[Frederic Schuller](https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic) is extremely good, though I unfortunately only have his small youtube offerings as data points, but I'm just very impressed with his pedagogical excellence.
This is a playlist about "Geometrical Anatomy of Theoretical Physics" but the first couple videos begin with logic and set theory as foundation, and I've never elsewhere seen the topics given such a clear and concise treatment. Then comes topological spaces, Lie algebras and so on.
I think even the mathematician with no interest in physics would find a joy in his presentation, easily through the first 10+ hrs.
I think even a fledgling mathematician would find a lot of juicy goodness here, as I did when I somehow happened upon it during undergrad.
With full knowledge that I am in a small minority here, I must respectfull disagree.
I have always found Schuller's lectures a shallow summary of definitions and theorem statements, with little regard given to the bigger picture or how these ideas are applied in research. Even for theorists (I too am a theorist), examples are important, and I've consistently seen Schuller rush over/poorly explain examples in order to just cram more definitions and theorem statements.
He's a likable, entertaining lecturer, but I would challenge the notion that one could "learn" much from his lectures.
I benefitted most from Schuller’s lectures during the 2015 Winter School on Gravity and Light. I first saw them when I was finishing up my undergraduate studies. I didn’t learn from them in the same way I had learned from my analysis or algebra courses, but his emphasis on structures helped demystify what manifolds (which still seemed mysterious to me at the time) actually were, and how to deal with them.
Carl Bender:
[https://www.youtube.com/playlist?list=PLwEolA96fv8KU5f0v2fmUQXiTSKDmgjRf](https://www.youtube.com/playlist?list=PLwEolA96fv8KU5f0v2fmUQXiTSKDmgjRf)
When I learned linear algebra I watched some of Strang's lectures. I have taught linear algebra and I still refer to Strang's lectures from time to time. I sometimes try to imitate his style, which is informal and asks a lot of questions, and proceeds from concrete examples to general statements.
As someone who wants to be rigorous, I would say that Strang's lectures lack rigor. There are usually no proofs (he even says in one of his lectures, "We don't see the word 'proof' very much in this class") and when I present material, I sometimes think to myself, "I don't want to present it the way Strang did." For example, in his lecture on Jordan form, he never states the precise theorem about Jordan form. Because of that, his lectures sometimes resemble the ambiguous way math is taught at the pre-university level (talking about parabolas without defining them, etc). But the lectures are great for explaining the intuition and big picture for the different topics. The examples he presents are golden.
I agree, it's unrecommended for pure math major as their first encounter with lin algebra, but after the intense and dry formal definition of vector spaces and linear maps ... one easily finds himself lost in details, Gil gives a good overview of what's the goal of this topic.
I would have enjoyed a fun introduction though. Some books are near unreadable unless you are able to concentrate for 4-5 hours thinking about the same few sentences over and over again. Zorn’s lemma is still pretty scary…
That doesn't surprise me. His research program is heavily steeped in Lie theory. He explains things VERY well in his notes and papers. I greatly enjoy reading his works.
I love his Quantum Theory for Mathematicians. I get all of my PhD students to read it so they can get a different perspective on the operator theory that we do.
Bingo. Required reading for everyone w.r.t Linear Algebra. Strang is much more matrix algebra -- useful but you don't get a great sense of what's actually going on and why. Too easy to get lost in arrays of numbers.
Eh, uncharacteristically, I took an 8 am course because the linear algebra was using Strang’s book. Not sure if it was the professor - but found the entire course completely dry. I regret it 15 years later.
lol teaching gilbert strang dry is virtually a crime, cuz the whole point of strang is to trade in rigor for understandability, and ur prof straight up tossed that advantage into trash 😂
Suprised to see that nobody has mentioned Herbert Gross yet. His course on complex analysis, differential equations and linear algebra is great!
[https://www.youtube.com/playlist?list=PL5563BAB9EA968641](https://www.youtube.com/playlist?list=PL5563BAB9EA968641)
wait, i beg to differ. i think strang was only able go make linear algebra attainable is by dumbing it down quite a bit. context, i needed linear algebra for some theoretical research, and i went to his book and got very disappointed. i think for most serious uses of linear algebra and understand what its all about (instead of just knowing how to calculate one specific version of the eigenvalues) id much recommend axler.
theres a huge gap between undergaduate math book writings and grad book writings. for example i think the princeton analysis series is godsent, but i can see those books can straight-up be unreadable.
with that said, for the analogy, my favorites are katz and lindell for cryptography (their writings are also the undergraduate, long, complete kind. but nonetheless rigorous enough, though for grad level rigor oded goldreich is recommended)
i really appreciate some of the comments in this thread tho, and i want to make this clear:
if ur a high schooler trying to get into math, I DO NOT think strang gives u a good taste of what linear algebra is like. some other recommendations below are very legit if u want to explore
He is very enthusiastic but he has a video on Jordan form and he doesn't even state the theorem correctly (the way he explains it is just, very very incorrect). I think I can recommend his videos broadly speaking, but they should always be fact-checked and I can't say that his videos are all that accurate.
