5 ms·
Study HN: Linear Algebra 1, fortnight 2
A bunch of us are learning Linear Algebra together (previously at http://news.ycombinator.com/item?id=3060521) by working through one of Strang's textbooks. Here's a new post to kick off the next two weeks. Will two weeks be enough to finish Chapter 1?
Most of us are squeezing this into limited spare time - a packing problem in its own right - and several are working through all the exercises, so our pace has proven slower than a typical course would go. At this point, though, we should at least declare 1.2 and 1.3 done. Less than that is just embarrassing. If you made it past 1.4 long ago, good for you; please post some comments to goad the rest of us forward. If you're not on 1.4 yet, get your ass in gear! Here are a couple observations to entice you.
1.4 is about matrix multiplication, which would seem to be about as mechanical and shallow a topic as you could get. I was surprised at how deep Strang was able to make it. (But be aware that we're not reading Standard Strang, we're reading Hippie Strang. That's ISBN 0030105676.) I thought that mastering the element-wise way of multiplying two matrices was all you needed to have it down. But Strang doesn't want you to think of it that way; he even impedes that view. Instead, 1.4 takes the earlier theme of the "row picture" and "column picture" and develops it further by teaching the matrix product AB as a combination of the columns of A or (alternatively) the rows of B. This is the book's bias in general, to de-emphasize formal representations and focus on the meaning of the constructs instead.
The other point concerns the exercises for 1.4. I put them off for a while because I thought they would be rote. They're not. They're designed to trigger insights that don't come automatically from reading the text, and turn out to be surprisingly satisfying. I found myself alternating between "aha!" and "doh!" when completing them. Yeah, that says something about the obtuseness of the student, but more about the skill of the teacher. Clearly a lot of love was put into crafting those exercises. So so far, I'd say we lucked out in picking this book.
p.s. If this sounds like fun, you should join us. All you have to do is learn. And it would still be pretty easy to catch up.
- tptacek 15y agoSo as a data point: I found 1.2 remarkably painful. Here's why: my math sucks, and I didn't have basic algebra internalized (I flunked algebra in high school and so while I can reason my way through a system of equations, I couldn't do it instantly). On top of that, I was way overthinking the problems. The sense I got of the 1.2 problems is that they're largely refresher stuff; that is, when given a system of 2 equations in 2 variables, they just want you to go ahead and solve it, not graph it in row and column view. I grabbed an algebra cheat sheet and breezed through it once I realized that. 1.3 went much faster for me; it was clear what the "new" material was, and Gaussian elimination is straightforward. This was material I really just wanted to go ahead and code up in Ruby (I wound up using a spreadsheet instead). Incidentally, I've found spreadsheets extremely useful for these early exercises. I'm hoping to be done with 1.4 tomorrow night. By far the biggest issue I'm having is finding time to spend on this, so I'm really only a couple (literally) hours into this so far.
- gruseom 15y agoWhat exercises in 1.4 have stood out for you? Here are some that evoked insight for me (or made me go "doh"): 3, 4, 5, 8, 10, 12, 13. I'll do some more tonight. (#5 was particularly "doh".)
- jpallen 15y agoI just had a look at the first thread and this looks like an excellent project. I won't be following along since I did maths as an undergrad and am now doing a phd, but I'd be happy to lend an experienced hand if anyone is having trouble. I'm not sure what is the best format to offer my help, so please reply if I can be of assistance in some way. I've done some tutoring of similar subjects before.
- gruseom 15y agoCool. Want to be our TA? :) I think the best format to offer help would just be to post comments here. Explanations of subtleties, additional insights, suggestions for further thinking, reminiscences of your own encounters with the material, or mentions of how this stuff comes in handy in other contexts would all be of interest. And it would help the discussion flow more. One lesson from the first thread might be that we were holding back a bit too much.
