Linear Algebra

What Hacker News says

1. kingofalltradesMar 2024
   First step into math is to learn how math people think. Here below is an intro textbook meant to do just that[0]. It's a high quality, free introduction. You don't need to know much past 6th/7th grade level math which amounts to knowing how to do arithmetic with fractions and decimals and maybe an ability to in/equalities in one variable. I will also add a couple of google links that might be more relevant to what you might be trying to do[1][2].

   [0] Book of Proof by Richard Hammack https://www.people.vcu.edu/~rhammack/BookOfProof/

   [1] Linear Algebra: Step by Step by Kuldeep Singh https://www.google.com/books/edition/Linear_Algebra/BJNoAgAA...

   [2] Number Theory: Step by Step by Kuldeep Singh https://www.google.com/books/edition/Number_Theory/YZr9DwAAQ...

   [3] Undergraduate Analysis: A Working Textbook by Aisling McCluskey and Brian McMaster

   https://books.google.com/books?id=7U9HtAEACAAJ&printsec=copy...

   There's even more relevant book for you[4]

   [4] Foundations of Applied Mathematics by Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans

   https://books.google.com/books?id=CEc3DwAAQBAJ&pg=PR3&source...

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2. ufopilotFeb 2022
   My very first intro to math was through "Learning to Reason: An Introduction to Logic, Sets, and Relations" by Nancy Rodgers.

   Many books of this kind are usually called "Intro To Discrete Math" because I think discrete math is easier and neater than its continuous counterpart. Most such books consider a sampling of topics from number theory, combinatorics, graph theory and discrete probability. Obviously, they do not even scratch the surface of the subjects they teach as that's not the goal. For example, compare what's in the counting section of a typical intro to discrete math book to an intro to counting proper. For example, my very first discrete math textbook was "Discrete Mathematics with Applications" by Susanna Epp and for counting it was "Combinatorics: A Guided Tour" by David R Mazur. Epp's books covers much, much, much less. Similarly, currently one of the very best intros to elementary number theory is "Number Theory: Step By Step" by Kuldeep Singh. The latter also contains much more material than a typical intro to discrete math.

   Beyond that every author of an into to discrete math likes to include a little taste of their own personal favorites. For example, Susanna Epp has a chapter on algorithm analysis whereas, say, Edward Scheinerman talks a little bit about group theory in his book "Mathematics: A Discrete Introduction". Yet other authors include chapters on geometry, vectors spaces, generating functions etc.

   I noticed (almost) every math/CS adjacent thread on HN gets mention of Big O notation. IMO, the best place to learn that is a textbook on discrete math. For example, Susanna Epp has a whole chapter dedicated to this topic in her intro to discrete math book. Consider [0] from Epp's book. Since it's a math book it's very careful and thorough meaning if you don't understand something in that example reading the material that precedes will clarify everything; just a matter of backtracking far enough. Since it's an extremely introductory book, it's also very patient. I personally struggled with the motivation for defining total derivative through little-o notation.

   There are a ton more textbooks on discrete math like, say, " Essential Discrete Mathematics for Computer Science" by Lewis & Zax or "Discrete Mathematics: An Open Introduction" by Oscar Levin.

   Some intro books also attempt to include a tiny bit of real analysis and general topology. Topics include ways of using least upper bound, epsilon-delta arguments, open/closed sets, a little on metric spaces etc. Obviously, such books don't get carried away by ditching epsilon-delta/balls arguments in favor of more general open sets to define continuity, for example. One such book that comes to mind is "Mathematical Proofs: A Transition to Advanced Mathematics" by Chartrand, Polimeni, Zhang.

   I think the best book in this introductory genre is "Book Of Proof" by Richard Hammack. It's freely available. I see the latest edition added a few techniques for proving convergence and continuity on the real line: epsilon-delta and epsilon-N stuff.

   I call the books mentioned above "Intro To Math 1". After these books one can get started with more in depth books on the subjects considered in the aforementioned books or pick up an intro to real analysis, topology, abstract algebra, whatever.

