Preface This book is a presentation of the elements of Linear Algebra that every mathematician , an d everyon e wh o use s mathematics , shoul d know. I t covers the core material, fro m th e basic notion o f a finite- dimensional vector space over a general field, to the canonical form s of linear operators and their matrices, obtained by the decomposition of a general linear system into the direct sum of cyclic systems. Along the way it covers such key topics as: systems of linear equations, lin- ear operators an d matrices, determinants, duality, inner products an d the spectral theory of operators on inner-product spaces. We conclude with a selection of additional topics, indicating some of the directions in which the core material can be applied and developed. In it s mathematica l prerequisite s th e boo k i s elementary , i n th e sense that no previous knowledge of linear algebra is assumed. I t is self-contained, an d includes a n appendix that provides al l the neces- sary background material : th e very basic properties of groups, rings, and of the algebra of polynomials over a field. The book is intended, however, for readers with some mathematical maturity and readiness to deal with abstraction and formal reasoning. I t is appropriate for an advanced undergraduate course. As the title implies, the style of the book is somewhat terse. W e mean this in two senses. First, we focus with few digressions on the principal ideas and re- sults of linear algebra qua linear algebra. The book contains fewer rou- tine numerical examples than do many other texts, and offers almos t no interspersed application s t o other fields these shoul d be adapte d to the readership and, if the book is used in a course, provided by the teacher. IX
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