HN Books @HNBooksMonth

Here's how I would do it, my 2 cents:

1) Find a good source of information --- typically, this is either very good lectures (like on youtube), a good textbook, or good lecture notes.

2) Do problems. There is a fairly large gap between those that just watch the lectures and those that have sat down and try to go through each and every step of the logic, and that's what everyone here (on HN) is pointing out when they similarly mention doing problems.

2b) Have solutions to those problems. I make this a separate point because it's important to spend quality time on a problem yourself before looking at the solutions. At the end of the day, if you read the problems and then the solution right away, that's much closer to reading the textbook itself instead of the more rigorous learning one goes through when trying things themselves.

If you were to ask me what textbooks or lectures I recommend, I think that's a more personal question than many here might guess. What topics are you most interested in? Are you really just solely interested in a solid background? How patient are you when doing problems?

Regardless, I'll give my two cents for textbooks anyway. In no particular order:

1) Griffiths E&M: https://www.amazon.com/Introduction-Electrodynamics-David-J-...

2) Axler, Linear Algebra Done Right: https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Ma...

Good luck!

Axler's book is fantastic. But sadly Springer altered the typesetting on the 3rd edition. A really classic and clear LaTeX layout got turned into something much less clear. This freaked me out. Look inside and compare:

* Second edition: https://www.amazon.com/dp/0387982582

* Third edition: https://www.amazon.com/dp/3319307657

I wonder whether widespread adoption of his book pushed editors to make it look flashier and watered down. The contents are the same though.

⬐ nilkn

This was one of my favorite books as a math undergraduate. I'm sad to see the highly legible and clear layout has been replaced by something so gratuitous and distracting.

For better or worse, the 3rd edition formatting and styling is something I've come to mentally associate with low-quality cash-grab big-lecture-hall tomes designed and written by committee over the course of a dozen editions. I wonder if I would have written off the book when I was a student if I'd seen it in such a form.

⬐ leephillips

I looked: you are so, so right. The 3rd edition, with all the ugly colors and drop shadows, looks like a middle school textbook. If I ever buy this book, and chances are I will, I'm going to pick up a copy of the 2d edition. Thanks for the warning!

⬐ colechristensen

On this topic, I love older textbooks that read closer to prose than whatever passed for flashy textbooks (at least 10 years ago, I can only imagine things have gotten worse)

⬐ CalChris

Along these lines, I prefer black and white matte rather than glossy color.

⬐ mathgenius

Wow this is tragic. I'm guessing it serves whatever market that Springer has identified. But I'm not sure that's such a good thing: at some point the more details you add to the exposition the less clear it becomes. The reader needs to stand on their own two feet, especially in mathematics. Some people seem to be good at memorizing endless rules and details, so I can see this serving those people. But those are the people that can just follow the "determinants path" that this book was originally meant to disavow. Sigh.

⬐ DennisP

Doesn't seem that bad, I just checked inside both and the content does look identical other than the visual style and some examples added in the 3rd.

I always have to mention my all-time favorite introductory book on this subject:

Liner Algebra Done Right (http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat...)

Its breakthrough is its focus on the basic algebraic properties of vector spaces and linear maps between them. It de-emphasizes matrix computations and especially determinants (they are covered, but only insomuch as they are necessary).

In my experience, the result of a typical linear algebra course is most students don't fully understand the determinant and more importantly they don't understand the proofs of major theorems which involve long manipulations of the determinant. They also don't understand the more algebraic side of the subject because they aren't given a chance to--it's not covered in much detail. The result is they don't understand the subject overall much at all.

This book is based on the observation that the abstract algebra involved in linear algebra is actually remarkably easy, much more so than arcane determinant manipulations.

⬐ rahimnathwani

I'm curious - did you work through this book on your own, or as part of a class or with help from others?

I started reading it, attempting most of the exercises at the end of each chapter, and some of the 'as you should verify' parts. I found the 'aha' moments during the text or when solving the exercises really enjoyable, but I got stuck often enough that I began to only pick up the book when I felt super-alert.

I stopped about a third of the way through, a few pages before the introduction of eigenvectors and eigenvalues. I would like to pick it up again, but am worried about how to maintain motivation the next time I'm faced with a page where I'm stuck/confused for 30 minutes.

EDIT: Thank you to the people who replied. Your empathy (~"you're not alone in getting stuck") and encouragement (~"you don't have to grok everything the first time through") have given me new motivation to try this again.