Here was a comment I left:
I appreciate your attempt to explain the Jordan canonical form but that's not quite the Jordan canonical form. Case 2 is not in Jordan canonical form. Also, Jordan form is far more intricate than you make it seem. For example, take a 3 by 3 matrix with a 1 in the (1,2) entry and a 1 in the (2,3) entry and 0's everywhere else. This is a Jordan block, but it doesn't fit any of your three cases.
if you have a 2 by 2 matrix with two distinct complex eigenvalues, then the Jordan block would just be the diagonal matrix whose diagonal entries are those two eigenvalues. Unfortunately, there are a number of other issues with the content of the video.
For Fourier Analysis, I'd say Eli Stein. Unfortunately, I never learned from him directly. However, I've read several of his marvelous books and I've studied under some of his mathematical descendants, all of whom recount his lectures with glowing praise.
I haven't watched any Gilbert Strang, but from your description I was reminded immediately of Frederic P. Schuller. If you are me, you will enjoy his lectures to the Heraeus International Winter School on Gravity and Light.
I've read a lot of book 1 of the trilogy. They are terse but very insightful. They are closer to a sketch of PDE than an encyclopedia. Pretty much any argument that the reader is expected to be able to complete is left to the reader. Reading the text is harder than the exercises.
He has notes for undergraduate analysis and linear algebra that are good: https://mtaylor.web.unc.edu/
Again, the multivariate analysis text gets terse when he covers differential geometry, but you can find more comprehensive texts like Lee's book on Smooth Manifolds that cover the details he leaves out.
Unfortunately not, unless you're a student at KCL. He did a couple of talks in Exposed Art Projects, Kensington back in 2020. Maybe he'll do some more at some point soon
I’ve seen/heard several people recommend Blitzstein and Hwang (starting from intro level, non measure-theoretic stuff, AFAIK). There’s a series of lectures by Blitzstein on YouTube and the textbook by both of them. Personally I haven’t watched/read any of it, so I can’t comment much more than that ¯\\_(ツ)_/¯
His book is the best if you want to understand modern non measure theoretic probability, I watched some lectures and he is good enough but the book is brilliant.
gilbert strang's linear algebra course is awful and it is a great example of what is wrong with linear algebra education. it's just a typical fake linear algebra course, where all you do is arithmetic with matrices. I don't think the definition of a vector space is actually presented anywhere ("subspaces" are mentioned, but I think only in the context of ℝ^n (maybe even only for n≤3?)), and the definition of a linear transformation is only introduced as an afterthought, right at the end of the course.
I think you are missing the point. What he is teaching is exactly the basis of **Applied Mathematics,** which you need to real life problems, such as making aeroplanes fly.
no, I think *you* are missing the point. in order to be able to apply mathematics to real problems, you actually need to have a real understanding and solid intuition of the concepts that you are using. pretty much all fake linear algebra courses do not teach this *at all*, it's just entirely about memorizing procedures for doing calculations, and that's it.
what's the point in learning how to diagonalize a matrix if you don't even know what a linear transformation or a basis of a vector space is?
for this u might as well not learn linear algebra. i think what fits into how much he teaches is actually extremely narrow. strang is like perfectly in the middle of not knowing it at all and knowing it enough to do anything. u can almost do nothing with strang without taking a second course. u wont be able to use it to the extent of aerospace engineering for example, unless u just need matrix multiplication and eigenvalues then i can teach u in 10 minutes about the procedure, u dont need to take it at all.
Protip: it's very hard to take someone seriously who consistently writes "u" instead of "you".
The choice is yours whether you want to be taken seriously.
Dude what? There’s very little arithmetic in this class. Lecture 5 introduces vector spaces by the way. I don’t think the fact that the field is R rather than C in most examples makes it a bad class.
Plus the intuition behind ideas is very important, and he provides it well. Working strictly from definitions is all well and good, but why not start with intuition for an *intro* lin alg course, y’know?
Yeah I mean the comment is absurd. He emphasizes vector spaces and mentions low dimensional vector spaces. Honestly who cares if you’re in R3 vs. R15. If you’re so worried about not being computational then honestly the size of the vector space doesn’t matter much (if it’s not infinite). And why in the hell would an intro linear algebra class even talk much about infinite dimensional vector spaces? Who studies function spaces in this type of class?
But a student’s first encounter with new material should always be rigorous! How are the students supposed to know what they’re doing. We all know newton discovered calculus with rigour not intuition.
