In the study of geometric algebras, a -blade or a simple -vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a -blade is a -vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade .
In detail: A 0-blade is a scalar. A 1-blade is a vector. Every vector is simple. A 2-blade is a simple bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and :
A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors , , and :
In vector spaces of dimension ≤ 3, every k-vector is a blade. In dimension ≥ 4, there exist k-vectors (starting already at grade 2) that are not blades. This makes the distinction important, because in higher dimensions most k-vectors do not correspond to any subspace, which requires defining blades separately from k-vectors to identify the k-vectors that actually do. In a vector space of dimension , a blade of grade is called a pseudovector or an antivector. The highest grade element in a space is called a pseudoscalar, and in a space of dimension is an -blade. In a vector space of dimension , there are dimensions of freedom in choosing a -blade for , of which one dimension is an overall scaling multiplier.For Grassmannians (including the result about dimension) a good book is: . The proof of the dimensionality is actually straightforward. Take the exterior product of vectors and perform elementary column operations on these (factoring the pivots out) until the top block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower block. Compare also with the dimension of a Grassmannian, , in which the scalar multiplier is eliminated.
A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace. Indeed, a -blade is naturally equivalent to a -subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating -multilinear scalar-valued function), such a -blade may be normalized to take unit value, making the correspondence unique up to a sign.
Examples In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as pseudoscalars, in that they are elements of a one-dimensional space that is distinct from regular scalars.
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.
See also Grassmannian Multivector Exterior algebra Differential form Geometric algebra Clifford algebra
Notes
References
A Lasenby, J Lasenby & R Wareham] (2004) A covariant approach to geometry using geometric algebra Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
External links A Geometric Algebra Primer, especially for computer scientists.