See, this is what I love about this channel - it's just the same attitude I've always had. I can understand something and work out the formula when I need it much more easily than I can memorise a formula, and about half of my professors at uni just didn't get that. I flunked a whole class on calculus because I couldn't internalise the maths until a friend described it in terms I understood and it clicked.
5:49 _"usually a matrix takes a vector and transforms it to some other vector"_ The matrix takes actually a coordinate vector of some vector v and transforms it to the coordinate vector of f(v) for some linear map f (represented by the matrix). Note that the coordinate vectors of v and f(v) can be with respect to different bases, even if f maps to the same vector space. This becomes important when we want to understand that the matrix Q is a "real" matrix after all, it is the matrix representing the identity map with respect to different bases for domain and target (which are both the same space).
This series is truly amazing, you're doing an awesome job! Thanks to you after more than a year after I was introduced to linear algebra (by a series of 3blue1brown) I've finally understood the change of basis. You've been explaining everything very clearly and you put emphasis on intuition, which is really important in mathematics in my opinion, and not many people are capable of doing it as well as you do. Thank you for making this series!
As someone who teaches mathematics at university level (to math majors), I think you are doing quite fine. I (like to) think that most of my colleagues who teach math majors do actually explain this stuff, but probably in other STEM fields, due to the lack of time to cover it, lots of these intuitive explanations get skipped over. So, please keep making these videos, they are becoming a good, casual, reference to the subject. You are a huge help!
Thanks very much! I appreciate it a lot coming from someone who teaches this properly. I did the easier linear algebra course at university because at the time I thought I hated maths (actually linear algebra was one of the things that convinced me that I don’t and soon after I switched to a maths major). I haven’t got a clue what the harder linear algebra class did, but I think even my class was pretty good with intuition and all mostly. Still, a lot of students seemed to struggle and I think that this was probably because they were too concerned with answering questions correctly to listen to the intuition! I suppose it’d be pretty different in a linear algebra for pure mathematicians class?
Thanks very much!! Actually, even though I enjoyed linear algebra very much, after a few years of not using it I think I had a lot of similar issues which meant I had to relearn it. I’m really really glad these videos help you brush up!
Absolutely! I think it should be a mandatory thing in classes to go back to what you learn a couple of years ago. With your new and nuanced understanding, a lot of ‘basic’ topics are absolutely fascinating. I got interested in QM itself because I decided to go back and learn it again after undergrad.
Well, the first course in linear algebra can be arranged in many different ways and still make sense. The topics covered are usually vector spaces, matrices, linear equations, linear operators, spectral theory, diagonalization (all finite dimensional, ofc). In pure mathematicians class you'd cover a lot of theory and are required to learn the proofs, so exercises and formulas actually do make a lot of sense if you are paying attention. As I've said, you are doing a good job of explaining why stuff works the way it does, especially since you immediately established the connection between matrices and linear operators, which is the most important thing in the subject. Mathematicians often say that one can never know too much linear algebra, it appears literally in every other math field.
From the time I'm writing this comment I have yet to do an assignment due in 3 days. I haven't watched lectures (hard to follow), I'm out of hope and keep questioning my existence but this video has given me a slight glimmer, a slight chance. Thank you
This series made me literally understand all the unconnected facts i had from my uni course and it finally made SO MUCH SENSE. It makes so much sense it made me have my own amazing insights over lots of things you haven't covered!!
This is the best explanation of change of basis I've seen. Ever text book I've read talks about changing one vector into another. This is the first time I've seen someone say that it's the same vector described a different way. ( which is what change of basis is really about). Thank you!
I think having the multiple choice questions built in the video is an awesome idea, because it gets us to actively think about and apply what is being taught and to gauge how much we really understand. Instead of passively watching and possibly forgetting the content soon afterwards, it helps us retain the information. Thanks for the videos!
@@LookingGlassUniverse Sorry for the late reply, but it should absolutely work with QM videos, since QM is so famous for being unintuitive, and so it makes sense for viewers to check their understanding while learning it.
