Exterior algebras and the cross product
C.1 Exterior algebras
Suppose that E is a real vector space. An element of E, a vector, can be considered to have magnitude and direction. In the same way, if x and y are two vectors in E then they somehow relate to an area in span (x, y). If we wish to make this more specific, we certainly require that the area should be zero if and only if x and y are linearly dependent. A similar remark applies to higher dimensions. We wish to develop these ideas algebraically.
A finite-dimensional (associative) real algebra (A, ) is a finite-dimensional real vector space equipped with a law of composition: ...