⬐ jjoonathan

I started reading it for a class but I wasn't assigned all the problems. Later, I managed to propel myself through it by building confidence with the problems of the first few chapters (which were easy by that point) and then embracing the sunk-cost fallacy to get through the rest.

One problem stood out. I forget the chapter, but I'll never forget the problem: prove that normed vector spaces are inner product spaces IFF the parallelogram law holds. It took about half a day to crack that problem and I was completely lost for about 90% of it. (That's not a strategy. The first paragraph was my strategy. My only point here is that you aren't alone in getting stuck.)

⬐ nilkn

I actually skimmed the book over before I took my first course in linear algebra (doing some of the exercises on my own, but certainly not all--I skipped anything that seemed too hard or just too boring during this reading). At that point, there were definitely parts of the book that were challenging to me, and I just jumped over them.

Then when I took the actual course, I more or less ignored the main course textbook and instead tried to tie whatever happened in the last lecture with what I found in Axler.

I considered Axler my secret weapon. It honestly felt like I had access to secret insights that trivialized the class and granted me the same intuition the professor had.

⬐ michaelbarton

I agree. I am also working through this book and feel the same as you. I think also part of the problem maybe not being able to discuss this with other people. I'm trying very hard to get through it but I think I miss being able to discuss problems with other students in a traditional classroom setting. These sorts of discusses can really help clarify things rather than just staring at the page for a long time.

⬐ ptr

A side note: how do you feel when you've lost your motivation? Is it common to barely be able to keep your eyes open? Or is that just me!

⬐ jjoonathan

Not just you. I'm procrastinating on one of those too-damn-hard math problems right now!

⬐ rahimnathwani

It's not just you. It's probably easier to accept that I'm sleepy and should take a rest, than to acknowledge that I need to focus and think harder about the material.

⬐ j2kun

Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can.

I suggest you don't worry too much about verifying every claim and doing every exercise before moving on. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. It's healthy, if you get stuck on such a problem, to think about other problems or come back to them later. It's not uncommon to find that by the time you revisit them you've literally grown so much (mathematically) that they're trivial.

⬐ vsbuffalo

> Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can.

This needs to be emphasized more in learning maths. Once I got over this fact and was guided by my interest rather than wanting to grok everything, I actually started understanding a lot more.

⬐ epaladin

Someone should tell us that sooner then. I've been trying to get over math anxiety for a long time, and having professors that say "it's simple, what don't you understand?" when you try to get help didn't help much.

⬐ j2kun

A common joke among mathematicians.

Two mathematicians are working on a problem and one makes a claim to which the other replies, "That's trivial!" The pair stare at the claim for ten minutes. The other says, "Is it trivial?" They go home, think about it all night, publish three papers about it over the next two years, and then they agree, "Yes, it's trivial."

⬐ akater

Oh yes, +1. Linear algebra is about vector spaces; matrices are a separate topic.

Unfortunately, Axler does not define determinant as a volume scaling coefficient and goes the still arcane way of “(-1)^\deg…” which does not really do justice to the concept.

Check out Sergei Winitzki's “Linear algebra via Exterior Products” https://sites.google.com/site/winitzki/linalg It's free for download and treats determinants in a more geometrically inclined way. BTW, it's written by a physicist, not a mathematician.

⬐ j2kun

No, he defines the determinant as the product of the eigenvalues, and this is almost the entire point for his writing the book!

⬐ j2kun

This book treats linear algebra like any other mature mathematical subject. It just so happens that it's easier to understand than most mature mathematical subjects. But yeah this is just how any pure mathematician would approach the subject (which is why I liked it so much).

⬐ shriphani

That particular series contains some of the best math texts I've read. My instructor used the Abbot text ( http://www.amazon.com/Understanding-Analysis-Undergraduate-T... ) for our real analysis course. The books are very aptly priced and are extremely well written.

⬐ billspreston

What is the equivalent book for other topics in math (I'm trying to self study) e.g. Calculus, Abstract Algebra, Probability, Statistics?

⬐ abhgh

Any idea how this compares to Hoffman-Kunze? I liked Hoffman-Kunze because, despite being dense, its no-nonsense and very rigorous.

⬐ enupten

I do so agree. Axler's book is also by far the most Geometric book on Linear algebra that I've ever read.

⬐ jjoonathan

This, a thousand times this.