Belated reply to the third homework question: I knew some linear algebra before, but this was still informative and also really inspiring. The problems were all straightforward in a good way, and I was utterly delighted when I realized you stuck an information-destroying transform into the initial video on transforms to prepare everyone for when matrices with no left-inverse turned up later on. Speaking of which: the concept of a left-inverse was incredibly clarifying. Thank you for including that.
This is how I was taught to think about linear algebra by my great prof in undergrad, and ever since, I've felt like I've had a leg up on everyone who didn't take the time to think about the subject as much or didn't have him as a prof. Seriously, I still run into people in grad school (currently a physics grad student) who struggle on certain problem sets because they never were taught to think about linear algebra the "right" way years ago. I just wanted to share this so that other viewers know how lucky they are to have your videos. Really great stuff!
That's so kind of you to say- thank you! I agree, linear algebra makes so much sense this way, but most people don't seem to know this version. Good luck with grad school!
@@LookingGlassUniverse It's also interesting to think about how the change of basis matrix may be interpreted as representing its own abstract linear transformation, in addition to the perspective you take here where it's just a machine that changes basis but leaves the abstract vector the same. I'm still not really sure if this is a widely useful way of thinking about things. I suppose it's the difference between an active and passive transformation?
I absolutely hated matrices and vectors before your series, now I've found myself playing around with them voluntarily. These are great videos, thanks for putting in the effort to make them.
Linear Algebra is the most fun subject of my first semester in college and your serie of LA is helping me a lot to absorb the subject and have fun with. To understand the concept behind helps a lot and Ive been indicating the series for all my friends. Thank you!
5:46 There is nothing strange about this matrix. Q_A->B never acts on the original vector but its representation in basis A. Suppose the vector Alice wants to communicate to Bob is v. Put the basis A into columns of matrix A, do the same for B: v = A [v]_A = B [v]_B Q acts on [v]_A and transforms it to [v]_B. Q doesn't act on v. Usually a matrix takes a vector and transforms it to some other vector, same thing happening here ([v]_A and [v]_B definitely qualify as vectors, though their nature depends on the underlying scalar fields, whereas v doesn't.) So I think this unnecessary distinction might be counterproductive, though it's probably worth mentioning that Q is the matrix of the identity transformation v -> v from its domain using basis A to its range using basis B, this is probably relevant to the point you're trying to make here. Which led me to think that the necessary distinction is between the case of change of basis for a vector vs change of basis for a linear transformation. To represent a vector you need one basis, so the process of changing basis for a vector involves a pair of bases (say, A and B), this video mostly deals with this. To represent a linear transformation you need two bases (the input basis for the domain, the output basis for the range), so the analogous process involves two pairs of basis (so 7:22 is a bit handwavy). (As mentioned above Finding the change of basis matrix for a vector can be thought of as finding representation matrix for the identity transformation using the pair of bases A, B.) So at this point it kinda annoy me whenever people talk about "changing basis of a matrix", what basis? do you mean changing from one pair of bases (where the input and output bases are usually, conveniently, implicitly assumed to be the same) of a LINEAR TRANSFORMATION to another pair (also usually, conveniently, implicitly assumed to be the same) and find out how the representation matrices are related? The most general case was covered in the last exercise in this video but I think it should be the main point of the video, introducing the special case first confused me 😞 and it took me awhile to realize that by allowing input and output bases to be different any linear transformation can be represented by a diagonal matrix, which should have been where the video ends. #END RANT 😤😤😤 TLDR Love this video, I would never thought about these stuffs if it weren't for your exposition. Wonderful!
Thank you! I was so confused by this in class bc they just gave a formula and explained in a way that I totally didn't understand, but now it makes sense! yay! I think it goes to show how when people think they suck at maths, it is probably just bc they aren't learning/being taught in a way that works for them.