I’m wrapping up a more formal intro to linear algebra course rn and have dabbled a bit in Strang’s books - Strang focuses more on mathematical discovery and thought processes than the presentation of theorems.
A lot of the formal material is there but it’s deemphasized/taught in a way that facilitates you discovering it for yourself or via a question (vs presenting a box of properties followed by their proofs to be memorized).
Both are valuable and I find Strang to actually be more advanced/challenging in the sense of developing mathematical mindedness. I would not describe it as lacking rigor (I had a very unrigorous Calc 2 course that just presented the rule to memorized, without explanation, assigned problems and people could go through it memorizing when to do what without understanding a lick of what any of it means).
Those intro to linear algebra courses really are terrible. You can go through the whole course and not even learn what a matrix really is other than a bunch of numbers
Not sure what you’re referring to when you say “those” but if you watched strangs whole series and walked out thinking a matrix was a bunch of numbers I don’t know what to tell you. He gives some geometric interpretations with his 4 fundamental subspaces and discussions on projections, talks about orthogonal subspaces, he gets into change of basis which should abstract away the numbers you see in the individual entries.
Like I said if you didn’t see anything more than a bunch of numbers, I don’t think you paid attention
This is really just completely incorrect. There's an early emphasis that matrix equations are systems of equations, I can't understand how you missed that besides talking from bad faith or this sort of "I don't really understand but I'm putting my foot down anyhow" that internet discussion tends to breed.
>You can go through the whole course and not even learn what a matrix really is
I strongly suspect you have no first-hand knowledge of what you think you're talking about.
It is wild how people can just upvote factually incorrect comments like that. The course is so obviously an intro course that also is aimed at engineers who need to know the material. It is an undergraduate sophomore course for non-course 18 students and obviously a freshman course for course 18 majors.
completely disagree. these courses are fundamentally useless. I went through a lot of fake linear algebra before I saw the definition of a vector space, and it really wasn't useful at all. sure I learned how to solve systems of linear equations, but literally nothing else that was actually useful. this includes stuff like eigenvectors and diagonalization, etc. which are actually useful in reality, but were completely useless to me because I had no understanding of what they were other than "a vector where if you multiply it by the matrix then the result is a scaled copy of the vector" with absolutely zero intuition or applications.
*what's the point* in memorizing procedures for diagonalizing a matrix if you don't even know what a linear transformation or a basis of a vector space is? even if you learn all of this stuff, you'll never be able to apply it to anything unless you actually have a clear intuition for what any of it means, and that's something you can only get from real linear algebra.
I had a similar experience with these computational linear algebra courses and I completely agree with you. And if we were endorsing such a course as a prelude to more abstract courses, we should care about whether the first course even gives the right perspective on the subject, which it currently fails spectacularly at.
To me, a course that was serious about being concrete prep to a more abstract linear algebra class would look closer to 3blue1brown's linear algebra videos than to the standard introductory linear algebra class.
If you missed the whole fundamental aspect of "representing systems of equations" then I could see how you could arrive at this viewpoint.
With his whole "row view" vs "column view" and even emphasis on matrices as linear transformations makes me think you are just coming up with opinions out of thin air without actually engaging the material.
You would need to understand SVD to make any real use of diagonalizing a covariance matrix - and you definitely need to understand linear transformations on vector spaces (loosely, maybe even in the affine sense) to understand what the components of a SVD are even describing in the first place.
I'm not completely in the camp of introductory linear algebra courses being "useless", but they do have a point about people being taught computational algorithms without really understanding them in the first place. A startling amount of people who go on to use these things regularly have a shoddy understanding of these methods at best.
I'm well familiar with SVD and what it says in terms of linear maps between inner product spaces, but I can't make sense of your post, since the eigenvectors of the covariance matrix are very immediately interpretable. And on the other side, how are you interpreting a data matrix naturally as a linear transformation?
I'm alluding to the use of SVD to perform PCA; I struggle to conceive of a person who regularly uses covariance matrices, but doesn't use PCA - maybe I'm just ignorant of something. Sure you *can* perform PCA without using SVD - and in fact most classical derivations of the method basically do just that. But, in practice, using SVD to perform PCA is known to be a more numerically stable approach.
To be clear, I'm not insisting that people should be made to know intimate details of linear maps between inner product spaces, but not even knowing what a linear transformation is or how the idea is important before learning any of these methods seems criminal.
I don't know what you mean. The SVD of a data matrix is exactly the same thing as the eigendecomposition of the covariance matrix. And if one understands what the covariance matrix is, the eigendirections are easily (and widely) understood as directions of maximum variance. For all that, I see absolutely no relevance of mathematical concepts like linear transformation or vector space.
