Study HN: Linear Algebra 1, fortnight 2
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.
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.
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".)
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.
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?
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.
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.
Yes, ROT13 (or maybe ROT47) might be enough to stymie lazy solution-seekers. Might do that here if an even-numbered exercise seems to warrant some discussion.
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 :)
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!
n avargl qrterr ebgngvba naq n ersyrpgvba nobhg, nf sne nf V pna gryy nal, yvar guebhtu gur bevtva frrzf gb tvir gur evtug erfhyg.
Are we sure this is the correct meaning for "A = -B", where A and B are matrices? Some things I've read seem to be saying that that should be equivalent to AB = identity matrix, rather than Av = -1(Bv), where v is some vector.
Also, WLOG (without loss of generality) the top left entry of C and D is either 0 or 1, because you can multiply either matrix by any number other than 0 without changing whether the equation is true.
Your WLOG is elegant, though, in that it gives a simpler starting point that's guaranteed to work if anything does. I'll have to remember that trick.
Prove me wrong, and I will be tripping over myself to upvote you ;)
Did you look at my solution (http://news.ycombinator.com/item?id=3129023)? There's no higher-level theory in it. Is it brute force? Not sure. At least it doesn't require solving any element-wise equations.
If you were talking about solving the general case (For any C, find D such that CD = -DC) I can see why you wanted more powerful tools.
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.
Keep going and I'll catch up someday!
(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.)
For whatever it's worth, PDF versions of the book are all over Scribd. I bought it (twice) but am actually working from a "stolen" PDF.
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!
Correction: I take the last sentence back. I have "Standard Strang" instead (it was cheaper).