This linear algebra series is very fun, educational and useful for everyone interested. On basic QM theory I hope you get to talk about von Neumann entropy but also entanglement and entanglement entropy. And product spaces, s.a. the tensor product of Hilbert spaces etc. Now on the video... I have a question that I sort of know the answer to, but I want to put it out there just in case. There is an assumption about the underlying space, as in position space, not the vector space, in which the vectors are embedded. It's sort of assumed to be Euclidean. The norm of a vector is assumed to be the Euclidean norm. What happens if these vectors live on a space with curvature, such as a sphere or high-genus surface (for 2 dimensions) among countless others. Would we not need to introduce a metric? What happens if the space has creases, tears, holes or conical points - or their higher-dimensional analogues. Oh and I have a second question, that is the vector space itself can be a topological space with a non-trivial topology. And the set of linear transformations also form a topological space. You get the idea. How is it best to exploit that. I'm so sorry.
_"There is an assumption about the underlying space, as in position space, not the vector space, in which the vectors are embedded."_ In general, there is no "underlying space".
Michael Sommers Yeah there is, in two completely different senses as well. One is implied by the definition of the vector norm, the other from the whole set of vectors.
You are assuming that all vectors are geometrical objects; they are not, at least not the way you mean. I suggest you look up the definition of a vector space. If you do, you will see that there is no "underlying space" required.
Ok, first of all some friendly advice, don't tell people to "look up" basic information, it comes across as rude and patronising. Secondly, and you seem to not have understood that there are two ways in which vectors can be related to geometry. First of all, normed vector spaces can be construed as metric spaces. So there is a link there. Secondly, vectors can act on objects that exist within a metric space. These are two different ways in which vectors can be related to geometry. (Although it's not exactly geometry, but that's a different matter)
Another thing that is rude and patronizing is unsolicited "friendly advice". I said to look it up because it is a bit too complicated for a TH-cam comment, and so that you would not have to believe what I said. What you have not understood is that I am not saying that no vector spaces can be interpreted geometrically. I am saying that not all are or can be. Finding examples of geometric interpretations of particular vector spaces does nothing to disprove my statement. Again, look up the definition of a vector space. If you do, you will see that there is nothing in it that requires a geometrical interpretation.
For some reason your channel is my favourite math/physics one on yt. And there are a lot of great ones. Guess it's the mix of questions, problems, solutions and your nice voice plus alice and all of the funny characters. Are you in postgrad? How's it going? What's your research field (I know it has something to do with quantum, but could you be more specific)? Could you do a video about this? Bye! I'm getting back to my rabbit hole. I'm late. :)
I just don't get it. Why can't they just have the same basis? Then they would not need all this. How can there be to different sets of basis vectors? In which real world application do we use this?
Wow. Even after Highschool + Undergraduate education and a lot of physics and maths videos, I couldn't answer one of your questions.. now my curiosity is sparked again!! (y) SUBed.
Hi, I have a question. At 7:28, why did we have to take a vector from Bob's language, convert it into Alice's language, track the transformation, and then convert it again to Bob's language? Why couldn't we just take any vector from Alice's language, apply the transformation and then covert it to Bob's language? Wouldnt the latter convey the same information i.e., the transformation applied in Alice's language? Thanks in advance.
I really like this series on linear algebra. I have done this stuff during my studies and a lot of the things you make clear - especially the (hidden) role of bases, eg Matrices being basis dependent - caused massive confusion when I was studying. Keep up the great work!
Hi there. I got hooked on your quantum videos and I was waiting until your next video to mention something in hopes you would be able to read this. You have by far provided the best, easiest to follow, and funniest series to explain quantum mechanics and so many details surrounding it. I hope you don’t mind me saying even though that isn’t the focus on this video. I still have to finish this one really. Looking forward to it.
I really liked this video. It's very fortuitous that you made this video because I have a test next week on change of basis. Thanks for helping me out!
It's interesting how there's an infinite regress problem in the example scenario, because in order to perform the transformation Alice must know what Bob's basis is, which requires them to communicate vectors from one to the other, which requires them to solve the original problem already before continuing.
I know its too late for a feedback,but still the pace was perfect, the illustration was to the point ,i liked when you make M_a said to co-ordinates of in base B as "no thanks" ,and take the base A. I really dream that one day you will push the series towards advance linear algebra topics ,like complex vector spaces or even upto spectral theorem,who knows !
