4 ms·

You might work yourself through the exercises in "baby Rudin" to learn analysis, and ask questions on Stackexchange, but it'll probably often be very frustratin

by sva_ 4y ago

You might work yourself through the exercises in "baby Rudin" to learn analysis, and ask questions on Stackexchange, but it'll probably often be very frustrating. I think there are so many great books on linear algebra, that it is difficult to single one out. I think LA is also picked up much easier by most people, especially with a programming background. It isn't really about the book imho, it's about being disciplined and do some practice exercises every day, like you're training a muscle. Edit: and I recommend you never consult solutions, except to verify yours. "Illusion of competence" is a big thing you might accidentally step into in mathematics. My experience is that looking up solutions gives you a short strong eureka feeling, but you learn almost nothing from it. You have to go and arrive there yourself.

moomin 4y ago
Came here to say Rudin. His presentation is “odd” in that he introduces powerful concepts and whacks large numbers of problems on the head with them, whereas the usual pedagogical approach is to teach techniques that correspond to these concepts but never explain the concept. I also really liked Herstein on algebra but I believe it’s out of print now.
- tzs 4y ago
  His "Topics in Algebra", "Abstract Algebra", or both?
  - moomin 4y ago
    I’m thinking Topics. Never read the other.
- MikeBVaughn 4y ago
  "Topics in Algebra" is fantastic! It also is in print, if you don't mind paying roughly $240 for a print-on-demand paperback. It's an obscene price, but it is available!
  - moomin 4y ago
    Grief, I picked it up in a bargain bin. Lovely book, goes so deep so clearly.
    - MikeBVaughn 4y ago
      I'm a big fan of Herstein's exposition. The text is plenty rigorous, without too getting bogged down in formalism, and he also manages to come across as personable and funny in a way that doesn't feel condescending. The people who update/glue stuff onto the 23rd edition of those 1200-page doorstop undergrad calculus texts would do well to revisit books of that era and learn the difference between "engaging" and "patronizing." That's not a dig at the books not being Spivak or Apostol - there's absolutely a place for that kind of undergrad textbook (though maybe not 1200 pages of it, updated every three years) - but in terms of attempts at being 'relevant' that transparently read as 'Hi, fellow kids!' I also dig Herstein's problem sets, there's a good breadth of difficulty, and they really elucidate key ideas, or point to other topics not covered in the text. Some of the problems are absolutely wicked, but in a good way. (IIRC, doesn't the introduction say that some are meant more to be attempted than actually solved?)
      - newsoul 4y ago
        I am searching for a good calculus book. Not those 1200 pages of boring stuff. Something reasonable that teaches in good depth and breadth.
        MikeBVaughn 4y ago
        There are lots of cool options, depending on what you want to do with it. If you're interested in rigorously proving how calculus works, starting from the structure of the real numbers, I hear nothing but fantastic things about Spivak. I've not worked much of it myself, but the exercises seem very well chosen from the sections I've read. It exists in an interesting space between a calculus textbook and an introduction to real analysis, and you'll see stuff like the intermediate value theorem proven rigorously. If you want to practice computational applications, you may have to supplement it with some other exercises and material, from what I've heard, though. Apostol also exists in the same space as Spivak, and was (possibly still is) the calculus textbook of choice at Caltech. For a more applied approach in line with the general content of the enormous undergrad omnitext, this seems cool: https://ocw.mit.edu/courses/res-18-001-calculus-online-textbook-spring-2005/pages/textbook/ https://ocw.mit.edu/courses/res-18-001-calculus-online-textb... Dover and Springer have some pretty neat slim volumes that occupy the 'less rigor/intended for applications' side, though the names and authors escape me right now. (Edit: Morris Kline was the Dover book I was thinking of, and I've heard good things about Serge Lang's introductory calculus book.) (Second Edit: Kline doesn't seem particularly slim at 900ish pages, but it is well-liked. Let's go with 'comparatively slim.' In my defense, most Dover books ARE comparatively concise.)
gituliar 4y ago
Absolutely agree. Exercises are the key for learning. Like with riding a bicycle -- you learn by trying not reading or watching. Composing an exercise section is hard, so thoughtful books usually have one. Also, I'd pick just one book for a subject and stick to it, pick it carefully to fit your style and language.
mikevm 4y ago
I couldn't disagree more. I studied Analysis in the Uni, and even in that environment Rudin is pretty bad. For a total newcomer that book will leave you completely helpless. Also, solutions are a must have, without them you are almsot totally lost. In their absence, it is OK to ask on StackExchange or #math on EFnet. First let's start with a few books to prep you for college-level maths: * https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/dp/B00B9CIORY https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/... * https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical-ebook-dp-B00I8XF6TQ/dp/B00I8XF6TQ https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat... ; or * https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Allenby/dp/0340676531 https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Al... ; or * https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-dp-1108439535/dp/1108439535 https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-... (I believe you can find solutions to the 2nd edition online) For Single-Variable Analysis * https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539 https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0... * https://www.amazon.com/How-Why-One-Variable-Calculus/dp/1119043387 https://www.amazon.com/How-Why-One-Variable-Calculus/dp/1119... * https://www.amazon.com/Mathematical-Analysis-Straightforward-Approach-2nd/dp/0521288827 https://www.amazon.com/Mathematical-Analysis-Straightforward... (contains solutions to exercises) * https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics-ebook/dp/B00XWDQUH4 https://www.amazon.com/Understanding-Analysis-Undergraduate-... (there are solutions online for the 2nd edition) * https://www.amazon.com/Numbers-Functions-Steps-into-Analysis/dp/0521788366 https://www.amazon.com/Numbers-Functions-Steps-into-Analysis... (this book is a brilliant exercise-guided approach that helps you build up your knowledge step by step + solutions are provided).
- Koshkin 4y ago
  Agree, Rudin is excellent as a second course on real analysis, but as a first one it is absolutely terrible.
  - bell-cot 4y ago
    +1, but nitpicking: There are situations where baby Rudin ("Principles of Mathematical Analysis") can be very good for a first course...but those are corner cases, that you'd want a really good, experienced instructor to sign off on.
d4rkp4ttern 4y ago
Besides practicing the given exercises (which is a must of course) I find it helpful to try to prove the Lemmas and theorems myself, or at least struggle with them for a while, so that (a) you understand the proof better and (b) the struggle plus reading helps concepts stick far better. In other words try to “own” the concepts as if you had developed them.
fn-mote 4y ago
I disagree with all of these recommendations about studying analysis. tl;dr: Study a less careful subject where you can more easily develop a useful intuition. Abstract algebra and linear algebra both fit in that category. (What follows is a caricature.) I tell people that analysis was born in the 19th century when Weierstrauss killed intutition as a tool for understanding calculus. Analysis might teach you what is "really true" but unless you are going to be some kind of professional I would HIGHLY recommend developing an intuition and understanding of things based on what should be true. It's a terrible thing to learn that what "should be" true is not, and for a long time that doesn't help you make progress.
- lupire 4y ago
  This is because analysis is built upon the fantasy axiom of "real" numbers (of which, according to the measure theory of real analysis, 100% are impossible to even have names, even in principle, let alone compute with), instead of the solid foundations of algebra.
- newsoul 4y ago
  Can you recommend some books?

