A (Terse) Introduction to Linear Algebra (Student Mathematical Library) by Yitzhak Katznelson and Yonatan R. Katznelson (PDF)

Description

Linear algebra is the study of vector spaces and the linear maps between them. It underlies much of modern mathematics and is widely used in applications. A (Terse) Introduction to Linear Algebra is a concise presentation of the core material of the subject–those elements of linear algebra that every mathematician, and everyone who uses mathematics, should know. It goes from the notion of a finite-dimensional vector space to the canonical forms of linear operators and their matrices, and covers along the way such key topics as: systems of linear equations, linear operators and matrices, determinants, duality, and the spectral theory of operators on inner-product spaces. The last chapter offers a selection of additional topics indicating directions in which the core material can be applied. The Appendix provides all the relevant background material. Written for students with some mathematical maturity and an interest in abstraction and formal reasoning, the book is self-contained and is appropriate for an advanced undergraduate course in linear algebra.

User’s Reviews

Editorial Reviews: Review The self-contained, well-written work assumes mathematical maturity…It is a concise, thorough introduction to linear algebra. –ChoiceThis little book is a pleasure to read… In short, there are many who can get a substantial amount out of this little book, and I am definitely glad to have read it… –MAA ReviewsThe book is written in an elegant, condensed way. It contains many exercises, mostly of theoretical character. The main advantage (in particular for teachers and talented students) is that basic ideas are carefully isolated and presented in a simple, minimal and understandable way. –European Mathematical Society Newsletter

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐…then this is your kind of book. Personally, I love it. A lot of my friends hate it. It’s incredibly dense. So much so that many of the proofs are only outlined, leaving the reader to fill in the details. By doing so, this book has managed to engage me in reading it and keep me focused better than any other math book has. Unpacking each theorem definition-by-definition keeps me versed on all the definitions and other theorems I need to know. It’s self-reinforcing. I wouldn’t recommend it if you’re not adept at parsing thick blocks of jargon and symbols. Definitely keep a supplementary text handy (like Wikipedia or Schaum’s Outlines).

⭐I like how it is for advanced students.

⭐As the title suggests. The book is very terse. So is this review.

⭐This is a very nice little book on linear algebra. As the title says, the text is (very) concise. The main part of the book is covered in only about 160 pages with a large typeset. This has been achieved by short and elegant proofs and by leaving much of the work to the student. In addition, routine numerical examples are omitted. Quite a few proofs have been left as exercises and as it’s remarked in the preface, some proofs are not complete to every detail and few are actually more like detailed lists of hints. Because of this, the book requires, and more importantly, exercises mathematical maturity. In my university it was offered as an advanced course for mathematicians but in Stanford, where the book comes from, the course is apparently aimed for second year students or so.The text starts from the very basics and requires no prior knowledge on anything but an ability to read mathematical text. It certainly helps if you’ve taken a prior (numerical) introduction course to linear algebra. Because of the quick pace, theory goes surprisingly far I think (see table of contents from the book preview). There has also been room for additional topics for the interested. The presentation in the book is great and the text flows well. The chapters are divided into subchapters each of which has at maximum only one theorem, proposition, corollary and lemma, so that references are easy to follow. Proofs are short, intuitive and easy to remember once you get them.The exercises are well chosen and certainly solvable even though the book has no solutions to them. There are practically no numerical examples or exercises, only proofs. Doing some of the exercises is an important part of studying the book properly.The current first edition has some misprints and lots of inconsistent notation. However, I did not find this a very big problem. Due to the nature of the book, it might not be well-suited for self-study but makes an excellent course book. If you wanted to solidify your previous knowledge on the subject, you might enjoy studying the book on your own. Of course you can also benefit from book without going into all the details and read it as a quick introduction to the subject or as a reference.Unfortunately, I cannot compare this book to other books on linear algebra as I haven’t read any. This book taught me a lot about doing mathematics on my own and I enjoyed reading it. The author has won an exposition prize with another book of his.

⭐This text is written for advanced undergraduates (and beginning graduates as well). It comprehensively covers the theory in an abstract setting. Being around 200 pages, it’s presented in an elegant, modern, streamlined, and concise fashion. It leaves the reader plenty of things to fill in, as well as a nice selection of good exercises. I bought this book on an impulse, just to have it in my library (already being well-versed in the subject). It would have been very nice to have learned from this book.Comparing/Contrasting with other noteworthy Linear Algebra texts:(1) Serge Lang – Linear Algebra; I learned from this book. Though Lang has written a bad book on just about every subject in mathematics, his Linear Algebra book is lovely, and comparable to this book. The main difference is that most of Lang’s exercises have an “Algebra” flavor, in that they mostly require you to exploit the group structure of vector spaces; Katznelson’s book (the book I’m reviewing) seems to take a more unified approach with its exercises.(2) Georgi Shilov – Linear Algebra; it’s a translation of an older Russian book. The notation is old/annoying, and (as is true for most translations of Russian books) the language/logic can be contorted and inefficient, though the theory is covered in a very classical way.(3) Paul Halmos – Finite Dimensional Vector Spaces; it’s another good book, though a bit out-dated, wordy, and geared with a more “Analytic” flavor.(4) Sheldon Axler – Linear Algebra Done Right; this is another modern book. It’s my favorite, behind the Lang and Katznelson (this) books. It seems to be a bit less abstract and down-to-earth, but nevertheless effective.In summary, among other noteworthy Linear Algebra texts, I feel that this book is the best because of its broad coverage of the theory, concise exposition, and its modern and unified approach. So, in my opinion, for students of pure math, this is THE book to have for Linear Algebra.

⭐I read a bit of this book to understand and refresh basic concepts of linear algebra. Being a Computer Science major, I did not need the details of linear algebra, and this reading was sufficient for a course on quantum algorithms. The contents are accessible to any non-mathematical majors with basic high-school level math. The book is perfect for CS/EE grad students who might need some linear algebra knowledge for topics like coding theory. Would highly recommend to anyone as a quick introduction/reference.

⭐Clear and concise. One of the best I have come across.

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