Alice and Bob are working with vectors. Alice and Bob speak different languages. Bob needs Alice's help with a problem e.g. he wants to rotate a vector by 30 degrees, but the resultant vector must be in his language. 3 key steps: 1. Translate the problem from Bob to Alice's language (Q); 2. Alice uses her version of 'rotate 30 degrees' matrix to left multiply onto that translated-vector to rotate the vector (A) (yes still legit bcos the vector is in space. Alice and Bob are looking at the same vector in space. It's just that the x/y/z axes are different written in a different 'language'); 3. Translate the vector back into Bob's language. (Q^-1) By language, I mean the basis. Written in different language means the vector is a coordinate vector of a different basis/the x,y,z axes are different
So basically a change of basis is essentially just a transformation matrix to transform a vector from one coordinate grid to another. Not really into the maths of quantum mechanics, but this is also very crucial for doing any sort of 3D work as you are guaranteed to be transforming coordinates from world space to view space and understanding the difference and why you do it is pretty crucial in order to further understand the tricks you use to reduce the amount of objects/points that are rendered and how objects are given depth despite not really having any.
I really like the multiple choice checks. Complex enough to check understanding but not so complex that I need to go find paper. Also I think you still owe me a promised answer to why gravitational interaction doesn't count as an interaction that would collapse a superposition.
🙈someone remembers I’m supposed to answer that! I worked out the answer at some point- but I think I need to go figure it out again. But yes, I’ll cover it in the measurement video :)!
I think this can be improved by explaining/demonstrating how can this be used, not with just a linear algebra problem but with a physics/"real world" problem that doesn't begin with "bob is using a different basis" but with more context as to why he's doing that and what is it good for..
That is coming (and if you watch the video carefully you will see a few hints of how it is applied). You can't put everything into a ten-minute video. Have patience.
Thank you very much for the insightful lecture. If it is not too much for asking, I was hoping you could please have a lecture regarding SVD. As I understand (probably incorrectly and hence requesting a lecture) as it closely ties in with the idea of changing the basis vectors which makes the original vector is an Eigenvector such that when the matrix multiplication is performed with the vector in the new space, the direction remains the same. Then we come back to the original space by taking an inverse of "change of basis vector". Thank you.
Your videos have always been good, and they are just getting better and better. This linear algebra series has been especially good. I have (I think) three linear algebra books. Two are quite abstract, and one is very concrete (the first two chapters are on Gaussian elimination and systems of equations, and linear transformations are relegated to a very short appendix). You are steering a course between those two, which is just right for physics.
Thank you very very much- that’s so kind. I really do appreciate you saying that since you’ve been following my videos for so long. That’s the way I was hoping to go! Add enough of the pure side to clarify but still have the intuition from examples. Glad it’s striking that tone!
wow thank you really much I'm a physic student and i really only plugged in the formula for basis chance until now, telling myself that i already understood it and just don't want to go through another time. really thank you verry much sorry for bad English
So one and the same Matrix can have different meanings depending on context ? Either its a linear transformation and transforms a vector or it tells you what the same vector would be if it were expressed in the new basis vectors had the old ones been transformed ?
Some really carefully considered pedagogical notation, illustration, disguised as whimsical fun! Great stuff! A lot of work hidden in these videos, I think. ( BTW basis, plural > bases. 'basees')
I really love this series; I'm not sure if there's too much more to say. I'm sure you're getting lots of comparison to 3B1B's linear algebra series, but it's always good to see things explained multiple times in multiple ways, especially since this series has an underlying quantum mechanical slant to it that makes it different from that series (and I can't place my finger on exactly why I feel that way - maybe it's just a psychological bias that I _know_ you're going to be linking back to quantum mechanics so I'm more alert for references, or maybe there's something really there, like the focus on vectors "really" being just a linear combination of basis vectors, or something). Either way, always looking forward to future videos!
Thank you so much for the kind words! I’m glad you think that there’s a quantum slant in these videos. I think QM has quite a different emphasis in linear algebra than the usual one (like you were saying with the matrices being linear operators etc) so that’s why I wanted to make them, even though 3b1b’s series is already excellent. Hopefully the quantumness will start to become more apparent soon!
