

Study HN: Linear Algebra 1. We're on. - gruseom

A couple of weeks ago, a bunch of us decided to put our internet time to better use and (re-)learn linear algebra together (http://news.ycombinator.com/item?id=2993321). Most have the book by now, so... let's do this! If you missed that thread and want to join, please do so. There's nothing you have to do other than learn.<p>The book is "Linear Algebra And Its Applications" by Strang, 4th ed., ISBN 0030105676. We'll work through one chapter every two weeks.  There are a <i>lot</i> of exercises. Let's discuss ones that are interesting, nontrivial or both. Questions and insights about the material are welcome. The first 5 chapters have Review Exercises at the end. Let's call those "mandatory" and discuss them at the end of the two weeks.<p>Housekeeping:<p>1. We'll do this on HN unless it becomes intractable or the landlord asks us to leave.<p>2. Put "Study HN: Linear Algebra 1" in the subject line. Then we can find posts like this: http://www.hnsearch.com/search#request/all&#38;q=%22linear+algebra+1%22<p>3. If you want to be emailed when something relevant happens, email me at the address in my HN profile and I'll put you on the list.<p>Finally, not all of us have the above book. Some are using an earlier edition or a different text altogether. Others have the 4th edition, but in a cheaper Indian format that is sold as identical, but - warning! - is not. Presumably to thwart students from getting away with using the cheaper edition for courses, the publisher has tastelessly scrambled the order of the exercises. How, then, are we going refer to exercises? Let's experiment to see what works. In the meantime:<p>4. To avoid confusion, only reference exercises by number (e.g. "Ex. 10 of 1.2") when they are from the official book (the 4th edition, expensive non-Indian format). If you don't have that text, and want to play along, email me and we'll figure something out.<p>So, HN: what's interesting in the early sections of Chapter 1? I have a few things, but let's hear from others. How about we stick to 1.1, 1.2, and 1.3 at first - up to Gaussian elimination, but not yet matrices.
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tptacek
Just for what it's worth: I'm playing catchup on 1.1-1.3 tomorrow and Tuesday,
and I'm sure others haven't started yet too, so feel free to keep posting here
and we'll try to keep paying attention to the thread.

~~~
tptacek
(it took me an hour to get through the first 9 in 1.2. I'll get there! but
console yourself with the thought that you, whoever you are, are far from the
worst at this in the group.)

~~~
gruseom
Whether or not that's true, and I doubt that it is, we all know who is hands
down the wittiest.

I'm stuck on 1.3.21. What the hell is "the 1,2,1 pattern or the -1,2,-1
pattern"? Does he mean a pattern in the columns? I see 1,2,1 but not -1,2,-1.
Is this like one of those duck-rabbit optical illusions?

Edit: there is, however, a -1,2,-1 pattern in 1.3.24, and the system there is
very similar to the one in 1.3.21.

~~~
tptacek
Yep: mechanical_fish. :P

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fuzzythinker
For me, I don't think we should do the calculations by hand just for the sake
of doing it. It should be done only as a means for understanding the core
concepts. That is I much prefer to do exercises where the understanding of
core concepts is brought to the attention.

So the question is "What are the core concepts up to 1.3"? I'm sure singular
vs. non-singular is one. Are pivots a core concept? I think open-ended
questions in the 2nd half of 1.3 questions are good candidates for deeper
understanding.

~~~
gruseom
_Are pivots a core concept?_

Good question. They don't feel that way to me. They seem more like the
implementation detail of an algorithm (Gaussian elimination). Is the algorithm
itself a core concept? Hard to say no to that, I guess, but I wonder.

What about what Strang calls the "row picture" vs. the "column picture"? There
you have two fundamentally different geometric interpretations of linear
equations. That was a revelation to me. When I studied linear algebra a long
time ago, I don't remember encountering the column interpretation. Perhaps I
did and didn't pay attention; I was more interested in symbolic manipulation
at the time, whereas now I want a sense of what the symbols mean
geometrically.

~~~
mechanical_fish
Amusingly, I'm in the entirely opposite boat. I managed to get an entire Ph.D.
in solid state physics without taking an actual course in linear algebra (a
procedure that I _do not_ recommend, but everyone is young and foolish once).
So I've got some idea of what vector spaces are about geometrically --
otherwise, graduate quantum mechanics would have been impossible to navigate
-- but I'm really bad at manipulating the actual symbols.

And, incidentally, I'm planning to attempt most of the problems. I've cherry-
picked my way through this subject before, and I was ultimately displeased
with the results. They say that the problem with an autodidact is that he has
a fool for a teacher, and with linear algebra I learned that the hard way.
Fortunately, now they've invented Open Courseware, Amazon.com, and HN, so I'll
give the fool another chance.

