Titlebook: New Foundations in Mathematics; The Geometric Concep Garret Sobczyk Textbook 2013 Springer Science+Business Media New York 2013 Clifford al

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Non-euclidean and Projective Geometries,We investigate the relationship between conformal transformations in ., studied in the previous chapter, and orthogonal transformations acting on the . in .. The horosphere is a nonlinear model of a pseudo-Euclidean space in a pseudo-Euclidean space of two dimensions higher.

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Lie Groups and Lie Algebras,We have studied linear transformation on . using the traditional matrix formalism in Chap. 7 and more generally in Chaps. 8–10, using the machinery of geometric algebra. This chapter explains the . interpretation of a general linear operator and offers a new proof of the Cayley–Hamilton theorem based upon this interpretation.

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Vector Spaces and Matrices,a column vector. Since the rules of matrix algebra over the real and complex numbers are identical to the rules of the addition and multiplication of geometric numbers, it makes sense to consider matrices whose entries are geometric numbers alongside the more traditional matrices of real and complex numbers.

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Linear Transformations on ,,f geometric algebra, such as the .-derivative and the simplicial .-derivative, are used to study its basic properties. We introduce the adjoint linear transformation and use it to derive the inverse of a nonsingular transformation.

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Structure of a Linear Operator,tral basis of idempotents and nilpotents which was developed in Chap. 1 for modular polynomials. The . of a linear operator, while technically more difficult, is just a refinement of its more fundamental spectral decomposition.

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Linear and Bilinear Forms,oint. Instead of talking about the . of a given basis, we introduce the concept of a .. The relationship between a bilinear and a quadratic form is discussed, and Sylvester’s famous law of inertia is proven. The material lays the foundation for studying geometric algebras of arbitrary signatures in later chapters.

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