The recent developments in canonical transforms, matrix theory, block Kronecker multiplications, and other areas are applied to extend and simplify results in the theory of first order systems and special relativity. Especially noteworthy are the author's results on Fourier transforms in dimensions lower than the surrounding space and his approach to the Doppler effect, which has never been published previously and supersedes previous work on this topic, which failed to solve the Doppler effect exactly. Some of the goals of this work are: to develop the theory of complex symmetric matrices as the rigorous foundations of first order systems, to exhibit in full generality the author's method of duality, and to discuss the neglected area of three dimensional effects in special relativity. The section on special relativity has been especially simplified so that it may be used as a beginning graduate text in this area. It includes the first full discussion of the Lorentz group in a book since Silberstein's pioneering 1913 treatment.
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