A Path to Advanced Algebra

This book offers a fresh route through the classical territory of linear algebra, built around one guiding question: when are two matrices similar? Rather than reaching first for eigenvalues, the authors begin with the language of modules and diagrams, developing an algorithmic, field-independent answer to this problem before turning to the more familiar theory of characteristic and minimal polynomials.

What sets this book apart is its third strand: the topology of similarity classes, a subject rarely treated at this level, which reveals how the algebraic and reduction-theoretic viewpoints interact through continuity, conditioning and Gaussian elimination's numerical pitfalls.

Throughout, morphisms take precedence over objects, and abstraction is kept firmly in service of concrete computation: matrices remain the constant thread linking modules over rings, Jordan and Frobenius forms, semisimplicity, and eigenvalue localisation. Numerous worked examples and exercises encourage active engagement, while starred sections point towards more advanced horizons – K-theory, resultants, connected solvable subgroups – without disrupting the main narrative.

Assuming only a first course in linear algebra and basic ring theory as prerequisites, this volume is designed for advanced undergraduates and graduate students seeking a rigorous, geometrically-minded introduction to invariant theory, canonical forms, and the continuity properties that govern them.