xii Preface Each chapter starts at a point where I think there are a fair number of seniors. In a certain sense this was the most difficult part of writing this book determining where to start so that most students would find it accessible. In standard courses it is easy to decide where to start. Here I feel that wherever I start I am certain to be ahead of some and behind others. In other words, every possible starting point is a compromise. This places a greater onus on the teacher to add material or delete sections so as to tailor the presentation to the students in the room. To help a little, there is an Appendix with seven sections, arranged not in the order in which they are needed in the book but in a quasi-logical development. The purpose of the Appendix is not to act as a course on these subjects, but to set a starting point, bridge gaps, act as a handy reference for this material, and help increase the comfort level of the students. In fact most of the Appendix covers material only needed as background for Chapter 6 on Modules, which I think is the most difficult in the book. If that chapter is not to be part of the course, there is little need to worry about it. The first two sections of the Appendix are on groups and rings just a taste of these topics is provided. In fact Chapter 6 develops a fair amount of ring theory, though only what is needed for its objectives. In the Appendix I tried to be a little more careful in discussing quotient groups and rings as even those students who have studied algebra seem uncomfortable with these concepts. I didn’t want a student to avoid a chapter like the one on modules just because they hadn’t encountered groups and rings, when all they need is the barest of familiarity with such concepts. Starting in §A.3, I discuss vector spaces over an arbitrary field, which is mildly dependent on the material on rings. Nothing deep is required to digest this as such concepts as linear independence using a field are the same as that idea for a vector space over the real numbers. The idea here is again to be thinking ahead to Chapter 6 where I want to prove the Artin-Wedderburn theorem, and this requires the idea of a vector space over a division ring. I felt the transition from fields to division rings was a bit less abrupt than starting with a linear space over the real numbers. §A.4 deals with linear transformations, again on vector spaces over an arbitrary field. This can be regarded as being sure that all readers have easy access to some basic facts concerning linear transformations and, at least as important, the examples that are used repeatedly in Chapters 3 through 6. §A.5 discusses lattices, though only the definition and some examples are presented. This language is used in discussing invariant subspaces of a linear transformation. The final two sections of the Appendix state results on triangular representation of a linear transformation over a finite-dimensional vector space over the real or complex numbers. This is used in Chapter 5 on Matrices and Topology. No proofs are presented, only the statements
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