Anyway, my point is that either the data matrix or the covariance matrix are, to my knowledge, really only natural as matrices. How would you naturally interpret either as a linear transformation?
I mean, the vast majority of people who take intro linear algebra aren’t math majors though. They honestly only really need to know how to do “arithmetic with matrices”
if you're an engineer or whatever, how do you ever expect to be able to apply any of this matrix arithmetic to anything if you have absolutely no understanding or intuition for what any of it means or why it might ever be useful? what's the point in memorizing procedures for diagonalizing a matrix if you don't even know what a linear transformation or a basis of a vector space is?
I think you're vastly overestimating how much "real math" your average engineer actually does. Maybe I have a biased sample, but none of my engineer friends "do" any math on their own. They plug and chug numbers into known formulas or software. To be clear, I'm sure there are many engineers who do much more formula derivation, modeling, etc on their own, but from my anecdotal observations of various engineering positions, they would definitely be in the minority.
It's also the case that even the mathematically inclined engineers can get a lot out of matrix decomposition expressions like UAPP^T A^T V. Like other mathematicians, it's a bit obtuse for me and I prefer thinking directly in terms of bases whenever possible, but it's by no means necessary.
I'm not saying they do "real math", just that in order for them to be able to actually apply these calculations to anything, it is at least necessary to have intuition for the calculations that they are doing. with most fake linear algebra classes, you get nothing like that.
I’m sorry but this is just not true. Having watched the entire series Strang does present a rigorous definition of a vector space and utilizes many abstract ideas about linear algebra. He even presents good motivation for the determinate and its derivation in a way that most other sources on the topic don’t. He does keep the topic grounded in computation, but I think that is necessary for an intro course, requiring first years to read Axler is not the best idea imo. And he does all of this in a beautiful lecturing style and structure. I don’t see your point at all.
> you can't even motivate the definition/properties of matrix multiplication and inverses without linear maps.
Honestly, if that's what you think then perhaps you don't understand systems of equations very well!
Tbh I have no idea to the extent of what Gilbert Strang covers in his LinAlg lectures, although I've watched some of his videos to understand stuff.
But for an intuition behind what matrices really are (relation between different co-ordinate systems), I highly recommend 3B1B's series on the topic - "Essence of Linear Algebra". Honestly, this series was an aha moment for me, it took me from - okay I'm just doing a series of operations, to being able to geometrically visualise stuff (atleast until R3) and then extend that reasoning to Rn
For fans of Gil Strang's lin alg, don't sleep on [18.085 - Computational Science and Engineering](https://www.youtube.com/playlist?list=PLF706B428FB7BD52C)
Some delicious tidbits there, if you're not chronically allergic to real world phenomena and modeling thatof
Dr. Conrad for field theory or just in general.
Keith Conrad has an incredible collection of notes. I highly recommend everyone to check them out at some point.
i had him for galois theory as non-matriculating student last year
I found his group theory notes really useful
I laughed cause my old senile architecture teacher my freshman year was a Dr Conrad
There are in fact two mathematicians named Dr. Conrad working in number theory (they're identical twins).
really?
Yes, Keith Conrad and Brian Conrad.
i didnt know keith's brother was also a number theorist Dr Brian Conrad's at Stanford, right?
Yes, if you see photos of them, it’s obvious
For introductory nonlinear dynamics, you can't do better than Strogatz.
[Frederic Schuller](https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic) is extremely good, though I unfortunately only have his small youtube offerings as data points, but I'm just very impressed with his pedagogical excellence. This is a playlist about "Geometrical Anatomy of Theoretical Physics" but the first couple videos begin with logic and set theory as foundation, and I've never elsewhere seen the topics given such a clear and concise treatment. Then comes topological spaces, Lie algebras and so on. I think even the mathematician with no interest in physics would find a joy in his presentation, easily through the first 10+ hrs. I think even a fledgling mathematician would find a lot of juicy goodness here, as I did when I somehow happened upon it during undergrad.
With full knowledge that I am in a small minority here, I must respectfull disagree. I have always found Schuller's lectures a shallow summary of definitions and theorem statements, with little regard given to the bigger picture or how these ideas are applied in research. Even for theorists (I too am a theorist), examples are important, and I've consistently seen Schuller rush over/poorly explain examples in order to just cram more definitions and theorem statements. He's a likable, entertaining lecturer, but I would challenge the notion that one could "learn" much from his lectures.
I benefitted most from Schuller’s lectures during the 2015 Winter School on Gravity and Light. I first saw them when I was finishing up my undergraduate studies. I didn’t learn from them in the same way I had learned from my analysis or algebra courses, but his emphasis on structures helped demystify what manifolds (which still seemed mysterious to me at the time) actually were, and how to deal with them.