~~~
gruseom
I propose that we explicitly mention any exercises that we stumble over. Yeah
it may make us look like idiots but we're destined to look like idiots anyway,
and there's a major upside: we're going to stumble over different things,
which means we can provide good explanations to one another, and learn better.

I have a few examples of this sitting at home but am hacking at the library at
the moment. It is heavenly quiet here. I will post them later.

~~~
mechanical_fish
So far I'm stumbling over the part where I actually sit down and read the book
and do the problems. ;)

But my book only arrived last week, so I'm not feeling too guilty yet.

~~~
gruseom
One function of this group is to turn guilt into shame. :)

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gruseom
How's everybody doing? I'm barely into 1.4, so I'm planning to work through a
fair bit this weekend in order to be done the chapter a week from now. Should
we discuss specific solutions from 1.2-1.3? Did anybody do the exercise marked
"very optional" (1.3.29)? Do we need to discuss ass-kicking tactics to keep
ourselves on track?

~~~
mechanical_fish
I'm through the first 16 exercises of the 1.3 problems. Probably won't find
time for the rest for a couple of days.

We definitely want to make sure that everyone gets the "Recommended" problem
from 1.3. It's a very important concept.

~~~
gruseom
Do you mean 1.3.18, which asks why it is impossible for a system of linear
equations to have exactly two solutions? This is one where people would
probably come up with a variety of different explanations.

Algebraically, if X and Y are solutions, then aX + bY is a solution if a + b =
1.

Geometrically, if two planes intersect at a line, and you try to add a third
plane that shares two points with them, you're forced to "capture" the entire
line.

Comments and/or additional ways of looking at this?

~~~
mechanical_fish
Yes, that's the one I mean, and those explanations pretty much capture it.

I might phrase the first explanation more simply as: If X and Y are solutions,
then the point halfway between X and Y (i.e. X/2 +Y/2) is a solution. Then you
can deploy Zeno's "paradox" to conclude that there is in fact an entire line
of solutions. ;) But, of course, your statement is the more general one.

This is important because it captures the vital quality of linear systems, the
quality that makes linear algebra worth studying in the first place: Once
you've got more than one solution to a linear system, any weighted sum of
those solutions is also a solution. This leads to the exciting possibility
that you don't need to carefully seek out every single solution (of which
there is often an infinite number); you just need to find a handful of
(hopefully simple) solutions that are sufficient to produce any of the other
solutions by constructing weighted sums. This is pretty much the fundamental
strategy of physics: Fourier analysis is the classic example of that strategy
at work, but it's also the basis of quantum mechanics.

~~~
gruseom
Fabulous comment. Makes me think you're sandbagging us with this "I never
learned linear algebra" schtick :)

I was musing about this off and on yesterday and another way of looking at it
struck me. Linear systems are flexible in some (well, few) ways and rigid in
others. The reason you can't build a model that just captures two points and
then absconds without getting stuck with infinitely more is that, being
linear, this stuff can't "bend". Another way of putting it (the sort of thing
someone who has spent too much time around computing and not enough around
geometry would say) is that the language of linear systems is expressive
enough to express certain things but not others. The possible interactions are
regimented. You can add dimensions to gain degrees of freedom, or equations to
collapse them, but both "freedom" and "restriction" here have a pretty narrow
meaning. I'm looking forward to learning about some of the magic things this
theory can be made to do. It makes sense that something with such a regular,
yet complex structure (complex by our standards anyway - mathematicians would
no doubt chuckle) could yield some beautiful results, but that it found such a
variety of real applications is surprising to me. More surprising in this
respect than its older cousin, calculus, which starts off much closer to our
naive sense of how reality works.

Now for those matrix exercises.

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ColinWright
Clickable:

[http://www.hnsearch.com/search#request/all&q=%22linear+a...](http://www.hnsearch.com/search#request/all&q=%22linear+algebra+1%22)

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gruseom
Has anyone tried the "very optional" exercise (1.3.29)?

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aamar
Yes, it was somewhat interesting though it does seem to me basically off-
topic.

My answer is here, if it's helpful: <http://pastebin.com/aPB1v2j9>

~~~
gruseom
Good for you! I haven't decided yet if I want to do that one. It seemed
tangential to me too.

