6 ms·

2b1b does a video on how you raise e to a matrix (hint: raising something to e is converted into another function where it's not nonsensical to do such a thing.

2b1b does a video on how you raise e to a matrix (hint: raising something to e is converted into another function where it's not nonsensical to do such a thing.) In about 10 minutes he explains something my math methods instructor struggled with over two weeks (4 classes). I was resistant to the view that YouTube could host "decent" instructional content for a long time. That video was one of the first that was CLEARLY superior to the instruction I received at a tier 1 research institution.

codetrotter 3y ago
> a video on how you raise e to a matrix Video link https://youtu.be/O85OWBJ2ayo https://youtu.be/O85OWBJ2ayo
Juicyy 3y ago
true for most of calc and a lot of electrical. Institutions failed to discuss why you need to do things. so much of it is here is Taylor Series or integration by parts. but nobody really knew what the application for this stuff was. just follow some rules and plug in some numbers.
pdpi 3y ago
My "ten minutes to understand something I never quite got" moment was his video on Fourier Transforms.
- suremarc 3y ago
  I understand Fourier transforms fairly well but his video blew my mind. He has a totally original way of distilling mathematical concepts intuitively.
- robertlagrant 3y ago
  Yes. The visualisations and the narration make it so clear.
- eimrine 3y ago
  Just listen to some music with spectrum visualization, this is a Fourier transform of what you are listening right now.
  - pdpi 3y ago
    Well, yes. That part is obvious enough. The interesting part is how you build the mapping from time domain to frequency domain. That is the part that never clicked for me.
    - eimrine 3y ago
      Maybe that's because a sinusoid of any frequence has its "roundness" and that is a matter of "hunting" which that formula performs every infinite-small moment to an x/y graph of your favourite music composition.
  - jameshart 3y ago
    That doesn’t provide the full understanding you might think it does. For example: define ‘right now’. Instantaneously, music is a single sound pressure measurement. That doesn’t have a Fourier transform. It doesn’t have a frequency. It’s just a single sample. Fourier transforms work on functions. Typically functions in the time domain. And typically (but not always) on that function within a bounded range of time. And the result is another function, this one of frequency. A spectrum analyzer, though, is showing the Fourier transform of a short snippet of some music. Then a moment later it’s showing you the transform for the next snippet. Looking at a spectrum analyzer makes you think a Fourier transform is itself a function of time (to some vector of numbers perhaps?). That is not the case. So looking at a spectrum analyzer can give you an incorrect intuition for what Fourier does. But you can do a Fourier transform on the whole of a piece of music. You’ll pick up frequency components like the overall beat, the bar structure, the verse/chorus alternation.
    - beambot 3y ago
      I think I get what you are trying to say... but the intuition about "this moment in time" is perfectly reasonable for a spectrum analyzer, since it's actually doing a DFT (not continuous from +-infinity) with the last sample (i.e. "now") defining the end of the window.
- geokon 3y ago
  This probably won't be a very popular take.. but to me Fourier Transforms are the perfect example of the opposite problem - where the educators are so hellbound on constructing a visual explanation that they end up brain damaging the students. I've seen this come up several times in my math education. For the Fourier transform there are primarily two issue: 1. The "complex plane" - this tries to make complex numbers somehow less scary and more intuitive? But it's actually a useless crutch that gives people a false sense of understanding. The core issue is that you can multiply two complex numbers and get another complex number. If I give you two vectors or two points on a map and tell you to multiply them.. you can't - b/c that is not a defined operation. We have no intuition about how to multiple 2D numbers (you might think, oh dot products and cross products! but those are something unfortunately unrelated) 2. "Frequency Space" and an the constant suggestion that your signal is somehow being broken up into it's magical innate hidden frequencies. This is also very deceptive (or doesn't hold up in the discrete case). The basis is selected to be mathematically convenient (ie. orthogonal) and is based on your sampling window and properties of complex numbers. These may correspond to some underlying natural frequency that's occurring - or it may not. But it's not helpful to try to confound the two If you instead approach the whole problem from a purely mathematical perspective of projection on an orthogonal basis and how to construct complex values that are convenient - then the whole setup is less "fun" but it actually becomes a lot more understandable. You can then move on from there and start asking yourself much more interesting questions about aliasing, ringing, fm/am, phase etc. I feel it took be ~5 attempts of learning Fourier Analysis to unwire my brain and unlearn these bad visual intuitions that send you down the wrong path
