Summer of Math Exposition

1:22 this was one of many scenes where there was too many formulas

1:49 It seems one could simply start the video with the linear approximation of the pendulum; this also cuts down the physical formulas needed to make sense of the original system

2:39 I liked the declaration of who the audience is, and the “skip to” claim

3:19 Somehow “matrices = linear transformations” seems incongruent with the level of abstraction vector spaces are discussed later

4:22 nice animations

5:14 Will there be heuristics introduced for AM and GM of an eigenvalue?

5:48 I understand that you are trying to fast forward the formalism, but this is too many definitions / words in one “slide”. Also why an arbitrary field K? There is also too much notation for the layperson, in my opinion, so that the people who can decipher this slide would not need to read it. Also you probably want to include that f is linear here.

6:57 In the previous slide you don’t mention anything about V being finite dimensional, but in the next slide now we can identify matrices with transformations?

7:22 Probably for “Why?” you want to mention “square matrix non-invertible iff det is zero”

8:04 “kernel” is too advanced a jargon in my opinion.

8:40 It would be better to state a_{ij} is the ij-th entry of A, or better yet, use a,b,c,d as entries of A

9:08 K[lambda]_n is too advanced a notation. Of course the coefficients of the char poly are also special, and even if you’re not going to go into it it might have been nice to at least mention this

10:08 it might have been nice to have some voiceover for the calculation (or cut it out completely)

12:13 this was an opportunity to mention “if triangular, eigenvalues are diagonal entries. if eigenvalues distinct, then diagonalizable”; and hence bypass the calculation

18:06 It feels like the video should have started with this part. It would have been nice to define what control theory is, and that you’ll focus on linear control theory, as opposed to nonlinear, stochastic, etc.

23:58 too many formulas in one slide, logical symbols seem overwhelming for a general audience. it would have been better to skip the formalism and describe the animation you show next, which I think would have been great if you could animate different eps’s and deltas.

26:47 You started out with the promise to make intuitive eigenstuff, however by now you introduced so much more advanced math to achieve this, that it seems like your presentation is self-defeating

33:34 It seems by just focusing on discrete time systems you could bypass many technicalities

36:23 While nice, the complexity claims about matrix factorizations seem a little out of place here

37:08 The personal statements are really nice, maybe you should have introduced yourself in the very beginning to properly orient the audience

Motivation: 3/9 - Imo the focus was brought on wrong things at the beginning. Too much detail was shown about the pendulum (a lot of definitions that mean nothing to the viewer), which we would come back to almost 20 minutes later. The video contained more cool stuff and it could be used in the intro to engage the viewer.

Clarity: 7/9 - I think that the topic is great and the title + thumbnail make it interesting. You explained it well, with a lot of examples, which I really liked. Good job!

Novelty: 5/9 - The topic was already covered by 3b1b as you’ve said in the vid, at least to some extent. Giving it a 5 because of the non trivial examples

Memorability: 5/9 - decent animation, that doesn’t bother but also doesn’t stand out. Props for title and thumbnail

Overall: 6/9 - I would say your video is on the level of typical YT math explainer, which is great for your first vid. Good job and keep it up!

My only suggestions: -Around 12:00 it would have been clearer to use a different letter than B for the basis sets, maybe C or D, etc. -When discussing Lyapunov stability and asymptotic stability, I like the Manim example you used to demonstrate but it would have been even better if you had one Manim example for each possibility: not Lyapunov stable, not asymptotic stable, Lyapunov stable but not asymptotic stable, etc.

Keep it up, this is a very valuable introduction to eigenvectors that I could see helping many people, and the application was interesting as well. Your presentation flowed and the concepts were well-organized and structured in a helpful order. Good visual style too. Thanks for the effort you put into this video, it shows.

Hm… I think that you may have been too ambitious about what you wanted to show. I think that the end result is at the same time too long and not detailed enough to help a student who would struggle with any of the numerous concepts that you present in the video (differential equation, eigenvalues/vectors, matrices as linear transformation, change of bases, linear approximation of a system, stability etc)

There are too many concepts in the video, it’s overwhelming. I know it is hard to leave some details out / things unexplained / state a result that is not entirely true, but in the case of your subject (control theory), it would have been necessary.

I see that the content is also suggested for high school, but it seems more like a university-level topic. From the perspective of a university student, the video can be a good introduction to a rational mechanics course. I recommend making shorter videos if you want them to be accessible to everyone. Since you cover many different topics, you might consider splitting them into multiple videos (some topics could be explored in depth and would make very useful videos for university students).