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An Intuitive Guide to Linear Algebra (2012)

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Re: An Intuitive Guide to Linear Algebra (2012)

#31

Post by JadeNB »

alacombe wrote: post #21

Speaking for Linear Algebra, I learnt more reading for a few hours the appendix of "The Design of Rijndael: AES - The Advanced Encryption Standard" than I did in 6 months of theoretical university teaching full of useless technical terms and solutions in search of problems...

> solutions in search of problems...

This sounds like it was meant to be pejorative, but it's what (applied) linear algebra, and applied mathematics more generally, is. Anyone can learn about a certain mathematical topic upon realising it's the one relevant to the problem they're facing—and learn it way more quickly, due to motivation and focus, than they would in a general-purpose course on the topic; the art is in recognising what mathematics is relevant, and you can't do that if you've never heard of it before. Having a library of (conceptual, not cookbook) solution hooks on which to hang your problems is how you get to be good at using mathematics.

Re: An Intuitive Guide to Linear Algebra (2012)

#32

Post by mhh__ »

sushisource wrote: post #3

I love explainers like this, but it frankly makes me a little angry that the vast majority of the math teachers I had in highschool and college taught in the awful way described in the setup to the piece. Why is that? Has anyone studied it, or is there even a solid anecdotal explanation? The best one I can imagine is many of these professors simply don't care much for teaching and are more focused on their research,…

Have you ever taught?

I taught Astronomy to the people in the years below me (a school tradition because it was optional), and it was absolutely exhausting trying to plan good lessons that didn't involve getting them to memorise stuff by wrote. I ""derived"" Kepler's laws for a bunch of 14-15 year olds who didn't even know logarithms yet and it was pretty brutal intellectually.

Also, I think teaching mathematics "intuitively" requires a bit of cooperation from the student - not in the sense of intelligence, just that for for every guy (or girl) who watches 3Blue1Brown (and looks deeper into the pure mathematics) there's another who's just along for the ride. I think the frequencies of those personalities are a product of how they were taught, but it's very difficult to convince people in "teaching time" as opposed to naturally (I was in the lowest maths group for years until I picked up a calculus book on a whim, but no teacher could've convinced old me that mathematics can be beautiful)

Quick Edit: To quote Tim Minchin, "Be a teacher". I might come across as moaning in the above but it's really interesting to explain things (in my experience at least - I love writing documentation!), and a good bullshit-test on yourself

Re: An Intuitive Guide to Linear Algebra (2012)

#33

Post by snicker7 »

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 a shared field across multiple disciplines. So I'm sure that there are many valid and useful interpretations as to what "linear algebra" is essentially about.

However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra).

Re: An Intuitive Guide to Linear Algebra (2012)

#34

Post by jfarmer »

Swizec wrote: post #27
jimhefferon wrote: post #17

Earlier quoted context omitted.

For sure there is an effect on Reddit and other places where someone will post a question such as, "I'm having trouble with my Calc class, what is a good book?" and people seriously answer Calculus on Manifolds . Now, CoM is a classic, a real great book, but it is useful only to people who have reached a certain level of mathematical maturity. That, presumably, is not the questioner. A version of this is that I also…

Once you understand monads, you lose the ability to explain monads. Hence the number of monads tutorials grows at an exponential rate as every new understander tries to explain them and fails. it's a fun problem in teaching

Ironically, I think the original paper that introduced Monads as a useful computational abstraction is the clearest explanation I've seen.

https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/b...

Re: An Intuitive Guide to Linear Algebra (2012)

#35

Post by Koshkin »

Swizec wrote: post #27
jimhefferon wrote: post #17

Earlier quoted context omitted.

For sure there is an effect on Reddit and other places where someone will post a question such as, "I'm having trouble with my Calc class, what is a good book?" and people seriously answer Calculus on Manifolds . Now, CoM is a classic, a real great book, but it is useful only to people who have reached a certain level of mathematical maturity. That, presumably, is not the questioner. A version of this is that I also…

Once you understand monads, you lose the ability to explain monads. Hence the number of monads tutorials grows at an exponential rate as every new understander tries to explain them and fails. it's a fun problem in teaching

> Once you understand... you lose the ability to explain

Sorry, this does not make sense to me.

Re: An Intuitive Guide to Linear Algebra (2012)

#36

Post by adamcharnock »

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 shouting, "why for the love of god would we be doing this"? Based on this reaction I suspect that the content neither has intrinsic appeal to me, nor does it have obvious use in my work, projects, or life.

Pre-degree maths though, I love. My A-level maths really changed how I saw the world, and I make use of it reasonably often (well, often enough to not forget it).

I think I'm writing this here because most other commenters seem to really grasp this subject, or feel that they grasp it better having seen these videos. I'm honestly happy for you. However, if anyone is reading this who doesn't feel like that, then know you're not alone :-)

Re: An Intuitive Guide to Linear Algebra (2012)

#37

Post by zelly »

formalsystem wrote: post #2

Viewing Linear Algebra as the study of linear operators instead of matrices makes everything so much simpler. Of course AB != BA Composition makes sense Inverse makes sense This is the book that helped me get it http://linear.axler.net/

This is a great book. I also recommend Halmos "Finite Dimensional Vector Spaces". The typical way linear algebra is introduced does not present a matrix as a linear transformation first and foremost.

Re: An Intuitive Guide to Linear Algebra (2012)

#38

Post by nearlynameless »

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 the supplied value? Is this notation specific to multiplication, to this expression, or what?

Positing that something is 'intuitive' when it depends so much on additional contextual knowledge seems ever so slightly disingenuous as best, and slightly harmful at worst; it can make the reader feel as though they must be dumb for not understanding this 'intuitive' material.

I do acknowledge that this is linear algebra, and if one doesn't have a really solid grasp of notation of regular algebra it is likely to go over their heads, but the practical explanations (such as the slope rise/run example) are quite clear and relatively simple to follow; it follows that a simple explanation of the notation might be helpful too.

Re: An Intuitive Guide to Linear Algebra (2012)

#39

Post by [deleted] »

xscott wrote: post #26
nightcracker wrote: post #24

Earlier quoted context omitted.

> 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…

I think this is spot on. Depending on what you're doing, a matrix can be: - A linear transformation - A basis set of column vectors - A set of equations (rows) to be solved - (your example: parity equations for coding theory) - The covariance of elements in a vector space - The Hessian of a function for numerical optimization - The adjacency representation of a graph - Just a 2D image (compression algorithms) ... (I'…

[deleted]

Re: An Intuitive Guide to Linear Algebra (2012)

#40

Post by salty_biscuits »

sushisource wrote: post #3

I love explainers like this, but it frankly makes me a little angry that the vast majority of the math teachers I had in highschool and college taught in the awful way described in the setup to the piece. Why is that? Has anyone studied it, or is there even a solid anecdotal explanation? The best one I can imagine is many of these professors simply don't care much for teaching and are more focused on their research,…

Here are two reasons I think are important, there are more but these stand out. (1) Teaching is hard, and the amount of training you get in teaching higher level maths (rather than the just learning higher level maths itself) is very limited. (2) intuition in maths will lead you on a merry path to very wrong ideas. There are functions that are continuous everywhere without a derivative anywhere, Russell's paradox, classically zenos paradox, etc, etc. A big part of higher level maths training is in being precise with what you mean so that you can learn to do proofs and not trick yourself. I think a synthesis of these approaches is the sweet spot. Be precise but show applications to motivate the material.
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