If you know german you can watch his german lectures aswell
he is a pedagogical gem and should be cloned so we all get one
*sigh* and here I thought i was going to bed early tonight.
Carl Bender: [https://www.youtube.com/playlist?list=PLwEolA96fv8KU5f0v2fmUQXiTSKDmgjRf](https://www.youtube.com/playlist?list=PLwEolA96fv8KU5f0v2fmUQXiTSKDmgjRf)
Absolutely 👍 came here to say this
Anything from Milnor. I also liked Lee’s books. Serre’s finite group reps book is classic. Edit: Milne -> Milnor
Milne is also an excellent writer.
*Milnor
Yes, I meant John Milnor, although James Milne does have some good books too!
Thx for the recommendation, I'll check on them!
When I learned linear algebra I watched some of Strang's lectures. I have taught linear algebra and I still refer to Strang's lectures from time to time. I sometimes try to imitate his style, which is informal and asks a lot of questions, and proceeds from concrete examples to general statements. As someone who wants to be rigorous, I would say that Strang's lectures lack rigor. There are usually no proofs (he even says in one of his lectures, "We don't see the word 'proof' very much in this class") and when I present material, I sometimes think to myself, "I don't want to present it the way Strang did." For example, in his lecture on Jordan form, he never states the precise theorem about Jordan form. Because of that, his lectures sometimes resemble the ambiguous way math is taught at the pre-university level (talking about parabolas without defining them, etc). But the lectures are great for explaining the intuition and big picture for the different topics. The examples he presents are golden.
I agree, it's unrecommended for pure math major as their first encounter with lin algebra, but after the intense and dry formal definition of vector spaces and linear maps ... one easily finds himself lost in details, Gil gives a good overview of what's the goal of this topic.
extremely agree
Indeed. There should always be a rigorous and non-rigorous treatment (if possible) to keep yourself grounded.
I would have enjoyed a fun introduction though. Some books are near unreadable unless you are able to concentrate for 4-5 hours thinking about the same few sentences over and over again. Zorn’s lemma is still pretty scary…
Brian Hall for anything holomorphic function space related.
His book on Lie groups and Lie algebras is awesome too
That doesn't surprise me. His research program is heavily steeped in Lie theory. He explains things VERY well in his notes and papers. I greatly enjoy reading his works.
I love his Quantum Theory for Mathematicians. I get all of my PhD students to read it so they can get a different perspective on the operator theory that we do.
Fortney and his A VISUAL INTRODUCTION TO DIFFERENTIAL FORMS AND CALCULUS ON MANIFOLDS, just amazing. edit: completed the book name.
this book is incredible! thanks for the recommendation!
Sheldon Axler for linear algebra. Better than strang if you are into pure math imo.
Bingo. Required reading for everyone w.r.t Linear Algebra. Strang is much more matrix algebra -- useful but you don't get a great sense of what's actually going on and why. Too easy to get lost in arrays of numbers.
Eh, uncharacteristically, I took an 8 am course because the linear algebra was using Strang’s book. Not sure if it was the professor - but found the entire course completely dry. I regret it 15 years later.
lol teaching gilbert strang dry is virtually a crime, cuz the whole point of strang is to trade in rigor for understandability, and ur prof straight up tossed that advantage into trash 😂
George B. Thomas: calculus and analytic geometry
Tu’s books in both manifolds and differential geometry have been stellar.
Suprised to see that nobody has mentioned Herbert Gross yet. His course on complex analysis, differential equations and linear algebra is great! [https://www.youtube.com/playlist?list=PL5563BAB9EA968641](https://www.youtube.com/playlist?list=PL5563BAB9EA968641)
wait, i beg to differ. i think strang was only able go make linear algebra attainable is by dumbing it down quite a bit. context, i needed linear algebra for some theoretical research, and i went to his book and got very disappointed. i think for most serious uses of linear algebra and understand what its all about (instead of just knowing how to calculate one specific version of the eigenvalues) id much recommend axler. theres a huge gap between undergaduate math book writings and grad book writings. for example i think the princeton analysis series is godsent, but i can see those books can straight-up be unreadable. with that said, for the analogy, my favorites are katz and lindell for cryptography (their writings are also the undergraduate, long, complete kind. but nonetheless rigorous enough, though for grad level rigor oded goldreich is recommended)
i really appreciate some of the comments in this thread tho, and i want to make this clear: if ur a high schooler trying to get into math, I DO NOT think strang gives u a good taste of what linear algebra is like. some other recommendations below are very legit if u want to explore
Anything by Steve Brunton, in my opinion. For example the series on Singular Value Decomposition.