   One of the most effective textbooks on elementary math analysis is "Principles of Mathematical Analysis" by Rudin. But one will have a real tough time with it even after studying the aforementioned books. But there are fantastic preparatory books like "Linear Algebra: Step by Step" by Kuldeep Singh which one needs because Rudin considers Euclidean Spaces proper, not just the reals. Other helpful books include "Yet Another Introduction to Analysis" by Bryant, "How to Think About Analysis" by Lara Alcock, "A First Course in Mathematical Analysis" by Brannan, "Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings" etc. None of these books are perfect. For example, none of them (but that of Cummings) include the necessary topological concepts. Also, even though the book by Bryant is chatty, his exercises can be tough. Case in point, he expects you to prove that rationals are incomplete in the very first chapter. To his credit, he gives a lot of hints and every one of his exercises have detailed solutions. When I first saw Alcock's book I got the impression that her book was an excuse to explain how we think about the convergence in reals. The rest of the book felt like a filler, even though the book is worth it for that chapter alone. It's good to look into many different books because the authors make different choices. For example, Cummings proves Bolzano-Weierstrass theorem through the fact that every sequence has a monotone subsequence whereas Brannan does it by way of bisection argument. Other authors might make yet other choices. In math the more techniques you have in your tool chest, the easier it is to attack problems. If you solve a problem one way, but the solution given is different it's worth memorizing the alternative solution as it's another technique that will come in handy in a different situation.

   Ok, I didn't say anything about abstract algebra and other subjects, but this post is already turning into a novel.

   After making very first and wobbly steps into math through the intro to discrete math books, the other path is to study what I call "Intro To Math 2". There are books like "Mathematical Concepts" by Jost or (especially) "Essential Mathematics for Undergraduates" by Giossi. Wanted to say more about this or the other topics above. Maybe next time.

   Please, excuse the grammar and inconsistent notation (& naming convention) as it is very late here.

   [0] https://ibb.co/Q81b96R

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3. mathhhNov 2021
   > You can work through the concepts and delude yourself into thinking you understand something when really you're just hand waving it

   I keep seeing this and I don't quite understand how this one works.

   If I were studying math on my own (which I've done and still do), I'd do the following:

   1. Pick a book. Say, Rudin's Principles of Mathematical Analysis[0]. Read a section, then attempt problems. Pick a problem. Say, "prove {1/n: n is natural} U{0} is compact directly from the definition(not using Heine-Borel)". It's guaranteed your "proof" is not a proof.

   2. Compare your solution to existing solution manuals or ask a question on MSE[1]. Since the given book by Rudin is super-massively famous, each question has probably been asked/answered about a bagillion times each on MSE, so just searching MSE alone would likely to spit out many answers to your questions. People on MSE will tell you exactly why your "solution" is wrong and where you tripped up. Sometimes even the clarifying answers are hard to understand. But then you can ask new questions, think more, correct your misconceptions until it all finally clicks. Do that with all the rest of the problems[2].

   I don't see how the process above is delusional.

   [0] This book is not a realistic fit for a novice, though. Instead, one would start with books like [3], [4], [5], [6] to learn how to prove things and think like a mathematician would.

   [1] https://math.stackexchange.com/

   [2] In reality, the more math you see and do, the more mathematically mature and less dependent on others(to check your work) you become. In fact, if you can solve any problem in "B@by Rudin" and some famous abstract algebra textbook (say, Dummit & Foote's "Abstract Algebra") cold, you're way ahead of most any undergrad math major in the world! Because standards on undergrad math majors are not that high, nor that brutal the world over no matter what they say. If, additionally, you can solve any given problem in a book like, say, Hatcher's "Algebraic Topology" or any other famous grad level textbooks on, say, differential geometry or, uhh, functional analysis, you're officially in the big leagues. Again, if you're worried about being delusional about your proofs, you can always present them on MSE.