by sva_ 4y ago

moomin 4y ago

Came here to say Rudin. His presentation is “odd” in that he introduces powerful concepts and whacks large numbers of problems on the head with them, whereas the usual pedagogical approach is to teach techniques that correspond to these concepts but never explain the concept. I also really liked Herstein on algebra but I believe it’s out of print now.

tzs 4y ago
His "Topics in Algebra", "Abstract Algebra", or both?
- moomin 4y ago
  I’m thinking Topics. Never read the other.
MikeBVaughn 4y ago
"Topics in Algebra" is fantastic! It also is in print, if you don't mind paying roughly $240 for a print-on-demand paperback. It's an obscene price, but it is available!
- moomin 4y ago
  Grief, I picked it up in a bargain bin. Lovely book, goes so deep so clearly.
  - MikeBVaughn 4y ago
    I'm a big fan of Herstein's exposition. The text is plenty rigorous, without too getting bogged down in formalism, and he also manages to come across as personable and funny in a way that doesn't feel condescending. The people who update/glue stuff onto the 23rd edition of those 1200-page doorstop undergrad calculus texts would do well to revisit books of that era and learn the difference between "engaging" and "patronizing." That's not a dig at the books not being Spivak or Apostol - there's absolutely a place for that kind of undergrad textbook (though maybe not 1200 pages of it, updated every three years) - but in terms of attempts at being 'relevant' that transparently read as 'Hi, fellow kids!' I also dig Herstein's problem sets, there's a good breadth of difficulty, and they really elucidate key ideas, or point to other topics not covered in the text. Some of the problems are absolutely wicked, but in a good way. (IIRC, doesn't the introduction say that some are meant more to be attempted than actually solved?)
    - newsoul 4y ago
      I am searching for a good calculus book. Not those 1200 pages of boring stuff. Something reasonable that teaches in good depth and breadth.
      - MikeBVaughn 4y ago
        There are lots of cool options, depending on what you want to do with it. If you're interested in rigorously proving how calculus works, starting from the structure of the real numbers, I hear nothing but fantastic things about Spivak. I've not worked much of it myself, but the exercises seem very well chosen from the sections I've read. It exists in an interesting space between a calculus textbook and an introduction to real analysis, and you'll see stuff like the intermediate value theorem proven rigorously. If you want to practice computational applications, you may have to supplement it with some other exercises and material, from what I've heard, though. Apostol also exists in the same space as Spivak, and was (possibly still is) the calculus textbook of choice at Caltech. For a more applied approach in line with the general content of the enormous undergrad omnitext, this seems cool: https://ocw.mit.edu/courses/res-18-001-calculus-online-textbook-spring-2005/pages/textbook/ https://ocw.mit.edu/courses/res-18-001-calculus-online-textb... Dover and Springer have some pretty neat slim volumes that occupy the 'less rigor/intended for applications' side, though the names and authors escape me right now. (Edit: Morris Kline was the Dover book I was thinking of, and I've heard good things about Serge Lang's introductory calculus book.) (Second Edit: Kline doesn't seem particularly slim at 900ish pages, but it is well-liked. Let's go with 'comparatively slim.' In my defense, most Dover books ARE comparatively concise.)

