Grassmann Algebra



What is Grassmann Algebra?  Grassmann algebra is a mathematical system which predates vector algebra, and yet is more powerful, subsuming and unifying much of the algebra used by engineers and physicists. It has remained relatively unknown since its discovery around 1832, yet is now emerging as a potential mathematical system for describing such diverse applications as robotic manipulators and fundamental physical theories.


Where does its power come from?  The intrinsic power of Grassmann algebra arises from its fundamental product operation, the exterior product. The exterior product codifies the property of linear dependence directly into the algebra. Simple non-zero elements of the algebra are products of linearly independent elements. For example, a simple bivector is the exterior product of two independent vectors; a line is represented by the exterior product of two independent points and a plane is represented by the exterior product of three independent points. Exterior products of linearly dependent elements are zero. These properties generate a geometric algebra par excellence.


  The best original source for Grassmann's contributions to mathematics and science is his collected works:

Grassmann, Hermann
Hermann Grassmanns Gesammelte Mathematische und Physikalische Werke.
Teubner, Leipzig. Volume 1 (1896), Volume 2 (1902,1904), Volume 3 (1911).

  Grassmann's algebraic writings are predominantly developed in two books recently translated by Lloyd C. Kannenberg:

Grassmann, Hermann
A New Branch of Mathematics: The Ausdehnungslehre of 1844 and other works.
Open Court, Illinois. (1995) ISBN 0-8126-9275-6


Grassmann, Hermann
Extension Theory
American Mathematical Society. London Mathematical Society. (2000) ISBN 0-8218-2031-1

  A brief bibliography of earlier writers in the Grassmannian tradition is given in the Grassmann Algebra book.


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