   [3] "Book of Proof" by Richard Hammach. It's online free.

   https://www.people.vcu.edu/~rhammack/BookOfProof/

   [4] "Discrete Math" by Susanna Epp

   https://www.google.com/books/edition/Discrete_Mathematics_wi...

   [5] "How to Think About Analysis" by Lara Alcock

   https://www.google.com/books/edition/How_to_Think_about_Anal...

   [6] "Linear Algebra" by Kuldeep Singh

   https://www.google.com/books/edition/Linear_Algebra/BJNoAgAA...

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4. teleforceJun 2020
   Coincidentally, I've spent countless hours during the Covid-19 isolation looking for good to excellent textbooks on Engineering and Computer Science. Perhaps I can share them in the near future. These are some excellent books based on my findings for your particular requirements.

   1) For general applied and engineering Maths book perhaps you can look into both textbooks by Prof. Kuldeep Singh and they uses Matlab but Octave should be fine [1].

   a) Engineering Mathematics Through Applications b) Linear Algebra

   2) For signal processing especially for communication I'd highly recommend "Digital Signal Processing in Modern Communication Systems" by Andreas Schwarzinger. It introduces the modern communication systems from the first principles to the more applied OFDM systems with examples that can be run using either Matlab or Octave. Just get the book it's really good even if you're not interested in communication systems.

   3) If you want to learn more advanced applied engineering books on signal processing I'd recommend reading books on time-frequency techniques both linear and non-linear. For introduction check the book "Explorations in Time-Frequency Analysis" by Patrick Flandrin. Time-frequency analysis is most probably going to be pervasive very soon for increasing the accuracy of AI and ML by providing better intermediate processing of data features.

   [1] https://maths-for-all.co.uk/

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5. newqazMay 2020
   Discrete math topics teach amazing way to think that most people never get to see in high school. Knowing stuff in free a book like [0] is immensely helpful. There are a ton of decent introductory discrete math books like the ones by Susanna Epp, Ed Scheinerman, Goranko bros and Gary Chartrand. Just google "list of discrete math book". What google spits out won't even scratch the surface of what's available out there. But modern CS folk will have to know much more than the basics of discrete math. For example, math analysis and probability theory are very helpful. This free book [1] gives a sampling of such topics. As preparation, one can start by looking at pre-real-analysis books like the ones by Lara Alcock and Jay Cummings, Linear Algebra by Kuldeep Singh and Probability Theory by Dimitri Bertsekas/Tsitsiklis. These books are very easy to read. There are also introductory books that give a bare-bones sampling of most undergrad math subjects from abstract algebra to topology like the ones by Gary Chartrand (separate from his discrete math book) and Steve Warner. Such books are designed to be as hand-holdy as possible. The more I type the more I realize there's more (much, much, much more) to say about the math side of things. Anyway, for another thing, google category theory just to be aware of it. There are a few undergrad/high school level books on the subject, but I am not sure how useful that is to a freshman.

   [0] Book Of Proof by Richard Hammack

   https://www.people.vcu.edu/~rhammack/BookOfProof/

   [1]Foundations of Data Science by Avrim Blum, John Hopcroft, and Ravindran Kannan

   https://www.cs.cornell.edu/jeh/book%20no%20so;utions%20March...

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6. mixtureMar 2020
   Disclaimer: please, excuse the typos, grammatical and stylistical mistakes below as I am pretty freaking drunk at the moment.

   Friendly reminder:

   while the material in the linked books might appear trivial to many, the style might be discouraging to those who've never seen "actual math" before. If you're in the latter group, try out the books below (some are not free, but they're still helpful [libge*n might be helpful]):

   1. BOOK OF PROOF by Richard Hammack. This FREE book is your passport into mathematics proper. It teaches you how to read math books. After this book you'll stop fearing the word "proof" and start demanding it wherever you go.