gituliar 4y ago

Absolutely agree. Exercises are the key for learning. Like with riding a bicycle -- you learn by trying not reading or watching. Composing an exercise section is hard, so thoughtful books usually have one. Also, I'd pick just one book for a subject and stick to it, pick it carefully to fit your style and language.

mikevm 4y ago

I couldn't disagree more. I studied Analysis in the Uni, and even in that environment Rudin is pretty bad. For a total newcomer that book will leave you completely helpless. Also, solutions are a must have, without them you are almsot totally lost. In their absence, it is OK to ask on StackExchange or #math on EFnet. First let's start with a few books to prep you for college-level maths: * https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/dp/B00B9CIORY https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/... * https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical-ebook-dp-B00I8XF6TQ/dp/B00I8XF6TQ https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat... ; or * https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Allenby/dp/0340676531 https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Al... ; or * https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-dp-1108439535/dp/1108439535 https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-... (I believe you can find solutions to the 2nd edition online) For Single-Variable Analysis * https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539 https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0... * https://www.amazon.com/How-Why-One-Variable-Calculus/dp/1119043387 https://www.amazon.com/How-Why-One-Variable-Calculus/dp/1119... * https://www.amazon.com/Mathematical-Analysis-Straightforward-Approach-2nd/dp/0521288827 https://www.amazon.com/Mathematical-Analysis-Straightforward... (contains solutions to exercises) * https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics-ebook/dp/B00XWDQUH4 https://www.amazon.com/Understanding-Analysis-Undergraduate-... (there are solutions online for the 2nd edition) * https://www.amazon.com/Numbers-Functions-Steps-into-Analysis/dp/0521788366 https://www.amazon.com/Numbers-Functions-Steps-into-Analysis... (this book is a brilliant exercise-guided approach that helps you build up your knowledge step by step + solutions are provided).

Koshkin 4y ago
Agree, Rudin is excellent as a second course on real analysis, but as a first one it is absolutely terrible.
- bell-cot 4y ago
  +1, but nitpicking: There are situations where baby Rudin ("Principles of Mathematical Analysis") can be very good for a first course...but those are corner cases, that you'd want a really good, experienced instructor to sign off on.

d4rkp4ttern 4y ago

Besides practicing the given exercises (which is a must of course) I find it helpful to try to prove the Lemmas and theorems myself, or at least struggle with them for a while, so that (a) you understand the proof better and (b) the struggle plus reading helps concepts stick far better. In other words try to “own” the concepts as if you had developed them.

fn-mote 4y ago

I disagree with all of these recommendations about studying analysis. tl;dr: Study a less careful subject where you can more easily develop a useful intuition. Abstract algebra and linear algebra both fit in that category. (What follows is a caricature.) I tell people that analysis was born in the 19th century when Weierstrauss killed intutition as a tool for understanding calculus. Analysis might teach you what is "really true" but unless you are going to be some kind of professional I would HIGHLY recommend developing an intuition and understanding of things based on what should be true. It's a terrible thing to learn that what "should be" true is not, and for a long time that doesn't help you make progress.

lupire 4y ago
This is because analysis is built upon the fantasy axiom of "real" numbers (of which, according to the measure theory of real analysis, 100% are impossible to even have names, even in principle, let alone compute with), instead of the solid foundations of algebra.
newsoul 4y ago
Can you recommend some books?