I was listening to Information by James Gleick and suddenly when talking about quantum information there appears Alice and Bob. I thought "I heard this somewhere...". Maybe these are common names in examples in English language, but it was funny to remember this channel.
This series is amazing!! Please keep going with your great job about Linear Algebra. If you give us more exemples in physics would be awesome. Thanks anyway 👍
Just a little observation. The transformed vectors used in minute 3 a1, a2 should have their name switched, right? As it is, it is saying b1 is getting mapped to a2 and b2 to a1.
This series is great . I felt like I understood change of basis before but the explanation was so clear, I definitely would have taken WAY more time answering homework number 2 numerically instead of with the intuition you showed in the video . Keep it up ^u^
I'm surprised you didn't talk about a transposed matrix. In 3D, if you have a matrix of the three bases in global space (forward, up, right) and you want to transform a vector to be relative to that object, you just need to transpose the orientation matrix to get the inverse (flip it row to column). In plain english, in a flight simulator, if I want to make a cockpit gauge to show where the horizon is, I would use the transposed matrix to transform the gravity vector from world space to relative to the instrument panel and use that to display the artificial horizon. It's super simple and way faster than calculating the full inverse (which is a huge deal in a 3D game). After watching the next video, those would be called "orthonormal basis vectors".
Your videos are absolutely amazing and very interesting to follow, they ver really really good from the beginning and they get better and better every time. please do not change anything... I cannot wait till you get us to tensors! Or won't you?
I don't know what could make this series better. Possibly harder questions since almost everyone who answers them knows the answer easily? But that could just be a sign that we have internalized the lesson easily. I don't suggest harder questions for there own sake.
I really like this series. In fact, you inspired me to start the linear algebra course in brilliant.org! I have just one question about this video, and that is: what happens with diagonalization? I thought the point of all these change-of-basis stuff was to find the diagonal matrix. But that's all for me, I love your videos and I really appreciate your effort to give us some intuition on it!
Oh really- yay! Do you like the Brilliant course? If I do sponsored videos in the future I only want them to be ones you guys like, so I’m curious what your thoughts are. Good question about diagonalisation! That’s changing the basis of a matrix so that it is now written in its ‘eigenbasis’. Have you come across that idea in the Brilliant course?
honestly i ve never quite understood why people insist on distinguishing changes of basis from other linear transformations quite so rigidly-it is just another one of many perspectives on the same thing. as for the speed your going, i think the only people that can actually answer that adequately are those who have never learned about linear algebra before. as for me i really just learned this from the not dissimilar 3b1b series and i found the videos on basis matricies and linear transformations to be by far the easiest to follow throughout that series as they require much less intense geometric thinking which is often the most difficult part.
i do not really have a good answer to that.once you kind of understood what is going on the geometry is an asset, but getting to that point is much more difficult. i suppose good visualizations go a long way already. though context can be immensely helpful too. a great example of this is chapter 11 of 3b1bs series i mentioned before, by giving more context it makes things a little more complex but simultaneously allows for a deeper understanding which helps provide confidence in your understanding
i didn't studied in college and skipped through book pdf...i remember there were words like span..etc...i thought not important ..but wow it was really surprising to me
Okay I dunno if it was god Playing Dice or stuff but i was lucky to have these videos here.....i want you to continue making such videos.... definately i can pass my semesters....by your videos :)
That's true, he went too far with his "I won't show you calculations and exercises" approach, glad we get those here as they are so crucial to understanding..
@@borg972 in the end, it's practice and making mistakes that are the greatest teachers. although he does encourage us to practice on our own every chance gets
If I had to guess, I would say chemistry 1 & 2 (I don't think there's a 3 in most colleges), physics 1,2, and 3, calculus 1,2, and 3, differential equations, multivariable calculus, linear algebra, and perhaps some complex analysis. But a lot of it will be working with quantum mechanics, which will be either its own course or part of physics, depending on which college you go to. All this being said, if your college or any future colleges you're thinking of have quantum chem as a major, then you could just check the course requirements. A lot of these courses can be skipped if you take/took the AP versions in high school, so it shouldn't be as crazy as it seems, at least in the beginning.