- deleted 15y ago[deleted]
- bdr 15y agoSorry, kind of OT: I'm working through Arora & Barak's "Computation Complexity" textbook and would love some company. Get in touch if you're interested.
- gruseom 15y agoHope you find someone. There should be a startup for this!
- aamar 15y agoAfter doing nearly all the exercises through 1.3 I did approximately 1/4 (randomly selected) through 1.5. I agree that the exercises generate good insights not in the text, but 1/4 might still be a good amount for me to do, given time constraints. I just noticed that odd exercises helpfully have solutions in the back. So moving forward, I'm planning to do all exercises marked recommended/important, all numbered 1 mod 4, and a few others that seem helpful. For the exercises which don't have solutions, it would be nice to have a place where we can compare answers yet not post them to the public internet. Would anyone else find that useful?
- tptacek 15y agoHoly I didn't realize the odd answers were in the back. Best comment ever. Thanks! The gap between what I think I understand after reading the text and what I find I really understand when I hit problem 1-2 in each section is immense, so I'm going to stick with the problem sets. You can just ROT13 the answers you don't want to make public.
- gruseom 15y agoWhat the hell is wrong with you two? I thought the first thing every student did was check if there were answers in the back. Just think of how much harder you made college for yourself. Edit: to be serious, the solutions at the back (which I look forward to flipping to after every other exercise, kind of like a rat getting a food pellet in an experiment) are interesting in their own right. He plants little clues in there, to hint at what was important or why he asked the question.
- warmfuzzykitten 15y agoI'm full up with taking the AI and ML courses from Stanford. Both courses provide background material in linear algebra, so I guess I'll make do with that for awhile. Two notes, though: The book is rather expensive (over $100USD) and, notwithstanding the glowing comments here, has some pretty scathing negative comments on Amazon. The question that comes to my mind is: If a book is going to reach an internet audience, is it still appropriate to charge classroom prices?
- gruseom 15y agoNo argument on the price; it amounts to assault. But the reviews on Amazon are interesting. Most love it, some hate it. The first thing I noticed is that the reviewers who love the book are the ones who take pleasure in the material. There are two kinds of negative reviewer: students who found it confusing (I suspect because it didn't hold their hand in the usual way - its focus is on developing intuition, something no one who just-wants-to-get-this-over-with could care less about) and mathematically more sophisticated readers who don't like its informality and gushiness. In other words, the negative reviews pretty much sum up what's good about the book. Of course, we're barely into it. I'm assuming the rest is like the first. Here's a litmus test for whether you would like it. This is how Strang introduces Gaussian elimination: The problem is to find the unknown values of u, v, and w, and we shall apply Gaussian elimination. (Gauss is recognized as the greatest of all mathematicians, but certainly not because of this invention, which probably took him ten minutes. Ironically, it is the most frequently used of all the ideas that bear his name.) The method starts by subtracting multiples of the first equation from the other equations. The goal is to eliminate u from the last two equations. That colloquial aside is irrelevant and would drive readers who are gritting their teeth to begin with crazy. I love it. It reminds me of my favorite old math prof to whom Euler was a close personal friend. If you find such a style delightful, this is the book for you. Strang's goal in this book seems to be to get you as excited about the subject as he is, and share how he really thinks about it, convention be damned. That's why it's Hippie Strang!
- julsonl 15y agoBased from your excerpt, that's basically the type of book I would want to read. Reading a math book without having any background with what the heck I was grinding numbers for would just bore me to tears. I might have an ebook on that lying somewhere...
- untitledwiz 15y agoThis is a great idea! I'm so happy people are getting together in virtual classrooms! I would join but I'm currently doing linear algebra in college. Hopefully, when I graduate I can join/start something similar on another topic :)
- roversoccer18 15y agoI took linear algebra a couple semesters ago in college, and have some matlab/octave scripts that solve linear systems using gaussian elimination, guass-jordan, etc. If you want I can email them to you guys if it would help you guys.
