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An Intuitive Guide to Linear Algebra (2012)

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Re: An Intuitive Guide to Linear Algebra (2012)

#71

Post by photon_lines »

adamcharnock wrote: post #36

I... I still really struggle with this. I'm a smart person, I've got a bachelors of engineering, I've been a professional software developer for around 14 years now, and I've built a house. But there is something about degree-level maths and beyond that I find deeply unintuitive in a way that software development isn't. Through comments here I found 3blue1brown's (clearly much loved) videos. By the third video I was…

That's because the way that math is taught and presented makes it extremely hard to grasp and relate to, and this makes it hard for others to find the topics within it interesting. Mathematics, in it's very essence, is incredibly beautiful. The universe - yes, everything in it and everything that composes you, is in fact a form of mathematics! What does this mean?

Well, it means that understanding it means that you understand yourself! Well, not just that, but it might eventually lead to you understanding the architecture of the universe! After that? How about God? Or is God maybe a part of the architecture?

OK, so maybe I won’t try to go into the details there since opinions on it differ. Essentially, through this language though, you can master almost anything that you want! Yes, it might seem crazy, but the things that happen around you and the things which every poet and playwright and prisoner and savior ever composed can be explained through mathematics! Through it, you can also open up entirely new and utterly interesting worlds!

Let’s say that you dropped your pen this morning. To someone with no background in physics, or mathematics, this means nothing. On the other hand, to someone like me, it means quite a bit. The slight delay and movement altered the gravitational and electromagnetic field around you and echoed on into eternity. Also, it most likely changed your days structure and composition, and shifted your life into a new line (see quantum mechanics and chaos and linear dynamics) and had a profoundly large impact on everything else around you. You may have avoided a car crash, or you might have met a person who you wouldn’t have normally encountered all because of that small change. And this small change echoed on and effected everything else in turn. And this is all very mathematical, and extremely beautiful, but most people don’t know anything about it, but it does mean something to people like me: we’re all incredibly inter-connected, and our lives are ruled by chaos. Everything that you do, and everything that you say, and all the things you see and inter-related and ohh so close, but we tend not to see it and it all has to do with not understanding the fundamental mathematics!

The above is only touching on one small aspect of it though, as it only deals with physics and chaos. There are entire branches of math which make the world incredibly interesting which have nothing to do with the fields I just mentioned! Hey, did you know that standing next to someone who might look slightly like your wife will cause you to behave in a similar manner in which you behave when she’s actually around you? Yes, neurons that fire together wire together, and it’s very mathematical underneath but yet so simple! How about the fact that E = mc squared isn’t really true? Yes, the formula has an extra term (square root of 1 minus v squared over c squared) which makes it possible for massless particles to have energy and deals with relativistic effects. Did you know that you can summarize most of modern classical physics in just a few equations? (Yup, you can find most of them here: https://www.feynmanlectures.caltech.edu/II_18.html ). How about balance? Did you know that if you were standing at arm’s length from someone and each of you had one percent more electrons than protons, the repelling force would be so incredible that the repulsion would be enough to lift a “weight” of the planet! Yup – math is full of fun surprises!

Now, modern math is sort of like a constant tease which shows you the shell of this beautiful program and this excitement, and you know the beauty is there, but it’s not easy to understand and grasp! For one, most of mathematics is filled with jargon and language that is incredibly information dense, and it looks like it’s been written by a schizophrenic C programmer who’s paranoid about losing his job, so the information tends to be lumped into these incredibly dense formulas which hide the beauty and truth, but the beauty and truth will always be there! You just need to have a bit of persistence and dedication. It also doesn’t help that most teachers tend to not make things nice. They puke out the same old standardized stuff regurgitated and taught to them, and so round and round it goes.

Hopefully though, we’ll get better at teaching it and conveying it’s structure as we learn better ways of not making things cumbersome and uninteresting to other people! Wow, I need to stop writing – sorry for the large wall of text, but I hope you get what I mean!

Re: An Intuitive Guide to Linear Algebra (2012)

#72

Post by vecter »

adamcharnock wrote: post #55
greentimer wrote: post #50

Earlier quoted context omitted.

Higher math involves a very different way of thinking from the typical, useful things people do for a living. Exactness is important. The abstractions can run very deep. It's easy to get lost in the pure side of things without really understanding how to apply it.

I think this is possibly the crux of it for me. I've certainly got well developed abstract thinking for software development, but software has always had a clear application for me, so the abstract thinking developed as a matter of course. I've rarely found any higher math instruction which takes for the form, "so you have this specific problem X, here is how we can solve it with technique Y"[1]. But I suspect that i…

Software development never gets that abstract. Yes there are abstractions, but compared to college math, they're extremely simple. The most complex abstractions I've come across in software engineering don't hold a candle to some of the abstractions you would see in a typical undergraduate math degree. Heck, my university used Baby Rudin [0] for its introductory analysis class, which was often taken by freshmen or sophomores.

[0] https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_math...

Re: An Intuitive Guide to Linear Algebra (2012)

#73

Post by rustybolt »

chobytes wrote: post #70
id_ris wrote: post #61

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms,…

Yeah, this sort of stuff is fine supplementary material... but nobody really learns anything from a blog post or some videos.

