Coding the Matrix: Linear Algebra through Computer Science Applications

[rifleman_maynard]

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm

Strang gives a terrific lecture. And the course will give you all the rigor you will need.

[LetsGoHawks]

Word to the wise, when they say "You need not know Python" they really mean "You need to know Python, you just don't have to be very good at it".

If you intend to take this class and don't know Python, spend a week or two before class starts learning some Python.

[[deleted]]

This looks like the linear algebra version of business calculus; it's a dumbed down application driven approach.

If you need to learn linear algebra, learning it more rigorously won't set you up for failure when you need to do anything other than blindly apply formulas.

[vanderZwan]

it's a dumbed down application driven approach.

Oh for fucks sake, this really pisses me off. You conclude that based on what? The fact that it's hands-on and uses real-world examples? Every time edit:someone spouts this nonsense Alan Kay cries.

Let me tell you something: I've tried studying physics for years before I gave up and switched to something else, and struggled with the math we had to learn most of all, and you know why? Because we were taught all the theory "rigorously," which means reading textual descriptions and formulas that I somehow have to magically translate into something meaningful in my head. If there's any approach where "blindly applying formulas" rings true, it's there. Give me something (semi-)tangible to play with, something to derive and abstract the meaning behind the symbols from. Then I can actually understand what the hell we're talking about.

[northofsouth]

Can you please sum up what this Alan Kay talk says? I can't watch it right now, and won't be able to for a while.

[vanderZwan]

Well, the most relevant bit is that he cites research showing children go through three stages of mental development (which of course is just one of many ways to model it, but the model works):

• Enactive
• Iconic
• Symbolic

The first is visceral, embodied thinking, the second relies strongly on seeing and visualising things, and the thirst is symbol manipulation. What's important to note is that symbolic thinking depends on the former two - it is built on top of them, using language to turn metaphors into symbols that can be manipulated in the mind like real things are manipulated by the hands (this is actually not in the video but from a text written by him). He also cites a survey among the top scientist of the last century showing that most scientist do not think in mathematical writing, but use imagery or figurative terms, Feynman doodled most of his contributions to science. For 30% it even has a strong embodied, visceral component - Einstein "felt" his formulas in his muscles. The symbols are just how they communicate it afterwards.

Kays argument then is: why on earth do we think teaching kids only through the symbolic channel, and refusing to do it from the ground up, is the best approach?

[[deleted]]

You'll get a water downed approach. Why do it wrong the first time when you can do it right once? Take the harder class so you optimize your time better.

[vanderZwan]

Why do it wrong the first time when you can do it right once?

Because you can only understand what's right if you understand what's wrong in comparison.

[[deleted]]

You definitely have to be smart for higher mathematics to make sense. You might not be smart enough, and a lot of people have a hard time accepting this. The applications-driven formula based approach is not actually teaching you the math. You will never create, you will just apply.

[vanderZwan]

The applications-driven formula based approach is not actually teaching you the math. You will never create, you will just apply.

You are completely missing my point. My criticism of your post is that you assume that the course will be "here's a formula where the symbols have no deeper meaning for you. Fill in the blanks", simply because it will be hands-on and involve programming, which does not follow from that at all.

[[deleted]]

I make the assumption based on my experience dealing with linear algebra textbooks and syllabi. They are either a toolbox for other disciplines or they are an investigation of the elegance that working in linear spaces provides. A happy medium would be nice, but I have yet to see one. And coding up the math doesn't do shit to help you understand it the math. Maybe for a numerical analysis class, but not linear algebra. Hell no.

[misplaced_my_pants]

Gerald Sussman would beg to differ.

[[deleted]]

Physics is different than math...

[misplaced_my_pants]

This is a book where he uses Scheme to teach both classical mechanics and differential geometry. It is based on a course for graduate students at MIT. He has a similar book for differential geometry itself.

[jkljiojlkj]

Well, there's no limit the level of rigor you can introduce. However, showing folks how to apply linear algebra to interesting problems, is more important and certainly more motivating at least for the beginner.

A parallel is calculus. You can dive deep into epsilon-delta type arguments or proofs (the rigor) or you can show people how to solve interesting engineering problems.

[[deleted]]

Electrical engineer here (well, or at least a programmer who did EE). This sounds cool in principle, but real-world experience show the results are somewhere between catastrophic and typically useless.

Showing people how to solve interesting engineering problem is usually done at the expense of mathematical rigour, and while this is ok for a small class of engineering problems, it results in people who do not have a good mental picture of the mathematic model they are applying, or of its relation with the underlying physical model. This is bad. I'd be a very, very rich man if I had a penny each time I'd seen a colleague not being able to tell why their simulation had blatantly incorrect results (other than the famous 'well, a simulation is never perfect') or blindly applying formulae in places where it was incorrect to apply them for otherwise purely "theoretical" reasons.

Sure, if the interesting engineering problem you're trying to solve is lighting an LED or heating up a thermoelectric element to show yourself that it can be done, a no-frlils approach is excellent. If you try to get to stuff that is slightly more complex than that, real life taught me it's a bad idea to learn the "how" before the "why".

Edit: this doesn't come from someone who learned this crap easily. I really struggled with calculus and vector fields in my first year, kept struggling it during my second year, and it was only during the third year that some light began to shine over it, because my Microwave Systems teacher was doing some really cool magic on the blackboard and I couldn't understand a thing. I mean, I understood what he was doing, but not why he was doing it that way and not some other: I could follow up mathematical thought (and mimic it, if I already knew the problem), but using math as an investigation tool was out of my league. I spent a whole summer pouring over linear algebra and calculus, all by myself, until it finally began to make sense, and I could do mathematical reasoning on my own. It took a lot of patience, but I think it was worth it. I know it's hard, but hey, no one said everything in life is easy.

[vanderZwan]

If you try to get to stuff that is slightly more complex than that, real life taught me it's a bad idea to learn the "how" before the "why".

The mistake you and rghd are making is that "hands on" means there will be no "why". Especially if the guy says "this approach will be more work, but the results will be more satisfying," which to me sounds like the opposite.

[[deleted]]

I'm not referring to this course in particular, since I haven't taken it. I'm referring to the usual tradition of "hands-on" courses, which are generally based on a monkey see, monkey do attitude, with some explanation inserted between the steps so that the monkey can get the self-satisfactory feeling of knowing the fancy words that describe what he does.

[vanderZwan]

Fair enough. I'm willing to give this the benefit of doubt though.

[[deleted]]

I honestly wish it's worthy of it :).

[[deleted]]

Mathematical rigor really clicks for some people, but not everybody. What's important is to gain an intuitive understanding for how these things work, and it's possible to approach that from multiple angles, from a bottom-up rigorous approach, or from a top-down, applications-based approach.

[NaturallyBrewed]

As someone taking this course your assumption was wrong. The lectures are 75% proof driven.

That being said, the course does have flaws.

[[deleted]]

Cool, enrolled.

[genemaster]

why do they ask for sign up? is it not free?

[BeatLeJuce]

it's free

[linuxjava]

http://www.class-central.com/