- gruseom 15y agoConsider 1.4.13c: Find 2x2 matrices such that CD = -DC, not allowing the case CD = 0. Of course one can just take a shotgun approach, i.e. come up with some arbitrary possibilities and then manipulate their elements individually until the equation holds. But what's more interesting is to view the matrices as operations on columns or rows, come up with operations that have the desired effect, then encode them as matrices to answer the question. What such ways are there to look at this little problem?
- nialo 15y agoOne way is to view this as a geometry problem, 2d for convenience: a column represents a point in the plane, and matrices are transformations that can be applied to that point. Then the problem becomes, find a pair of transformations such that the point ends up in two opposite locations if the transformations are applied in different order. a solution, under rot13: n ersyrpgvba naq ebgngvba jbexf, V guvax ol naq nobhg nal natyr gung vfa'g avargl qrterrf be nal nkvf gung vfa'g gur k be l nkvf. I'd be interested in an explanation for exactly why the exceptions to my solution don't work, I don't really understand why they shouldn't. I read this this morning and haven't been able to get the problem out of my head since, glad this thread exists :)
- gruseom 15y agoThat's exactly the sort of interpretation I was looking for. I don't remember how to represent geometrical transformations as matrices, but that's really just a technical detail; one can understand your solution without it. That being said, I still don't understand your solution. Are you saying that any such pair of transformations as you describe has the desired anticommutative property, except for the exceptions? I don't see either part of that - the general claim or the exceptions. My solution exploits the fact that in CD, C combines the rows of D, while in DC, C combines the columns of D. Here's the rest: Yrg P or gur vqragvgl zngevk rkprcg jvgu -1 nf vgf obggbz-evtug ryrzrag. Gura PQ vf Q jvgu vgf obggbz ebj artngrq. Fvzvyneyl, QP vf Q jvgu vgf evtug pbyhza artngrq naq -QP vf Q jvgu vgf yrsg pbyhza artngrq. Fb gb trg PQ = -QP, jr whfg arrq gb svaq n Q fhpu gung artngvat vgf obggbz ebj unf gur fnzr rssrpg nf artngvat vgf yrsg pbyhza. Sbe 2k2 zngevprf, guvf vf nal Q jvgu mreb va gur gbc-yrsg naq obggbz-evtug cbfvgvbaf. Rot13 don't work so good for specifying matrices!
- gruseom 15y agoIt's been a few days. Anybody care to report where you are? I'm happy to say I'm still doing all the exercises, not so happy to say I've only just finished the ones for 1.4. I thought exercises 29, 52, and 54 were interesting, among others. Finding time continues to be an issue, but actually working through the material is fine. Many of the exercises are rather fun, as they mostly have some non-mechanical twist. Occasionally (as in 1.4.23) Strang's informality goes a little too far and I have no idea what he's asking.
- tptacek 15y agoI'm just starting 1.4. I agree with you about the informality (and also general terseness) but it's something I assume I'm going to acclimate myself to.
- mechanical_fish 15y agoTrying to catch up; just starting the exercises to 1.4. Keep going and I'll catch up someday!
- gruseom 15y agoI'm going to be offline for a few days this week, and might not be able to work on exercises. Will post when back. Keep going you guys!
- gruseom 15y agoI'm back and will get back into the exercises and post as soon as I can. What's others' status?
- gruseom 15y agoI have found 1.5 harder to follow, but pretty interesting once you figure it out. The heart of it is the proof (if you can call it that!) of A = LU on pp. 34-35. It is quite beautiful - just one simple idea. But I have a question. From p. 35: Both sides of (7) end up equal to the same matrix U, and the steps to get there are all reversible. What is the importance of "and the steps to get there are all reversible"? I don't see how the argument uses this fact. (On a practical note, I suppose a new thread is in order as we can no longer post top-level comments on this one. If no one else creates one, I'll do so after I've worked through the exercises in 1.5.)