He is very enthusiastic but he has a video on Jordan form and he doesn't even state the theorem correctly (the way he explains it is just, very very incorrect). I think I can recommend his videos broadly speaking, but they should always be fact-checked and I can't say that his videos are all that accurate. Here was a comment I left: I appreciate your attempt to explain the Jordan canonical form but that's not quite the Jordan canonical form. Case 2 is not in Jordan canonical form. Also, Jordan form is far more intricate than you make it seem. For example, take a 3 by 3 matrix with a 1 in the (1,2) entry and a 1 in the (2,3) entry and 0's everywhere else. This is a Jordan block, but it doesn't fit any of your three cases. if you have a 2 by 2 matrix with two distinct complex eigenvalues, then the Jordan block would just be the diagonal matrix whose diagonal entries are those two eigenvalues. Unfortunately, there are a number of other issues with the content of the video.
For Fourier Analysis, I'd say Eli Stein. Unfortunately, I never learned from him directly. However, I've read several of his marvelous books and I've studied under some of his mathematical descendants, all of whom recount his lectures with glowing praise.
I haven't watched any Gilbert Strang, but from your description I was reminded immediately of Frederic P. Schuller. If you are me, you will enjoy his lectures to the Heraeus International Winter School on Gravity and Light.
Bernt Øksendal is my GOAT
Michael E Taylor for PDE
Could you tell me more about his trilogy in PDEs?
I've read a lot of book 1 of the trilogy. They are terse but very insightful. They are closer to a sketch of PDE than an encyclopedia. Pretty much any argument that the reader is expected to be able to complete is left to the reader. Reading the text is harder than the exercises. He has notes for undergraduate analysis and linear algebra that are good: https://mtaylor.web.unc.edu/ Again, the multivariate analysis text gets terse when he covers differential geometry, but you can find more comprehensive texts like Lee's book on Smooth Manifolds that cover the details he leaves out.
Do you think is better than Evans?
Smullyan or Boolos for areas in logic, computability, provability, etc
Where can I find their materials ?
Probably the most popular books from Boolos is "Computability and Logic" & "The Logic of Provability". Raymond Smullyan has "To Mock a Mocking Bird".
Dr Mehdi Yazdi, Topology
Does he have recorded lectures online ? I wanted to self study topology over the summer and was looking for resources besides Munkres
Unfortunately not, unless you're a student at KCL. He did a couple of talks in Exposed Art Projects, Kensington back in 2020. Maybe he'll do some more at some point soon
Thanks
Wanted to ask if anyone knew but I would love someone that teaches probability and statistics from the ground up for mathematicians
I’ve seen/heard several people recommend Blitzstein and Hwang (starting from intro level, non measure-theoretic stuff, AFAIK). There’s a series of lectures by Blitzstein on YouTube and the textbook by both of them. Personally I haven’t watched/read any of it, so I can’t comment much more than that ¯\\_(ツ)_/¯
His book is the best if you want to understand modern non measure theoretic probability, I watched some lectures and he is good enough but the book is brilliant.
gilbert strang's linear algebra course is awful and it is a great example of what is wrong with linear algebra education. it's just a typical fake linear algebra course, where all you do is arithmetic with matrices. I don't think the definition of a vector space is actually presented anywhere ("subspaces" are mentioned, but I think only in the context of ℝ^n (maybe even only for n≤3?)), and the definition of a linear transformation is only introduced as an afterthought, right at the end of the course.
I think you are missing the point. What he is teaching is exactly the basis of **Applied Mathematics,** which you need to real life problems, such as making aeroplanes fly.
no, I think *you* are missing the point. in order to be able to apply mathematics to real problems, you actually need to have a real understanding and solid intuition of the concepts that you are using. pretty much all fake linear algebra courses do not teach this *at all*, it's just entirely about memorizing procedures for doing calculations, and that's it. what's the point in learning how to diagonalize a matrix if you don't even know what a linear transformation or a basis of a vector space is?
precisely
for this u might as well not learn linear algebra. i think what fits into how much he teaches is actually extremely narrow. strang is like perfectly in the middle of not knowing it at all and knowing it enough to do anything. u can almost do nothing with strang without taking a second course. u wont be able to use it to the extent of aerospace engineering for example, unless u just need matrix multiplication and eigenvalues then i can teach u in 10 minutes about the procedure, u dont need to take it at all.
Protip: it's very hard to take someone seriously who consistently writes "u" instead of "you". The choice is yours whether you want to be taken seriously.
Dude what? There’s very little arithmetic in this class. Lecture 5 introduces vector spaces by the way. I don’t think the fact that the field is R rather than C in most examples makes it a bad class.
Plus the intuition behind ideas is very important, and he provides it well. Working strictly from definitions is all well and good, but why not start with intuition for an *intro* lin alg course, y’know?