My first introduction to Linear Algebra came from Strang's OpenCourseWare videos and, a year later, a class that used his book. Despite having gone through the material twice, I struggled to apply it to actual problems. Connecting the real world (err, models of the real world) to the blocks of numbers was the sticking point. What are the matrix's units? How do I write down a matrix that does what I want? I looked things up and prayed that whatever black magic the mathematicians on wikipedia had invoked in their derivations would match the implicit assumptions I knew I had to be making.

Then I was forced to re-take linear algebra in college. We used Axler's book and suddenly everything made sense. The blocks of numbers that you type into matlab are simply shadows of the mathematical concepts of a vector and linear transformation. The abstract mathematical concepts are actually closer to the real world concepts they model because they don't have to depend on arbitrary choices of coordinate system, etc. Once you understand how to model real-world quantities with abstract vectors and linear maps (easy) and how to translate those coordinate-free equations into coordinate-dependant matrix equations (easy), THEN you understand the material enough to apply it without fumbling around trying to paste other peoples' formulas together (hard).

OP's book looks well written but it entirely follows the tradition of emphasizing the blocks of numbers over the abstract concepts. Linear maps don't appear until page 75, while blocks of numbers enter on page 5! That's nuts! Use Axler.

This is really basic. Noone(who uses the subject) should need a cheat sheet for this... Also there are really good books out there and linalg is a must nowdays. As for textbook: http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat... As for reference: http://www.amazon.com/Matrix-Analysis-Roger-Horn/dp/05213863...

This is my favorite introductory textbook on linear algebra which goes to great lengths to avoid matrix and determinant computations:

http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat...

⬐ nashequilibrium

Thanks, i have been doing a lot of machine learning hacks lately and almost everything involves "vector spaces", 249 pages should be fun, just finishing up "Think Bayes" by Alan Downey.

⬐ 11001

Axler is not a text in applied linear algebra, it is a book you should read after you are already familiar with both basic abstract algebra and basic linear algebra (i.e. you know how to multiply matrices etc.), it then provides a nice motivation for linear algebra as a subfield of abstract algebra. You may still, however, not find the ideas of determinant, trace and other practical concepts very intuitive after reading Axler.

⬐ j2kun

Basic abstract algebra? Definitely not needed to use that book to its full potential.

⬐ clebio

I read Axler's book quite recently, after pouring through reviews of linear algebra books. I did like it quite a lot -- it has a very clean, approachable narrative style. Having completed it, though, I feel there's maybe not enough operational knowledge gained from it. Meaning, it explains the theory, the motivations, well, but one might augment it with Schaum's[1], or some such guided exercises.

[1]:http://www.amazon.com/Schaums-Outline-Algebra-Edition-Outlin...

I'm in!

By the way, some less applied materials include:

- http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/...

- http://www.math.mcgill.ca/goren/AlgebraII07-08/CourseNotesMa...

I've read and would recommend Linear Algebra Done Right by Sheldon Axler. http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/...

For numerical aspects of linear algebra (and other subjects), I've found Numerical Recipes to be quite helpful.

⬐ phren0logy

I've heard good things about this one, as well as _Linear Algebra for Everyone_. http://amzn.com/8847018382

From the book's intro:

>But, suppose we asked a professional mathematician to step back a bit from his habitual way of speaking and write in a more linear fashion? And suppose we even asked more, for example, that he make his writing lively? ... The purpose of this book is to furnish the reader with the first mathemat- ical tools needed to understand one of the pillars of modern mathematics, i.e. linear algebra. The text has been written by a mathematician who has tried to step out of his usual character in order to speak to a larger public. He has also taken up the challenge of trying to make accessible to everyone the first ideas and the first techniques of a body of knowledge that is fundamental to all of science and technology.

⬐ Silhouette

While Numerical Recipes has certainly earned its place in the history of the genre, it is only fair to point out that it is very out-dated by now, and that the example code is neither particularly well-written (from a software development point of view) nor freely usable in your own projects.

As others have suggested, something like Golub and Van Loan is a better choice. The technical notes to go with the LAPACK library, and similar documents from those developing related software, are also likely to be of interest to anyone doing this stuff seriously.

While not explicitly mentioned in the introduction I think it's safe to say that the books title is a play off of the popular title "Linear algebra done right"

http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/...

Which is a pretty amazing text if you're delving in to the algebra side of linear algebra. Though I suspect significantly less useful than "Linear algebra done wrong".

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/...