Grassmann Algebra

Abstract

An algebra is a linear space in which, besides the usual operations of addition and multiplication by numbers, a product of elements is defined with the usual distributive law:

$$\begin{array}{*{20}{c}} {\alpha \left( {\alpha b + \beta c} \right) = \alpha ab + \beta ac} \\ {\left( {\alpha b + \beta c} \right)a = \alpha ba + \beta ca} \\ \end{array}$$

where α, β are numbers and a, b, c elements of the algebra.