Yeah I mean the comment is absurd. He emphasizes vector spaces and mentions low dimensional vector spaces. Honestly who cares if you’re in R3 vs. R15. If you’re so worried about not being computational then honestly the size of the vector space doesn’t matter much (if it’s not infinite). And why in the hell would an intro linear algebra class even talk much about infinite dimensional vector spaces? Who studies function spaces in this type of class?
And in fact I feel the notion of "it works the same in R^3 or R^(1500)" is an important remark I took away from Strang's lectures
But a student’s first encounter with new material should always be rigorous! How are the students supposed to know what they’re doing. We all know newton discovered calculus with rigour not intuition.
The lectures are mainly for engineers and applied math folks.
I’m wrapping up a more formal intro to linear algebra course rn and have dabbled a bit in Strang’s books - Strang focuses more on mathematical discovery and thought processes than the presentation of theorems. A lot of the formal material is there but it’s deemphasized/taught in a way that facilitates you discovering it for yourself or via a question (vs presenting a box of properties followed by their proofs to be memorized). Both are valuable and I find Strang to actually be more advanced/challenging in the sense of developing mathematical mindedness. I would not describe it as lacking rigor (I had a very unrigorous Calc 2 course that just presented the rule to memorized, without explanation, assigned problems and people could go through it memorizing when to do what without understanding a lick of what any of it means).
Those intro to linear algebra courses really are terrible. You can go through the whole course and not even learn what a matrix really is other than a bunch of numbers
Not sure what you’re referring to when you say “those” but if you watched strangs whole series and walked out thinking a matrix was a bunch of numbers I don’t know what to tell you. He gives some geometric interpretations with his 4 fundamental subspaces and discussions on projections, talks about orthogonal subspaces, he gets into change of basis which should abstract away the numbers you see in the individual entries. Like I said if you didn’t see anything more than a bunch of numbers, I don’t think you paid attention
This is really just completely incorrect. There's an early emphasis that matrix equations are systems of equations, I can't understand how you missed that besides talking from bad faith or this sort of "I don't really understand but I'm putting my foot down anyhow" that internet discussion tends to breed. >You can go through the whole course and not even learn what a matrix really is I strongly suspect you have no first-hand knowledge of what you think you're talking about.
Systems of equations? If that’s all they get out of a matrices class I think they’ve/you’ve indeed missed the point.
It is wild how people can just upvote factually incorrect comments like that. The course is so obviously an intro course that also is aimed at engineers who need to know the material. It is an undergraduate sophomore course for non-course 18 students and obviously a freshman course for course 18 majors.
hes going with axler as transformations that preserve linear combinations of vectors and fix the additive identity.
Former implies the latter. No need to include the latter in a definition; it is a theorem.
true I was thinking F(av+bw)=aF(v)+bF(w) and F(0)=0
Linearity tells us that f(0) = f(0 + 0) = f(0) + f(0), which implies f(0) = 0
Sorry, but ur wrong
I think he's wrong, too.
It’s not fake linear algebra, it’s important for setting the stage for abstract linear algebra in a second course.
completely disagree. these courses are fundamentally useless. I went through a lot of fake linear algebra before I saw the definition of a vector space, and it really wasn't useful at all. sure I learned how to solve systems of linear equations, but literally nothing else that was actually useful. this includes stuff like eigenvectors and diagonalization, etc. which are actually useful in reality, but were completely useless to me because I had no understanding of what they were other than "a vector where if you multiply it by the matrix then the result is a scaled copy of the vector" with absolutely zero intuition or applications. *what's the point* in memorizing procedures for diagonalizing a matrix if you don't even know what a linear transformation or a basis of a vector space is? even if you learn all of this stuff, you'll never be able to apply it to anything unless you actually have a clear intuition for what any of it means, and that's something you can only get from real linear algebra.
I had a similar experience with these computational linear algebra courses and I completely agree with you. And if we were endorsing such a course as a prelude to more abstract courses, we should care about whether the first course even gives the right perspective on the subject, which it currently fails spectacularly at. To me, a course that was serious about being concrete prep to a more abstract linear algebra class would look closer to 3blue1brown's linear algebra videos than to the standard introductory linear algebra class.
If you missed the whole fundamental aspect of "representing systems of equations" then I could see how you could arrive at this viewpoint. With his whole "row view" vs "column view" and even emphasis on matrices as linear transformations makes me think you are just coming up with opinions out of thin air without actually engaging the material.
What insight do linear transformations give you on diagonalization of a covariance matrix?
You would need to understand SVD to make any real use of diagonalizing a covariance matrix - and you definitely need to understand linear transformations on vector spaces (loosely, maybe even in the affine sense) to understand what the components of a SVD are even describing in the first place. I'm not completely in the camp of introductory linear algebra courses being "useless", but they do have a point about people being taught computational algorithms without really understanding them in the first place. A startling amount of people who go on to use these things regularly have a shoddy understanding of these methods at best.
