In the study of geometric algebras , a k -blade or a simple k -vector is a generalization of the concept of scalars and vectors to include simple bivectors , trivectors , etc. Specifically, a k -blade is a k -vector that can be expressed as the exterior product (informally wedge product ) of 1-vectors, and is of grade k .
In detail: [1]
A vector subspace of finite dimension k may be represented by the k -blade formed as a wedge product of all the elements of a basis for that subspace. [6] Indeed, a k -blade is naturally equivalent to a k -subspace endowed with a volume form (an alternating k -multilinear scalar-valued function) normalized to take unit value on the k -blade.