Exterior Algebra Quiz

The graded associative algebra ∧V in the exterior algebra of a vector space V is defined as $∧ V = k ⊕ V ⊕ ∧^2 V ⊕ ∧^3 V ⊕ ⋯$. The corresponding product ∧ on the exterior algebra ∧V is called the exterior product or wedge product.

A k-blade in the exterior algebra corresponds to the oriented parallelotope spanned by elements of the form v1∧...∧vk, where vi ∈ V.

In the exterior algebra, elements in ∧kV are called k-multivectors and are given by a sum of k-blades. They are an abstraction of oriented lengths, areas, volumes, and more generally oriented k-volumes for k ≥ 0.

The alternating property of the exterior product ∧ in the exterior algebra is given by the equation v∧v = 0 for all v ∈ V. By distributivity and linearity, this also implies it is antisymmetric: u∧v = - v∧u.

The exterior algebra is also called the Grassmann algebra after Hermann Grassmann, who made significant contributions to its development.