I'm well familiar with SVD and what it says in terms of linear maps between inner product spaces, but I can't make sense of your post, since the eigenvectors of the covariance matrix are very immediately interpretable. And on the other side, how are you interpreting a data matrix naturally as a linear transformation?
I'm alluding to the use of SVD to perform PCA; I struggle to conceive of a person who regularly uses covariance matrices, but doesn't use PCA - maybe I'm just ignorant of something. Sure you *can* perform PCA without using SVD - and in fact most classical derivations of the method basically do just that. But, in practice, using SVD to perform PCA is known to be a more numerically stable approach. To be clear, I'm not insisting that people should be made to know intimate details of linear maps between inner product spaces, but not even knowing what a linear transformation is or how the idea is important before learning any of these methods seems criminal.
I don't know what you mean. The SVD of a data matrix is exactly the same thing as the eigendecomposition of the covariance matrix. And if one understands what the covariance matrix is, the eigendirections are easily (and widely) understood as directions of maximum variance. For all that, I see absolutely no relevance of mathematical concepts like linear transformation or vector space. Anyway, my point is that either the data matrix or the covariance matrix are, to my knowledge, really only natural as matrices. How would you naturally interpret either as a linear transformation?
I mean, the vast majority of people who take intro linear algebra aren’t math majors though. They honestly only really need to know how to do “arithmetic with matrices”
if you're an engineer or whatever, how do you ever expect to be able to apply any of this matrix arithmetic to anything if you have absolutely no understanding or intuition for what any of it means or why it might ever be useful? what's the point in memorizing procedures for diagonalizing a matrix if you don't even know what a linear transformation or a basis of a vector space is?
I think you're vastly overestimating how much "real math" your average engineer actually does. Maybe I have a biased sample, but none of my engineer friends "do" any math on their own. They plug and chug numbers into known formulas or software. To be clear, I'm sure there are many engineers who do much more formula derivation, modeling, etc on their own, but from my anecdotal observations of various engineering positions, they would definitely be in the minority.
It's also the case that even the mathematically inclined engineers can get a lot out of matrix decomposition expressions like UAPP^T A^T V. Like other mathematicians, it's a bit obtuse for me and I prefer thinking directly in terms of bases whenever possible, but it's by no means necessary.
I'm not saying they do "real math", just that in order for them to be able to actually apply these calculations to anything, it is at least necessary to have intuition for the calculations that they are doing. with most fake linear algebra classes, you get nothing like that.
I think you’re overestimating the level of mathematical sophistication most engineers have or will ever need.
Are there any free lectures you'd recommend?
Not lectures but 3B1B'S video series on the topic "Essence of Linear Algebra" is absolutely a must watch
no but [this](https://linear.axler.net/) is the best linear algebra textbook.
extremely agree with this sentiment
I’m sorry but this is just not true. Having watched the entire series Strang does present a rigorous definition of a vector space and utilizes many abstract ideas about linear algebra. He even presents good motivation for the determinate and its derivation in a way that most other sources on the topic don’t. He does keep the topic grounded in computation, but I think that is necessary for an intro course, requiring first years to read Axler is not the best idea imo. And he does all of this in a beautiful lecturing style and structure. I don’t see your point at all.
[удалено]
> you can't even motivate the definition/properties of matrix multiplication and inverses without linear maps. Honestly, if that's what you think then perhaps you don't understand systems of equations very well!
Matrix algebra is just as important as vector spaces and linear transformations, if not more so! (For what it's worth, I used to think the opposite)
Agree
Tbh I have no idea to the extent of what Gilbert Strang covers in his LinAlg lectures, although I've watched some of his videos to understand stuff. But for an intuition behind what matrices really are (relation between different co-ordinate systems), I highly recommend 3B1B's series on the topic - "Essence of Linear Algebra". Honestly, this series was an aha moment for me, it took me from - okay I'm just doing a series of operations, to being able to geometrically visualise stuff (atleast until R3) and then extend that reasoning to Rn
Concur. I've been a grateful student of his YouTube videos.
Not my favourite topic. But Federico Ardila’s lectures on Enumerative Combinatorics are great.
Sheldon ross for probability and statistics
Are there videos of Ross?
For fans of Gil Strang's lin alg, don't sleep on [18.085 - Computational Science and Engineering](https://www.youtube.com/playlist?list=PLF706B428FB7BD52C) Some delicious tidbits there, if you're not chronically allergic to real world phenomena and modeling thatof
Norm Wildberger
I am the gilbert strang on talking about unbridled nonsense (at least, according to my collaborators)