Linear Algebra Done Right

Linear Algebra Done Right by Sheldon Axler for the following reasons: - I was revisiting a topic in greater depth, which is a common theme in university-level math courses. - It is a rigorous book, written in the style of definition, proposition, theorem, etc. - It was the first math book where the exercises don't just reinforce what you learned in the chapter, but teach you new material (another common theme in advanced math textbooks). - Linear Algebra is arguably the most important math subject these days.

To echo sibling comment but make it more prominent - "Linear Algebra Done Right" is a great second book, I wouldn't use it as a first source of learning Linear Algebra. (I'm pretty sure it says as much in the intro, IIRC.) I personally learned from Strang, then this book, which complemented each other very well. Strang is very focused on practicality, actually computing things, always seeing the matrix behind everything. "Done Right" is great at ignoring most of the "matrix view" of things completely in favor of pure linear transformations instead. Both views have their use - so learning one, then the other, is a great idea.

"Linear Algebra done right" by Sheldon Axler. He also has a beautiful series for videos on YouTube that can serve as a companion to the book.

The book is open access (https://linear.axler.net/), I am going to quote directly the author in the preface: << all linear algebra books use determinants to prove that every linear operator on a finite-dimensional complex vector space has an eigenvalue. Determinants are difficult, nonintuitive, and often defined without motivation. To prove the theorem about existence of eigenvalues on complex vector spaces, most books must define determinants, prove that a linear map is not invertible if and only if its determinant equals 0, and then define the characteristic polynomial. This tortuous (torturous?) path gives students little feeling for why eigenvalues exist. In contrast, the simple determinant-free proofs presented here (for example, see 5.21) offer more insight. Once determinants have been banished to the end of the book, a new route opens to the main goal of linear algebra— understanding the structure of linear operators.>> If you like mathematics, it is actually a pretty nice book.

I recently saw Sheldon Axler's Linear Algebra Done Right recommended here, and have been working through it. In my nonexpert opinion, it's really good, and has a great series of videos reviewing each section in a clear and understandable fashion. https://linear.axler.net/

Baby Rudin is indeed awesome. I feel like Linear Algebra Done Right is a great book, but it is a better second book about Linear Algebra, rather than a good first book. Actually, I think the whole subject deserves a fairly mechanical treatment first, and only then an explanation of the underlying math. I think it would make sense to introduce Matrix multiplication and related concepts to students in high school, so they become completely mechanical - much like we teach students how to differentiate functions as a purely mechanical process.

Baby Rudin, Principles of Mathematical Analysis, was that book for me. It was that book for a lot of people, which I guess means it felt "perfect" for people with a broad range of backgrounds. I never found a book like that for algebra. Algebra felt opaque, simultaneously hard and boring, and linear algebra felt dry and mechanical, more like a vocational skill than like mathematics. If I had encountered Axler's Linear Algebra Done Right in undergrad, I might have been more interested in algebra. By the time I encountered "computer science" in my mid-twenties, I don't think the quality of textbook mattered as much to me, but I wish I could remember the name of the formal languages and automata textbook I used to have. It was short and probably not very advanced, but the concepts are still vivid in my head. I see state machines everywhere, and things feel easy to me when I can tackle them as state machines.

Linear Algebra Done Right

Focusing on mathematics, I think about two kinds of foundational books. 1. Historically foundational: books that brought important ideas, introduced a new branch, progressed the maths significantly. You can read them these days, but their content, proofs, language, etc. is not up to date. Examples: Elements by Euclid, Philosophiae Naturalis Principia Mathematica by Newton, Disquisitiones Arithmeticae by Gauss. 2. Modern foundational books: The ones that can be used for studying now. Books that are clearly written, popular, cover their subject well. Examples: Visual Complex Analysis, Spivak’s Calculus, Linear Algebra Done Right, Hardy’s An Introduction to the Theory of Numbers.

Eh, I was thinking about something else, actually. What you listed are taught in every CS program, aren't they? It isn't what I imagined when I heard "rigorous" at all. Topology, number theory, abstract algebra (I mean, real one, not CS-course basics), statistics, tensor analysis? Isn't that "undergrad math"? For things like Set theory/combinatorics/logic basics I'd recommend Rosen's "Discrete Math and Applications"[1]. CS oriented, simple, interesting, broad. Covers all the basic stuff. Linear algebra — two books, "Linear algebra done Right" and "Linear algebra done Wrong". Second one more math-oriented, the first one — pretty simple, pretty clear, fun to read. Real analysis ("calculus" you mean?) — I personally learned from different sources and probably the most concise book I read is Fichtengolz's "differential and integral calculus", but I don't know if it's available in english. I guess, almost any book on topic is fine.…

I third the recommendation on Axler's LADR. I'd actually like to hear about alternatives to Rudin. My undergrad analysis class used it, but it's lack of diagrams was particularly bothersome. I'd spend an hour digesting a rat's nest of a paragraph only to discover that the underlying concept was simple enough that even a rough sketch ought to be able to get the gist of it across in seconds.

Thanks. Axler's LADR is indeed fantastic, I have already worked through 1/2 of it. I'm really happy to see my opinion seconded. It's really short and precise. Some people seem to prefer Halmos' Finite Dimensional Vector Spaces, but I found the presentation less didactic. Perhaps it's also more of an upper division text, and I'm not there yet. I was looking for a real analysis companion, perhaps baby Rudin. I was also wondering whether it'd make sense to proceed directly to real analysis, or to step down a bit and read something like Spivak's Calculus. I'd also like to hear about recommendations at undergrad level in the fields of logic and set theory, geometry, combinatorics, and probability theory. Those would complete my basic math curriculum.

Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra. As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fields, r-modules, and advanced topics (toward the end) with the unifying theme of how they work and relate as categories. It is very much a book that focuses on the why over the what and how. And there are many many many juicy exercises. I would recommend against Spivak's Calculus on manifolds. It's too dense and too focused on advanced topics (unless you're an ivy league undergrad, you don't learn cohomology). That being said I don't have an analysis reference that fits your request. As a CS person I can continue with book recommendations related to computing-related topics in math (computational algebraic geometry comes to mind). Let me know if you're interested.

If you have some decent background in reasoning and logic, I would suggest Linear Algebra Done Right by Sheldon Axler. It provides a general introduction in mathematical reasoning as well as providing a strong framework for maths needed in many fields. http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/... (http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/0387982582/)

I personally liked "A Logical Approach to Discrete Math" by Gries and Schneider which can be had for a good price on Amazon. As for statustics, I liked "Statistics: A Gentle Introduction" by Coolidge and "Principles of Statistics" by Bullmer which can be had for a great price even new. Although only slightly related to your chosen topics, "Linear Algebra Done Right" by Axler is awesome. -m