  - SkyBelow 3y ago
    I think there is a special trick that happens many places in math. When you are given a visual, and then some integral or derivative is applied to it, you need to pause and make sure to truly include that in the mental model. Often I find math videos go far too fast through this section and I have to re-watch that specific spot and even pause and just ponder on how does the integral or derivative change what I was working with.
  - 9erdelta 3y ago
    Can't speak to Fourier specifically -- but heartily agree that a lot of times in math, teachers try to make it "easy" and provide visuals that are actually a hindrance. Instead of understanding _the material_ you understand the _crutch_.
- hpcjoe 3y ago
  There's an old joke in math/physics student circles that covers this. Student goes to see the prof in their office hours to try to understand something the prof said is seen to be trivial. They spend 4 hours working on it, and the student returns to their friends in class. They ask how it went. The student says "yeah, it was trivial." For those who don't quite get the joke, 4 hours to show something is trivial, tends to not support that the thing is trivial to comprehend.
  - pdpi 3y ago
    To be fair there is a grain of truth to the joke. E.g. Cayley’s theorem is sort of like that — once you get it, it feels like it barely qualifies as a theorem, it’s just a natural consequence of the definitions involved.
frankfrank13 3y ago
Much the same experience for me, although sans tier-1 lol. He's clearly gifted, for a long time I think wrongly assumed everyone who makes content is not.
gumby 3y ago
> CLEARLY superior to the instruction I received at a tier 1 research institution. It was pretty clear at the R1 institution I attended (MIT) that it is really a huge research lab with a small school attached, and that the educational part, for undergrads at least, was definitely not a priority. The continuing stream of advertising (err, alumni updates) that I’ve received over the subsequent decades don’t seem to contradict this either. Still glad I went and learned a ton, but it’s not for everyone. ETA: don’t mean to imply that it might have been good or bad for you, dingosity (how would I know?); just responding in a general way to a singular anecdote/observation in your comment.
- locusofself 3y ago
  I've been watching some MIT lectures online recently, most of which are about 10 years old. Do they still use those huge a chalkboards?
  - deleted 3y ago
    [deleted]
  - webshit2 3y ago
    Yes usually blackboard or slides. Sometimes iPad + stylus.
- ChuckMcM 3y ago
  Interesting that we both saw the same thing. I visited MIT as it was on my list of schools I wanted to attend out of HS and realized after that visit that all the "action" there was in research/postgrad so it was a better graduate school candidate than undergrad.
  - webshit2 3y ago
    The undergrads are more involved with research than at most other schools (huge range in quality/difficulty/etc, but the option is there). The teaching quality doesn't really suffer from the focus on research. Blackboard style classes probably aren't much better than any other school, but lab classes were quite good.
rjh29 3y ago
> I was resistant to the view that YouTube could host "decent" instructional content for a long time. There is SO much good content on there though. I can only guess "the algorithm" was hiding it from you. Sadly we often have to rely on sites like reddit or HN to find these creators.
- VBprogrammer 3y ago
  While we're at it. Ben Eater is the equivalent for Computer / Electronics fundamentals. Check him out of you haven't before.
- aeternum 3y ago
  the algorithm is actually quite good but you need to give it signal. Most people prefer entertaining videos over educational videos. If that's what you tend to click, that's what it will show you. Entertaining is hard to resist, so just create a second account and curate the history of one of them.
  - rjh29 3y ago
