Blade (geometry)

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:
 * 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 $a$ and $b$:
 * $$a \wedge b .$$
 * A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors $a$, $b$, and $c$:
 * $$a \wedge b \wedge c. $$
 * In a vector space of dimension $n$, a blade of grade $n − 1$ is called a pseudovector or an antivector.
 * The highest grade element in a space is called a pseudoscalar, and in a space of dimension $n$ is an $n$-blade.
 * In a vector space of dimension $n$, there are $k(n − k) + 1$ dimensions of freedom in choosing a $k$-blade for $0 ≤ k ≤ n$, of which one dimension is an overall scaling multiplier.

A vector subspace of finite dimension $k$ may be represented by the $k × k$-blade formed as a wedge product of all the elements of a basis for that subspace. Indeed, a $k × (n − k)$-blade is naturally equivalent to a $k(n − k)$-subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating $k$-multilinear scalar-valued function), such a $k$-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.