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Rating: 
Summary: Little Value for Self-Preparation
Review: As an individual with a limited background in mathematics entering a technical graduate program, I was initially enthused when I came across this book. From its description I thought it would provide the reader with the ability to be introduced to the practical aspects of some key mathematical concepts. Instead, it is basically a stripped down book for mathematicians, focusing on proofs rather than examples. My two main gripes with this book:1) It is devoted almost entirely to theoretical proofs and nearly devoid of numerical examples. Again, this is of limited value unless you're entering specifically a theoretical mathematics program (I'm in physics).
2) Unless I'm missing the point, the target audience for this book is the individual reviewing this material on his/her own, outside of or in preparation for a classroom. Yet the book has no answers to any of the end-of-chapter exercises; this is a fatal flaw for a book of this type. I'm sorry to have to give this book such a negative review, but a more appropriate title would be 'Review of key theoretical concepts for Graduate Students in Mathematics".
Rating: 
Summary: Very helpful as a guide
Review: I found this book to be very helpful as a guide to self-study in mathematics. I did not rely on the chapters for understanding, but rather used them as a topic list for a several year course of study. I used the bibliography to find the best books for study and then later used the chapters as an essential review. When I finished, I felt I had a completely satisfactory undergraduate education in mathematics at a fraction of the usual cost. I now have an excellent library as well.
Rating:
Summary: A Good Tool for Diligent Self-Study
Review: There's no doubt about it -- this book designed for people who want to learn some real math. It doesn't take, as the title and description might lead you to believe, a "Math for Engineers" approach. Each chapter covers, in the span of 10 or 15 pages, what would normally be an entire semester's worth of material, and as a result, is quite dense -- there are alot of ideas crammed onto each page. But unlike traditional advanced math books (which are notoriously dense) the focus is more on developing intuitions than on long strings of equations. An important strength is that every chapter ends with suggestions on textbooks in that chapter's subject. This turns out to be quite helpful, since one can't reasonably expect to learn everything important about any of these subjects from a brief chapter in any book. I can envision three main ways in which this book might be useful: First, in combination with one or more of the books in listed in the bibliography for learning a new subject. Second, on its own for review of topics you've seen before. Third, as a reference for "basic" definitions and theorems, as in: "What's a Hilbert space again?" Overall, this will be a good book to have around, but not a substitute for real study.
Rating:
Summary: A Good Tool for Diligent Self-Study
Review: There's no doubt about it -- this book designed for people who want to learn some real math. It doesn't take, as the title and description might lead you to believe, a "Math for Engineers" approach. Each chapter covers, in the span of 10 or 15 pages, what would normally be an entire semester's worth of material, and as a result, is quite dense -- there are alot of ideas crammed onto each page. But unlike traditional advanced math books (which are notoriously dense) the focus is more on developing intuitions than on long strings of equations. An important strength is that every chapter ends with suggestions on textbooks in that chapter's subject. This turns out to be quite helpful, since one can't reasonably expect to learn everything important about any of these subjects from a brief chapter in any book. I can envision three main ways in which this book might be useful: First, in combination with one or more of the books in listed in the bibliography for learning a new subject. Second, on its own for review of topics you've seen before. Third, as a reference for "basic" definitions and theorems, as in: "What's a Hilbert space again?" Overall, this will be a good book to have around, but not a substitute for real study.
Rating: 
Summary: Good for a recap, bad for anything more
Review: This book has a very particular purpose: to recap some basic concepts from undergraduate mathematics so that you get the "big picture". In other words, for every math course you took as an undergrad, this book provides a good outline of the major ideas and how they fit together. But, it is only an outline; nothing more. If you actually missed out on some topic, or your knowledge of a subject is shaky, then this book won't help much. It will only help by providing a bibliography of some other references for that subject. This book is meant to organize your undergraduate math knowledge, not to supplement it. With that said, I'll mention a few words about the content of the book. It is quite well written and definitely extracts the essential ideas for your quick consumption. There are a few topics that I personally feel are missing, such as Gram-Schmidt and Jordan Canonical Forms for Linear Algebra, and UFDs and PIDs from Algebra. In general, it seemed like the book leaned a little more towards analysis than algebra, but the vast majority of important topics were indeed encapsulated in their synopsis. Good for a very specific audience, but otherwise not wonderfully useful.
Rating:
Summary: Good for a recap, bad for anything more
Review: This book has a very particular purpose: to recap some basic concepts from undergraduate mathematics so that you get the "big picture". In other words, for every math course you took as an undergrad, this book provides a good outline of the major ideas and how they fit together. But, it is only an outline; nothing more. If you actually missed out on some topic, or your knowledge of a subject is shaky, then this book won't help much. It will only help by providing a bibliography of some other references for that subject. This book is meant to organize your undergraduate math knowledge, not to supplement it. With that said, I'll mention a few words about the content of the book. It is quite well written and definitely extracts the essential ideas for your quick consumption. There are a few topics that I personally feel are missing, such as Gram-Schmidt and Jordan Canonical Forms for Linear Algebra, and UFDs and PIDs from Algebra. In general, it seemed like the book leaned a little more towards analysis than algebra, but the vast majority of important topics were indeed encapsulated in their synopsis. Good for a very specific audience, but otherwise not wonderfully useful.
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