    It's good for me too, but it took a very long time to train it to stop giving me 'YouTube face' videos, clickbait bullshit, Linus Tech Tips, politics, Amber Heard etc. And it still sometimes serves that up if there's a new topic. I also find my feed doesn't change that much, so the downside of being picky is that you might miss out on some good content.
jonahx 3y ago
I've had math instruction at the undergraduate and graduate level at top universities, with good instructors that I liked, and I can say without question YouTube is superior. There are many channels with outstanding content. This video is not an outlier.
- inetknght 3y ago
  I thoroughly enjoyed watching 3b1b's videos. They're very intuitive for me. > This video is not an outlier. I would argue: this video is an outlier in that it's "best of the best". I've seen other videos. While they're usually "good", they're also usually very dry.
  - dingosity 3y ago
    And I've absolutely seen some "meh" videos. Apropos of nothing, I met Jim Blinn a month ago which got me thinking about "the mechanical universe" (from '85-'86) which was one of the first videos I saw that really "nailed it" in terms of using video to teach technical concepts. So good videos aren't a new thing. But I think there's MUCH more decent and excellent content coming out these days.
- chrisweekly 3y ago
  See also https://betterexplained.com https://betterexplained.com
- veltas 3y ago
  It's much like open source vs corporate, but not quite. The bazaar really is better than the cathedral.
- remus 3y ago
  Personally I don't think it is so clear cut. The value of in-person instruction is in being able to ask questions, and a good instructor will then adapt the content to help answer your questions and give you a rounded understanding of the material. Youtube videos are amazing in several respects (thinking of the reach and production value of 3b1b, for example), but being able to ask questions of a knowledgeable expert is valuable imo.
jackphilson 3y ago
perhaps we should remodel our educational system towards content that had a great deal of time spent creating it rather than a slideshow created last night. if only there was this medium that allowed information to be duplicated all over the world instantly so that every professor doesn't need to create their own content
- jerf 3y ago
  I'm actually some combination of flabberghasted and annoyed that we're in 2023 and computers have still not been terribly successfully deployed into any educational setting. I suppose the idea that we should look to the results rather than some Ivory Tower education professor's ideas about what constitutes good teaching is just too foreign for the establishment as a whole. I've got armies of PhDs A-B testing whether they can extract .001 cent more per ad from me, but it's too much to ask of society as a whole to check into the question of what curriculum works better. Grant shouldn't have to be all but forcing his way uphill into this space.
gowld 3y ago
> hint: raising something to e raising e to something > is converted into another function where it's not nonsensical to do such a thing.) That function is the power series expansion, taught in your calculus class. https://en.wikipedia.org/wiki/Matrix_exponential https://en.wikipedia.org/wiki/Matrix_exponential > CLEARLY superior to the instruction I received at a tier 1 research institution. 3B1B is a tier 1 educational institution. But also, videos often feel better than classes, because videos don't have homework, so you don't actually have your feeling of understanding tested.
GuB-42 3y ago
What 2b1b, as well as other science YouTubers do is that they give you insight that more conventional teaching may lack. What YouTubers don't offer however is training, as in, the sometimes boring part where you do exercises. They are complementary, an entertaining YouTube video may first give you motivation to get into a seemingly boring subject. It may also help you after you took your classes, to help make elements fall into place, but it only works if you have learned the elements beforehand. As educative these videos are, they are entertainment, and they are driven by the rules of entertainment, they wouldn't get much views otherwise. So it leans heavily on accessible and clever explanations, surprising results and relatively short form, the boring stuff is often left out, and unfortunately, some of it is necessary to progress. Most science YouTubers admit it, and urge everyone to take school seriously. I have never learned a field just from YouTube videos. It either left me with superficial knowledge (which is better than nothing) or acted as a way to consolidate knowledge I already had.

