Exterior algebra

The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.

gives the exterior algebra the additional structure of a graded algebra, that is

These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.

Expanding this out in detail, one obtains the following expression on decomposable elements:

In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:

In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:

All results obtained from other definitions of the determinant, trace and adjoint can be obtained from this definition (since these definitions are equivalent). Here are some basic properties related to these new definitions:

In physics, many quantities are naturally represented by alternating operators. For example, if the motion of a charged particle is described by velocity and acceleration vectors in four-dimensional spacetime, then normalization of the velocity vector requires that the electromagnetic force must be an alternating operator on the velocity. Its six degrees of freedom are identified with the electric and magnetic fields.