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## NEW FOUNDATIONS FOR GEOMETRIC ALGEBRA

• New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, andNew foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea that Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n×n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multilinear quantities.  • Volltext Document Type: Conference Proceeding Ramon Gonzalez Calvet https://doi.org/10.25643/bauhaus-universitaet.2764Cite-Link https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20170314-27644Cite-Link http://euklid.bauing.uni-weimar.de/ikm2012 1611-4086 Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 04 - 06 2012, Bauhaus-University Weimar Klaus GürlebeckGND, Tom LahmerORCiDGND, Frank WernerORCiDGND English 2017/03/03 2012/07/04 2017/03/14 Bauhaus-Universität Weimar Bauhaus-Universität Weimar Bauhaus-Universität Weimar / In Zusammenarbeit mit der Bauhaus-Universität Weimar 12 Angewandte Informatik; Angewandte Mathematik; Computerunterstütztes Verfahren 000 Informatik, Informationswissenschaft, allgemeine Werke / 000 Informatik, Wissen, Systeme 500 Naturwissenschaften und Mathematik / 510 Mathematik 31 Mathematik / 31.80 Angewandte Mathematik 56 Bauwesen / 56.03 Methoden im Bauingenieurwesen Bauhaus-Universität Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar, 19. 2012 Creative Commons 4.0 - Namensnennung-Nicht kommerziell (CC BY-NC 4.0)