By Dieudonne Grothendieck

ISBN-10: 0387051139

ISBN-13: 9780387051130

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**Example text**

A + B = B + A. , for any three multivectors A, B, and C ∈ G, we have ( A + B) + C = A + ( B + C). , for any three multivectors A, B, C ∈ G, ( AB)C = A( BC). , for any three multivectors A, B, C ∈ G, A( B + C) = AB + AC. ( B + C) A = B A + C A. 7) are independent of one another because neither commutability nor anticommutability of the geometric product of multivectors is axiomatized. Axiom 7: There exists a unique multivector 0 ∈ G, called the additive identity, such that A + 0 = A = 0 + A. 8) Axiom 8: There exists a unique multivector I ∈ G, called the multiplicative identity, such that I A = A.

Datta and R. Datta J. 8. No. 2, 30–32. 7. L. Russo, La Rivoluzione Dimenticata, second edition, Feltrinelli Editore, Milano, Italia (1997). 8. T. G. Vold, Am. J. Phys. 61, 491–504 (1993). 2) where (AB) o = A · B = a · (b · B) = (BA) o , (AB) 2 = a ∧ (b · B) + a · (b ∧ B) = −(BA) 2 , (AB) 4 = A ∧ B = a ∧ b ∧ B = (BA) 4 . 5), we establish that (AB) o + (AB) 4 = (1/2)(AB + BA) = (BA) o + (BA) 4 and (AB) 2 = (1/2)(AB − BA) = −(BA) 2 . 7) a ∧ (b · B) + a · (b ∧ B) = (1/2)(AB − BA). 8) and The expression (1/2)(AB − BA) is called the commutator or commutator product of A and B.

As in the addition of real and imaginary numbers, numbers of different types are collected as separate parts under the name of complex numbers. We note in passing that the geometric product of vectors has, except for commutativity, the same algebraic properties as the scalar multiplication of vectors and bivectors. In particular, both products are associative as well as distributive with respect to addition. Now, in conformity with the development of geometric algebra for threedimensional space as a symbolic system, we develop the geometric algebra for the space of an arbitrary dimension by introducing the following axioms and definitions.

### Elements de geometrie algebrique by Dieudonne Grothendieck

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