A 2020 Vision of Linear Algebra • Hacker News Board

Re: A 2020 Vision of Linear Algebra

#201

Post by cashsterling » Thu, May 14, 2020, 5:25 AM UTC

I like all Strang's books... at least the ones I have. I don't have his Learning from Data book, yet... however.

I also really like the applied linear algebra book by Boyd Vandenberghe: https://web.stanford.edu/~boyd/vmls/ Free PDF is available on their website. There is Julia and Python code companions for the book and lecture slides from both Profs their websites. Also check out their other books, many of which have free PDF's available.

I can also recommend Data-Driven science and engineering by Brunton and Kutz. http://databookuw.com/ There used to be a free preprint PDF of the book but I can't find it now. Book is totally worth picking up... MATLAB and Python code available. Steve Brunton's lectures on YouTube are pretty damn good and compliment the book well: https://www.youtube.com/channel/UCm5mt-A4w61lknZ9lCsZtBw/fea...

Another really cool book is Algorithms for Optimization by Mykel Kochenderfer and Tim Wheeler: https://mitpress.mit.edu/books/algorithms-optimization. Julia code used in book.

Re: A 2020 Vision of Linear Algebra

#202

Post by Tainnor » Thu, May 14, 2020, 6:27 PM UTC

Koshkin wrote: post #124

madhadron wrote: post #121

Earlier quoted context omitted.

That depends on the part of linear algebra. In an abstract function space when you start calculating dimensions of kernels and the like and get ready to make the jump to infinite dimensions, Banach spaces, and Hilbert spaces, it's about as abstract as monads.

Well, to be fair, functional analysis is not part of linear algebra proper. (If you want to get more abstract, you go to rings and modules and from there to category theory.)

Typically, linear algebra is understood to be the study of finite-dimensional vector spaces, so functional analysis is not necessarily part of it.

However, things like the vector space of polynomials of degree at most n, the vector space of all homomorphisms between two vector spaces, the dual space of a vector space, etc. are all concepts that belong to linear algebra proper yet are more "abstract" than just "computations with matrices".

Re: A 2020 Vision of Linear Algebra

#203

Post by madhadron » Fri, May 15, 2020, 2:56 AM UTC

DagAgren wrote: post #197

madhadron wrote: post #122

Earlier quoted context omitted.

I've had good luck with explaining it as a characteristic of a programming language. In a language consisting of sequences of statements with bindings and function calls, we expect that f(x) is the same as a = x; f(a) and the same as g = f; g(x); That's the monad laws. Whatever craziness you want to put in the semantics, those are properties you probably would like to preserve in your language.

I think I understand monads less now.

Do you really understand them less, or has it dislodged ideas that you thought were true? Moving towards zero is not always decreasing.

Re: A 2020 Vision of Linear Algebra

#204

Post by apricot » Fri, May 15, 2020, 1:11 PM UTC

ZeikJT wrote: post #195

heymijo wrote: post #193

Earlier quoted context omitted.

Hi Sriram, a teaching system can incorporate both! Apologies if my original reply made it seem like it can't. Why don't teaching systems in America incorporate both the majority of the time? Two major reasons: 1. Cultural inertia. Most teachers emulate the pedagogy that they experienced in their schooling. Some are aware that you can try to mix conceptual+procedural and try to. I call them the "notch generation"- try…

#2 bothered me so much in school. The system gauges success via tests that check short term learning. It really, really isn't good at measuring learning.

I always did very well on tests at school, but I wasn't really learning anything, or more precisely, I wasn't learning how to learn. I was learning how to pass tests, but that's a rather useless skill to have. I had to learn learning as an adult, and it was more difficult than if I had to learn it as a child.

Re: A 2020 Vision of Linear Algebra

#205

Post by thr0w__4w4y » Sat, May 16, 2020, 11:26 PM UTC

romwell wrote: post #185

heymijo wrote: post #178

Earlier quoted context omitted.

Durable and flexible knowledge... After reading your comment and ansible's reply [0] I wanted to pause and comment on this. The United States Air Force Academy found that its cadets who took their first calculus class with a professor who focused on conceptual understanding helped those cadets create a durable and flexible understanding of the math [1]. The kicker is that the cadets got worse scores in Calculus I and…

When I taught Calculus, I taught understanding over memorizing steps to arrive at a solution for a particular type of problem. The very best students loved it, but most of the people didn't like it at all. With mathematics, like with gym, you gain when you put in effort. Most people don't enjoy either.

Yes indeed. Outside of work, I'm an endurance sports person, so basically performance is correlated strongly with training hard and suffering. There is a saying, "Pain is weakness leaving the body", I first heard it in high school (team went on to win a state championship in a highly competitive state). When I was suffering on workouts I just pictured myself getting stronger.

OK hopefully I didn't get too far afield. To me, the analogous concept in learning, particularly in technical fields, is that "learning is ignorance leaving the mind".

In college, particularly math and physics, I /always/ focused on understanding the underlying principles. Initially it was out of fear that if I forgot the formulas, I could re-derive them. But a strange thing happened... through that process, I developed an intuition and an ability to "see" what formulas and concepts to apply when. Once I got to that point in a problem, "seeing it for what it was", finishing to the solution became busywork.