An Intuitive Guide to Linear Algebra

Seconded. I was very skeptical of LADR when I was forced to use it freshman year. I thought I already knew all the practical stuff from Strang's book, but learning linear algebra from the mathy angle has paid off again and again and again. Choices of units & coordinate systems are impossibly confusing without having a foundation in how to reason independently of such arbitrary choices and add them in later.

I think the turning point came when I realized that "abstract" linear algebra was actually much closer to physical reality than "applied" linear algebra. Superficially, it seems like the exact opposite state of affairs from computer engineering, where grids of numbers take you closer to "bare metal" and "higher level reasoning" takes you further away. In math, it's the grids of numbers that are further removed from "bare metal" and the abstract reasoning that you can count on when all else fails!

I didn't need to be convinced that I would eventually come to regret having an insecure foundation so much as I was mixed up about up vs down and trying to build a foundation in the middle of the sky.

Anyone want to study group this book? I did Strang a few years back, along with the OCW lectures, but it wasn't very proofs-centric.

I can't recommend this book more. I read through about half of it before I took my first real course on linear algebra. The result? I got literally 100% in that class -- I didn't miss a single point on anything the entire semester. The teacher was flabbergasted, but I had my secret weapon -- Axler. The rest of the class was relying on the course textbook, some awful fat thing that I'm pretty sure nobody could ever really learn from. Axler's book made the concepts so crystal clear to me that it felt as if I'd known them since birth.

I can't guarantee that everyone is going to have the same experience, but yeah, it's a good book.

In my experience, Linear Algebra Done Right by Axler is a great second book on linear algebra. But it makes sense to most students only if they've seen and done concrete matrix calculations before. Giving it to a mathematical novice is likely to lead to baffelment. Too abstract.

I think I'm in that camp.

I got a D in linear algebra, my lowest grade in university and one of my lowest grades ever, but excelled in computer graphics, and even got permission to take a graduate level course in it for elective credit as an undergrad.

My recollection is that the problem sets in the linear algebra class seemed really abstract and vague, and I couldn't reason through it very well because I couldn't visualize what we learned, given the information from the class.

Computer graphics, however, let me visualize what we were covered, and suddenly everything made sense. I could see, literally, what the end result was supposed to be, and it just clicked.

Now that I think about it, it's kind of a shame I had the linear algebra class first. The CG experience would have helped me out quite a bit.

What are your thoughts on those revisiting linear algebra to instead try their hand at learning geometrical algebra instead?

This is incorrect.

A Lie _group_ is a group which is also a manifold, such that the group operations are smooth. (I don't know what it means for a set to be smooth.)

A Lie _algebra_ is a vector space together with a bilinear operation (the Lie bracket, written [a, b]) which also satisfies the Jacobi identity:

  [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0.

How are they related? If G is a Lie group, the tangent space at the identity is a Lie algebra. What does the Lie algebra represent? Well, unlike a plain manifold, in a Lie group, given a tangent vector a, there is a path starting from a the identity in the Lie group always in the direction a. Given two tangent vectors a, b, you can flow a little in the direction a, and then a little in the direction b, or vice versa. The Lie bracket can be thought of as a measure of how the flow fails to commute.

Ah! So its just a group that has a defined group of continuous transformations that can be applied to it?

p-adic numbers are actually quite simple; so simple, I've actually seen children invent it on their own!! They're just like ordinary decimal notation numbers, except instead of allowing these to extend infinitely to the right (as in 3.14159...), you allow them to extend infinitely to the left. So, in addition to familiar numbers like 45 or 67.8, you also have numbers like ...95141.3 or ...999 or what have you.

You add and multiply these according to the same rules you're already familiar with. So, for example, if you add 1 to ...999, the last digit of the output is 0, and you get a carry. Making the second to last digit 0, with another carry. And so on and so on, making the result ...0000 over all. Thus, ...999 acts like -1.

(If we were working in base two, this last example would be just like the "two's complement" you are perhaps familiar with from computer arithmetic!)

In fact, most discussion of p-adic numbers isn't done in base ten. Instead, people typically focus on p-adics in a prime base (hence the p). Why? Because in a prime base, you will find that every nonzero p-adic number has a multiplicative inverse, which is very convenient (while in a composite base, you will find that sometimes nonzero numbers multiply to zero). But there's nothing actually stopping you from making use of the notion for non-prime base, should you be interested in doing so; the notion is still perfectly coherent. (That having been said, another reason mathematicians focus on p-adics in prime bases is that the Chinese Remainder Theorem essentially allows one to reduce the study of all other bases to this case.)

There's a lot of beautiful further theory to explore here, but again, the basic idea is hopefully quite simple. Let me know if you have any questions and I'll be happy to try explaining further or more clearly.

You might enjoy the r/elimath subreddit:

https://www.reddit.com/r/elimath/

Someone already asked about the p-adics:

https://www.reddit.com/r/elimath/comments/2rhavm/explain_pad...

p-adic numbers are measure of necesary- resolution to describe. A pair of p-adics is very close when you can use a very coarse ruler (of tick size p^n, large n) to measure the exact difference (1 vs 10000000001). p-adic numbers are far away when you need a very fine grid (1.0 vs 10.000000000001) to measure the exact difference.

I'd request the same for whatever mathematical concepts Quants use for stock market trading. Or at least a list of concepts useful for such an endeavor.

Obligatory Schemes and Syzygies

I think so – it covers the fundamentals. Numbers, Algebra, Functions. Not exactly in depth, but if you've done high school math you should quickly remember and be able to get to the fun stuff.

Yes, I can't recall where I first read this (someone's comment on HN most probably) but it was very enlightening.

For context, I don't think I ever knew the thing we were studying was called linear algebra, so it took me another 17 years to realize that is the common name for what we studied in that class. Now I'm wondering if maybe one of the intents of the small engineering-focused school's tightly integrated math-science-engineering curriculum was to avoid an introductory linear algebra class. We had to understand matrices and vector spaces earlier for other classes and didn't do what you call the "applications" of linear algebra until sophomore year.