@@dipanitajana6213 You're welcome! And like I said, that's just me guessing. And if you do get a hold of the course load and it looks intimidating, don't worry - all of the subject matter these courses is pretty fun amd interesting and it isn't too hard to understand topics like these on this channel.
I feel that your Linear Algebra (LA) series is slightly better than 3blue1brown's, because not only do you cover the "intuitiveness" of LA, you also show much clearer examples to explain how the transformation (or anything else) is evaluated. 3b1b does a "pause and ponder" moment as well, but your mid-video polls are much more helpful in making the target audience understand LA, especially those who are weaker in it. Although your videos are much heavier in content and tend to be pretty long, someone who's determined to pick up LA is going to learn quite a lot from your videos. 3b1b attempts to do the same but sometimes he gets too far away from the theoretical aspects and doesn't return to the solving problems (mainly because he uses cool animations to explain which looks way better but when you think back about it you can't really remember much, whereas you use caveman methods like whiteboard marker and toothpicks to show your basis vectors - presentation is not that good but much more impactful). Though I've taken an LA course and hence not very qualified to judge your videos and his, your videos have made me learn a little more than his, and that's what I have to say. I've tried teaching LA to some people and I personally feel that watching your videos would have been way more effective than my own way of teaching. Anyway, all the best in your course and keep these videos coming!
Thanks so much for your detailed comment! I’m really flattered that you could compare my videos to 3b1b’s favourably at all so thank you very much! But as you pointed out, our videos have different purposes. I think his videos are aimed at people who get it at the level school taught them, but what to get it deeper and see the real beauty of the subject. I dig that. But I was aiming at a very different crowd (ironically, not my usual audience). I love teaching people who ‘hate maths’ and don’t get it and these videos are more for them- while at the same time I’m trying to give a perspective that isn’t usually taught so that even more advanced students can learn something small. I made these videos with the intention that I have to give clear simple take home messages and I needed to make them stick ....which I still need to work but I’ve been experimenting with things. The midroll polls where the main one- so I’m glad you think those help! But yeah, another was specifically the crappy ‘graphics’. There’s some research that suggests that when notes are given in hard-to-read fonts students retain more because they have to engage more. Similarly, if you give students some still images versus an animation of a process, they learn more from the images because they need to figure out themselves what it would look like animated. So while I don’t animate in (large) part because I can’t, the toothpicks that are hard to see and visualise were kind of a conscious choice. Again, that’s no disparagement of 3b1b, because his main aim probably isn’t to have it stick in student’s minds. It was to show how beautiful and visual these things are. Anyway, you seem to be someone who is able to see and judge how effective various teaching techniques are. Do you have any advice for me? I’m really keen to experiment
Hello, and thank you for this great video! I 've been strugling a little with the notations of the change basis matrix, can we say that (Q_A->B)^(-1) = (Q_B->A)^(-1), meaning that the index _A->B is just an indicator but the matrices are the same finaly, it's just Q?
Just before I watched this video, I was in the tram and apparently there was a girl who was nervous because she was about to write an exam in linear algebra. Maybe I should have calmed her down by saying that linear algebra is so fun that there's nothing to be afraid about.
I haven't been a tutor in linear algebra this semester, so I probably won't find out. I'm in Germany, so I'm far away from Melbourne, sorry. Why do you ask?
We have holidays here, too. That's when exams are usually written. (Officially, it's not called holidays, but instead lecture-free time, but I don't think holidays exclude doing maths.) I don't know how they do it in other countries though. I'm not a supervisor either, I'm just a university student. In the last winter term, I was one of the tutors for linear algebra. Would you believe that only a fraction of all students was able to solve a system of linear equations in the first exam?