   Link: https://www.people.vcu.edu/~rhammack/BookOfProof/

   2. The book by Don Shimamoto is super trivial if you know linear algebra. The best books in the genre (imo) are (a) Linear Algebra: Step by Step by Kuldeep Singh whose secret is that he uses all the exercises from other LA books as his 100% worked out examples (which seems to be a tradition for Indian authors; I have tons of examples from other branches of math). Indian authors teach you how to drive a car by putting you at the steering wheel instead of lecturing you about it. BTW, if you know math at the level of, say, 7th grader, then you're perfectly suited to study from Singh's book. If that isn't clear, do you know how to add and multiply fractions? If yes, then you're perfectly ready.(b) A modern Intro to Linal by Henry Ricardo. This book is great in its ability to explain certain introductory concepts. For example, the linked Shimamoto book starts out with (going by memory) "the elements of R^n are finite sequences of length n... bla bla bla". Well, Ricardo explains what that means. It also helps if you know what a "sequence" is. But we're getting there. Another author who does a good job of it is Sheldon Axler, I believe. He does have his own idiosyncrasies, but I digress. The only difficulty here is, probably, the idea of diferentiability of multivariable functions.

   Links:

   https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh-ebo...

   https://www.amazon.com/Modern-Introduction-Linear-Algebra/dp...

   http://sites.msudenver.edu/wp-content/uploads/sites/385/2017...

   3. Now to analysis of real kind as opposed to the complex one. If you've never heard anything about this, then you should hook up with Lara Alcock's "How To Think About Analysis". To the best of my recollection, half the book is about how to think about the term "sequences". She goes above and beyond in trying to explain you what that means. The rest of the book is so-so. Once you learn about sequences, you can move on to Jay Cumming's book. It's a bit more involved with proofs and shyt, but still extremely accessible. Laughably so. But note, he only stays grounded on the real line and does not venture into the Euclidean Spaces proper. However, there's another easygoing dude named Raffi Grinberg who'll take you through R^n by hand, no problem. If you master this book, then you're at the level of what's referred to as "Baby Rudin". Next, if you're feeling adventurous, there's a new book by Sheldon Axler that introduces you into the modern analysis which is FREE FOR NOW. GRAB IT WHILE YOU CAN CAUSE I SAW IT ON AMAZON.

   Now if you're a business type who ain't got no time for all this foolishness, try "Intro to Analysis: From Number To Integral" by Mikusinski's. It starts with axioms for reals and takes you straight through to Lebesgue Integral in no time. Every section has very few exercises. Say, about 4. It also skips through the traditionalist Riemann integral. Because why not.

   Links:

   https://www.amazon.com/Think-About-Analysis-Lara-Alcock-eboo...

   https://www.amazon.com/Real-Analysis-Long-Form-Mathematics-T...

   https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

   https://link.springer.com/content/pdf/10.1007%2F978-3-030-33...

   https://www.amazon.com/Introduction-Analysis-Number-Integral...

   -------------------

   Now to a few examples of Indian authors' propensity to give you everything you need to know about the subject unabashedly. Check out:

   -- "Sequences and Infinite Series" by N.P. Bali. It's 100% worked out proofs after proofs after proofs. If you read through this book, I imagine you could easily crush any "sequences and series" portion of any elementary real analysis exam in existence. The author also has a book on real analysis which is structured very similarly to the aforementioned book. No idea what N.P stands for, though.

   Now to a book you've never heard of. "Data Structures and Algos" by Gav Pai. It easily blows Cormens, Sedgewicks and Skienas out the park by giving you, as you guessed it, all the 100% worked out examples you need to understand the topic.

   "Fundamentals of Complex Analysis by Bhat" is another one. All its problems are immediately followed by solutions. Look, it understands you're not at 40K a year school with 10 profs to help you out with any of your questions. Books like this are written for students, not to impress the profs.

   I just remembered some other Indian textbooks on abstract algebra, discrete math etc, but I cannot remember their names. They are also all about giving you all the examples you need to grok the concept at hand.

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