Linear physics and the mathematics related to the study of linear physics is the foundation of most ongoing work in applied and theoretical physics and engineering research. A fundamental understanding of the axioms, definitions and theorems of linear physics are therefore essential for the study of the fields of analytical mechanics, quantum mechanics, quantum electrodynamics as well as the emerging science of string theory.;From analytical mechanics the representation of problems in phase space and the elegant properties of symplectic manifolds are explored and then shown in this dissertation to be expressible in the terse form of spinor vectors in a complex vector space. A two dimensional complex spinors is shown to represent a two sheeted cover of a Riemann surface in an Minkowski space. This dissertation shows how such a spinor is used to represent massless Fermions and then extends this model to represent massive Fermions in a Minkowski tangent space using an original geometrical model.;Within the body of this dissertation is found an original exposition of geometrical and matrix algebra which shall provide a watershed of future research in the fields of physics, linear mathematics and systems science.
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