codetrotter 3y ago

> a video on how you raise e to a matrix Video link https://youtu.be/O85OWBJ2ayo https://youtu.be/O85OWBJ2ayo

Juicyy 3y ago

true for most of calc and a lot of electrical. Institutions failed to discuss why you need to do things. so much of it is here is Taylor Series or integration by parts. but nobody really knew what the application for this stuff was. just follow some rules and plug in some numbers.

pdpi 3y ago

My "ten minutes to understand something I never quite got" moment was his video on Fourier Transforms.

suremarc 3y ago
I understand Fourier transforms fairly well but his video blew my mind. He has a totally original way of distilling mathematical concepts intuitively.
robertlagrant 3y ago
Yes. The visualisations and the narration make it so clear.
eimrine 3y ago
Just listen to some music with spectrum visualization, this is a Fourier transform of what you are listening right now.
- pdpi 3y ago
  Well, yes. That part is obvious enough. The interesting part is how you build the mapping from time domain to frequency domain. That is the part that never clicked for me.
  - eimrine 3y ago
    Maybe that's because a sinusoid of any frequence has its "roundness" and that is a matter of "hunting" which that formula performs every infinite-small moment to an x/y graph of your favourite music composition.
- jameshart 3y ago
  That doesn’t provide the full understanding you might think it does. For example: define ‘right now’. Instantaneously, music is a single sound pressure measurement. That doesn’t have a Fourier transform. It doesn’t have a frequency. It’s just a single sample. Fourier transforms work on functions. Typically functions in the time domain. And typically (but not always) on that function within a bounded range of time. And the result is another function, this one of frequency. A spectrum analyzer, though, is showing the Fourier transform of a short snippet of some music. Then a moment later it’s showing you the transform for the next snippet. Looking at a spectrum analyzer makes you think a Fourier transform is itself a function of time (to some vector of numbers perhaps?). That is not the case. So looking at a spectrum analyzer can give you an incorrect intuition for what Fourier does. But you can do a Fourier transform on the whole of a piece of music. You’ll pick up frequency components like the overall beat, the bar structure, the verse/chorus alternation.
  - beambot 3y ago
    I think I get what you are trying to say... but the intuition about "this moment in time" is perfectly reasonable for a spectrum analyzer, since it's actually doing a DFT (not continuous from +-infinity) with the last sample (i.e. "now") defining the end of the window.
geokon 3y ago
This probably won't be a very popular take.. but to me Fourier Transforms are the perfect example of the opposite problem - where the educators are so hellbound on constructing a visual explanation that they end up brain damaging the students. I've seen this come up several times in my math education. For the Fourier transform there are primarily two issue: 1. The "complex plane" - this tries to make complex numbers somehow less scary and more intuitive? But it's actually a useless crutch that gives people a false sense of understanding. The core issue is that you can multiply two complex numbers and get another complex number. If I give you two vectors or two points on a map and tell you to multiply them.. you can't - b/c that is not a defined operation. We have no intuition about how to multiple 2D numbers (you might think, oh dot products and cross products! but those are something unfortunately unrelated) 2. "Frequency Space" and an the constant suggestion that your signal is somehow being broken up into it's magical innate hidden frequencies. This is also very deceptive (or doesn't hold up in the discrete case). The basis is selected to be mathematically convenient (ie. orthogonal) and is based on your sampling window and properties of complex numbers. These may correspond to some underlying natural frequency that's occurring - or it may not. But it's not helpful to try to confound the two If you instead approach the whole problem from a purely mathematical perspective of projection on an orthogonal basis and how to construct complex values that are convenient - then the whole setup is less "fun" but it actually becomes a lot more understandable. You can then move on from there and start asking yourself much more interesting questions about aliasing, ringing, fm/am, phase etc. I feel it took be ~5 attempts of learning Fourier Analysis to unwire my brain and unlearn these bad visual intuitions that send you down the wrong path
- SkyBelow 3y ago
  I think there is a special trick that happens many places in math. When you are given a visual, and then some integral or derivative is applied to it, you need to pause and make sure to truly include that in the mental model. Often I find math videos go far too fast through this section and I have to re-watch that specific spot and even pause and just ponder on how does the integral or derivative change what I was working with.
- 9erdelta 3y ago
  Can't speak to Fourier specifically -- but heartily agree that a lot of times in math, teachers try to make it "easy" and provide visuals that are actually a hindrance. Instead of understanding _the material_ you understand the _crutch_.
hpcjoe 3y ago
There's an old joke in math/physics student circles that covers this. Student goes to see the prof in their office hours to try to understand something the prof said is seen to be trivial. They spend 4 hours working on it, and the student returns to their friends in class. They ask how it went. The student says "yeah, it was trivial." For those who don't quite get the joke, 4 hours to show something is trivial, tends to not support that the thing is trivial to comprehend.
- pdpi 3y ago
  To be fair there is a grain of truth to the joke. E.g. Cayley’s theorem is sort of like that — once you get it, it feels like it barely qualifies as a theorem, it’s just a natural consequence of the definitions involved.

frankfrank13 3y ago

Much the same experience for me, although sans tier-1 lol. He's clearly gifted, for a long time I think wrongly assumed everyone who makes content is not.

gumby 3y ago

> CLEARLY superior to the instruction I received at a tier 1 research institution. It was pretty clear at the R1 institution I attended (MIT) that it is really a huge research lab with a small school attached, and that the educational part, for undergrads at least, was definitely not a priority. The continuing stream of advertising (err, alumni updates) that I’ve received over the subsequent decades don’t seem to contradict this either. Still glad I went and learned a ton, but it’s not for everyone. ETA: don’t mean to imply that it might have been good or bad for you, dingosity (how would I know?); just responding in a general way to a singular anecdote/observation in your comment.

locusofself 3y ago
I've been watching some MIT lectures online recently, most of which are about 10 years old. Do they still use those huge a chalkboards?
- deleted 3y ago
  [deleted]
- webshit2 3y ago
  Yes usually blackboard or slides. Sometimes iPad + stylus.
ChuckMcM 3y ago
Interesting that we both saw the same thing. I visited MIT as it was on my list of schools I wanted to attend out of HS and realized after that visit that all the "action" there was in research/postgrad so it was a better graduate school candidate than undergrad.
- webshit2 3y ago
  The undergrads are more involved with research than at most other schools (huge range in quality/difficulty/etc, but the option is there). The teaching quality doesn't really suffer from the focus on research. Blackboard style classes probably aren't much better than any other school, but lab classes were quite good.

rjh29 3y ago

> I was resistant to the view that YouTube could host "decent" instructional content for a long time. There is SO much good content on there though. I can only guess "the algorithm" was hiding it from you. Sadly we often have to rely on sites like reddit or HN to find these creators.

