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A Polynomial Approach to Linear Algebra is a text which is heavilybiased towards functional methods. In using the shift operator as a centralobject, it makes linear algebra a perfect introduction to other areas ofmathematics, operator theory in particular. This technique is very powerfulas becomes clear from the analysis of canonical forms (Frobenius, Jordan). Itshould be emphasized that these functional methods are not only of greattheoretical interest, but lead to computational algorithms. Quadratic formsare treated from the same perspective, with emphasis on the importantexamples of Bezoutian and Hankel forms. These topics are of great importancein applied areas such as signal processing, numerical linear algebra, andcontrol theory. Stability theory and system theoretic concepts, up torealization theory, are treated as an integral part of linear algebra.Finally there is a chapter on Hankel norm approximation for the case ofscalar rational functions which allows the reader to access ideas and resultson the frontier of current research.