Abstract
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.
As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.
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References
Hestenes, D. and Ziegler, R.: Projective geometry with Clifford algebra, Acta Appl. Math. 23 (1991), 25–63.
Hestenes, D.: Universal geometric algebra, Simon Stevin 62 (1988), 253–274.
Hestenes, D. and Sobczyk, G.: Clifford Algebra to Geometric Calculus, D. Reidel, Dordrecht, 1985.
Whitehead, A. N.: A Treatise on Universal Algebra with Applications, Cambridge University Press, Cambridge, 1898 (Reprint: Hafner, New York, 1990), p. 249.
Clifford, W. K.: Applications of Grassmann's Extensive Algebra, Amer. J. Math. 1 (1878), 350–358.
Hestenes, D.: Spacetime Algebra, Gordon and Breach, New York, 1966, 1987.
Porteous, I.: Topological Geometry, 2nd edn., Cambridge University Press, Cambridge, 1981.
Salingaros, N.: On the classification of Clifford algebras and their relations to spinors in n dimensions, J. Math. Phys. 23 (1982), 1.
Li, D., Poole, C. P., and Farach, H. A.: A general method of generating and classifying Clifford algebras, J. Math. Phys. 27 (1986), 1173.
Dimakis, A.: A new representation for spinors in real Clifford algebras, in J. S. R. Chisholm and A. K. Common (eds.), Clifford Algebras and their Applications in Mathematical Physics, D. Reidel, Dordrecht, 1986.
Angles, P.: Construction de revêtements du groupe conforme d'un espace vectoriel muni d'une métrique de type (p, q), Ann. Inst. Henri Poincaré 33 (1980), 33.
Lounesto, P. and Latvamaa, E.: Conformal transformations and Clifford algebras, Proc. Amer. Math. Soc. 79 (1980), 533.
Ahlfors, L. V.: Clifford numbers and Möbius transformations in R n, in J. S. R. Chisholm and A. K. Common (eds.), Clifford Algebras and their Applications in Mathematical Physics, D. Reidel, Dordrecht, 1986.
Maks, J. G.: Modulo (1,1) periodicity of Clifford algebras and generalized (anti-) Möbius transformations, Thesis, Delft University of Technology, The Netherlands.
Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras, Kluwer Academic Pubs., Dordrecht, 1990, Chap. 12.
Kastrup, H. A.: Gauge properties of the Minkowski space, Phys. Rev. 150 (1966), 183.
Lang, S.: SL 2 (R), Addison-Wesley, Reading, Mass., 1975.
Clifford, W. K.: Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. IV (1973), 381–395.
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Hestenes, D. The design of linear algebra and geometry. Acta Appl Math 23, 65–93 (1991). https://doi.org/10.1007/BF00046920
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DOI: https://doi.org/10.1007/BF00046920