VBprogrammer 3y ago
While we're at it. Ben Eater is the equivalent for Computer / Electronics fundamentals. Check him out of you haven't before.
aeternum 3y ago
the algorithm is actually quite good but you need to give it signal. Most people prefer entertaining videos over educational videos. If that's what you tend to click, that's what it will show you. Entertaining is hard to resist, so just create a second account and curate the history of one of them.
- rjh29 3y ago
  It's good for me too, but it took a very long time to train it to stop giving me 'YouTube face' videos, clickbait bullshit, Linus Tech Tips, politics, Amber Heard etc. And it still sometimes serves that up if there's a new topic. I also find my feed doesn't change that much, so the downside of being picky is that you might miss out on some good content.

jonahx 3y ago

I've had math instruction at the undergraduate and graduate level at top universities, with good instructors that I liked, and I can say without question YouTube is superior. There are many channels with outstanding content. This video is not an outlier.

inetknght 3y ago
I thoroughly enjoyed watching 3b1b's videos. They're very intuitive for me. > This video is not an outlier. I would argue: this video is an outlier in that it's "best of the best". I've seen other videos. While they're usually "good", they're also usually very dry.
- dingosity 3y ago
  And I've absolutely seen some "meh" videos. Apropos of nothing, I met Jim Blinn a month ago which got me thinking about "the mechanical universe" (from '85-'86) which was one of the first videos I saw that really "nailed it" in terms of using video to teach technical concepts. So good videos aren't a new thing. But I think there's MUCH more decent and excellent content coming out these days.
chrisweekly 3y ago
See also https://betterexplained.com https://betterexplained.com
veltas 3y ago
It's much like open source vs corporate, but not quite. The bazaar really is better than the cathedral.
remus 3y ago
Personally I don't think it is so clear cut. The value of in-person instruction is in being able to ask questions, and a good instructor will then adapt the content to help answer your questions and give you a rounded understanding of the material. Youtube videos are amazing in several respects (thinking of the reach and production value of 3b1b, for example), but being able to ask questions of a knowledgeable expert is valuable imo.

jackphilson 3y ago

perhaps we should remodel our educational system towards content that had a great deal of time spent creating it rather than a slideshow created last night. if only there was this medium that allowed information to be duplicated all over the world instantly so that every professor doesn't need to create their own content

jerf 3y ago
I'm actually some combination of flabberghasted and annoyed that we're in 2023 and computers have still not been terribly successfully deployed into any educational setting. I suppose the idea that we should look to the results rather than some Ivory Tower education professor's ideas about what constitutes good teaching is just too foreign for the establishment as a whole. I've got armies of PhDs A-B testing whether they can extract .001 cent more per ad from me, but it's too much to ask of society as a whole to check into the question of what curriculum works better. Grant shouldn't have to be all but forcing his way uphill into this space.

gowld 3y ago

> hint: raising something to e raising e to something > is converted into another function where it's not nonsensical to do such a thing.) That function is the power series expansion, taught in your calculus class. https://en.wikipedia.org/wiki/Matrix_exponential https://en.wikipedia.org/wiki/Matrix_exponential > CLEARLY superior to the instruction I received at a tier 1 research institution. 3B1B is a tier 1 educational institution. But also, videos often feel better than classes, because videos don't have homework, so you don't actually have your feeling of understanding tested.

GuB-42 3y ago

What 2b1b, as well as other science YouTubers do is that they give you insight that more conventional teaching may lack. What YouTubers don't offer however is training, as in, the sometimes boring part where you do exercises. They are complementary, an entertaining YouTube video may first give you motivation to get into a seemingly boring subject. It may also help you after you took your classes, to help make elements fall into place, but it only works if you have learned the elements beforehand. As educative these videos are, they are entertainment, and they are driven by the rules of entertainment, they wouldn't get much views otherwise. So it leans heavily on accessible and clever explanations, surprising results and relatively short form, the boring stuff is often left out, and unfortunately, some of it is necessary to progress. Most science YouTubers admit it, and urge everyone to take school seriously. I have never learned a field just from YouTube videos. It either left me with superficial knowledge (which is better than nothing) or acted as a way to consolidate knowledge I already had.