade). Not every multivector is a single blade, but every multivector can be expressed as a linear combination of blades of various grades.

Relation to geometric algebra: The bladbasis also serves as a basis for the full geometric (Clifford) algebra, where blades span the algebra under the geometric product in addition to the wedge product. Applications include physics, computer graphics, and robotics, where oriented subspaces and their products are useful.