I disagree. Maybe you'll remember it not as well if you don't do exercises, but there is no reason you can't learn from a blogpost (which is trivial to prove since you can just copy-paste the contents of a book in the form of one or more blogposts).

Re: An Intuitive Guide to Linear Algebra (2012)

#74

Post by rustybolt »

nearlynameless wrote: post #38

While this explanation is certainly much clearer than what I remember of high school maths, I still have a pretty tough time following the formula examples. When I see A(x) = ax, I'm not entirely sure how to read it. Is A meant to be a function that accepts x? If so, why is the equivalent expression a * x? Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by th…

The particular example might be confusing because the same letter is used twice with different capitalization. There is no direct relation between them.

> Is A meant to be a function that accepts x?

Yes.

> If so, why is the equivalent expression a * x?

Because that how A(x) is defined.

>Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value?

Yes, it's an unspecified constant (a, b, c... are used to denote constants by convention), so you can really calculate A(x) for supplied values of x yet until the constant 'a' is specified.

> Is this notation specific to multiplication, to this expression, or what?

No. Functions might be defined using any expression. For example

    A(x) = b^x
is a valid function as well (again, we have an unspecified constant). Just don't expect to encounter it in an introductory course in linear algebra (since it would deal mostly with linear functions).

Re: An Intuitive Guide to Linear Algebra (2012)

#75

Post by yori »

rustybolt wrote: post #73
chobytes wrote: post #70

Earlier quoted context omitted.

Yeah, this sort of stuff is fine supplementary material... but nobody really learns anything from a blog post or some videos.

I disagree. Maybe you'll remember it not as well if you don't do exercises, but there is no reason you can't learn from a blogpost (which is trivial to prove since you can just copy-paste the contents of a book in the form of one or more blogposts).

What your parent comment meant was that it is not possible to learn well from typical blog posts like this that try to condense the subject into a 3000 word article. Of course, if you copy-paste the content of a book into a blog post, then parent comment's point no longer applies.

Re: An Intuitive Guide to Linear Algebra (2012)

#76

Post by robpal »

id_ris wrote: post #61

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms,…

My experience from a decade of doing professional maths is that there are no shortcuts. You only learn maths by doing hundreds (thousands) of exercises, both mundane and more exctiting.

Also, the concepts "mature" in the brain. I remember sleepless nights in the first year of undergrad spent on understanding the details of the proof of the Jordan decomposition and a few years later (when studying algebraic groups) it all felt trivial.

There's no shortcut to understanding maths, just a lot of time spent in solitude trying to make sense of all these abstract concepts (and they DO make sense).

Re: An Intuitive Guide to Linear Algebra (2012)

#77

Post by robpal »

nightcracker wrote: post #24
vecter wrote: post #12

This is ok but nothing is as intuitive as 3B1B's series on YouTube that has been posted hundreds of times on HN [0]. Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post. [0] https://www.youtube.com/watch?v=fNk_zzaMoSs

> Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post. I... disagree. Some of linear algebra is about that. And it's probably a good way to view it that way when learning. But some of my current work (coding theory) involves linear algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really at no point…

Linear algebra IS about linear transformations and vector spaces.

The thing is that the field over which the space is defined can be quite arbitrary (finite, infinite, not algebraically closed etc.) which has immense consequences on the behavior of such objects.

When one drops the assumption on finite number of dimensions, the story becomes wild (and is known as functional analysis, beautiful and extremely useful branch of mathematics).

Re: An Intuitive Guide to Linear Algebra (2012)

#78

Post by abdullahkhalids »

dfabulich wrote: post #68
id_ris wrote: post #61

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms,…

> While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. I'm not sure I understand your point. Are you just saying that this blog post isn't an adequate substitute for taking a course in linear algebra? (Of course it isn't. But who said it was?)

Unfortunately, for a lot of people, including undergraduates, the dopamine hit they get from watching a video or passively reading a textbook makes them believe that these are adequate substitutes for doing thousands of exercises.

In my university, undergraduates have admitted that they have done fewer than 50 questions throughout the entirety of my math course. Their grades obviously reflect that, but they will do the same next semester.

Re: An Intuitive Guide to Linear Algebra (2012)

#79

Post by ginnungagap »

yori wrote: post #75
rustybolt wrote: post #73

Earlier quoted context omitted.

I disagree. Maybe you'll remember it not as well if you don't do exercises, but there is no reason you can't learn from a blogpost (which is trivial to prove since you can just copy-paste the contents of a book in the form of one or more blogposts).

What your parent comment meant was that it is not possible to learn well from typical blog posts like this that try to condense the subject into a 3000 word article. Of course, if you copy-paste the content of a book into a blog post, then parent comment's point no longer applies.

I'd say it still applies, even when reading a textbook going through the exercises is crucial

Re: An Intuitive Guide to Linear Algebra (2012)

#80

Post by diffeomorphism »

This seems to be "what are matrices and how do you work with them" and not linear algebra. I mean that can be useful sometimes but seems more like something you would teach in a numerics course instead.

Actually, I think this way of explaining and motivating things (linear map==matrix) will get really, really confusing once you try to understand changes of bases or eigenvalue decomposition. A linear map is something that takes vectors and spits out vectors while preserving the vector structure (i.e. addition and scalar multiplication on the input give you addition and scalar multiplication of the output).

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