Ah that’s just cruel to have exams during ‘holidays’ haha! It seems sadly to be the case everywhere that people don’t really like or try to understand linear algebra in university :/
Wow, great sweet channel you have here. c: I wonder why you sometimes have a german Title. Bist du Deutsche? However, i guess i ll have to work on my vektor skills, if i m going to study soon. I also have a mathematical quastion. I just had a weird coincidence, when i was calculating something and it blew my mind. I wonder, if you can picture a pattern, why it happens how it happens, or if it actually stops. I calculated 20 : 19.6=1.020408[...] so you get out, what seems like an infinite 2^x sequence in the decimal place, which occurs every 10^-2x. You can also do this with 19,96 or add more 9s as you like infront of the 6, which just changes 10^-(2+y)x where y would be the number of 9s. I wonder, if there is an end to it, or (what would be much more exciting) if there isnt one, which would mean it would go infinite. Such a weird infinite thougth, because it would be infinite, but not really infinite, that it goes beyond the decimal place. I guess because 10^-2x gets smaller faster, then 2^x can grow. Also there seems to be a weird pattern behind this, which probably has to do with some diversibility rules, but i couldnt really figure out which, because what has all this 20 : 19.6 stuff to do with 2^x and 10^2x?? It just blew my mind, and i wonder, if you can help me out. Have a nice day. ^_^
The fraction 20/19.6 can be rewritten as 20/(20-0.4), then as 1/((20-0.4)/20), then as 1/(1-0.02). This form corresponds to the sum of a geometric series 1+r+r^2+r^3+... =1/(1-r) . The first terms will be far away enough so as to not overlap when you sum them, so you see them in the result of the division. But as you raise to higher powers, they begin to "collide". Btw, where have you seen german titles? Only some names appear.
Thanks for that nice explanation. :) Oh and these "german titles" are just a weird youtube translation. So dont mind that. :D Btw. i noticed, that when you take all "9s" away, like 20 : 16 or 20/20-4 or 1/(1-0.2) you ll just get 1,25. Which means all these collisions actually add up to 0,01, which is somehow really cool i think.
Homework: Well, I understood this 3rd video and I did OK on your 2nd (about inverses)... maybe with a small gap here and there. Why? Because surprisingly I struggled (don't know if I still do) in your 1st introductory video, which theoretically should have been the easiest (no?). Linear Algebra was always a subject that confused me, and, not to blame you, but your 1st video brought up those feelings buried long time ago, to the point that I was a bit afraid to watch your last 2 videos (but I'm glad I did it) and I'm still postponing re-watching video #1. In summary, to me, when it comes to linear algebra the slower and the more examples the better. Other than that where did the red nails go ? :)
Thanks very much for the feedback!! That’s very useful! Would you be able to explain what made that video difficult in your opinion? Then I can work on improving!
So, let's see, how much time do you have? :) Ok, 1st let me say sort of what "Halberdier" told before: have fun doing your videos and don't be so hard on yourself. I mean, it can be, probably is, totally my (others) fault not being able to understand this subject, and maybe you just can't do nothing about it. I know it can be really frustrating for you, but such is life. Imagine you'll be teaching monkeys linear algebra ? Yeah, right ! No matter how hard you try I doubt that any monkey would get it... Maybe I (we) are just not smart enough like monkeys. Having said that, and after overcoming my fear I re-watched the AB=/=BA video again. And the good news are that I am not a monkey! I feel much better now :). But it is still sinking in... What made the video difficult: 1) In college I ended hating the subject. Didn't understand the purpose, so I just memorized enough to pass, and even without studying I passed the exam, and after that I quickly wiped out the all (almost) subject. 2) In college I think their approach was introducing matrices as systems of equations. Your approach is radically different, and so their was maybe some resistance / brain rigidity on my side. 3) On your side maybe your speed was too fast, at least for me, but I understand this is youtube (max 15min videos ) and I can always rewind as many times as I wish. On the other hand you may consider breaking the subject into more videos, each covering less, but with more examples (not just breaking the actual video into 2 shorter videos) 4) The 1st time I saw the AB=/=BA video I lost you completely at the 1st pop quiz at 3:20. What confused me then was that you all the suddenly introduced a V2. And I remember I was already confused cause I was not grasping what did you mean by applying a matrix (due to #2 above), such that I thought the rotation pop quiz was not linear. I guess I was interpreting it more like a change in coordinates instead... 5) At 11:46 I still struggle a bit. Why A(b11 b21) = b11(a11 a21) + b21(a12 a22) ? (this is a rhetorical question only) I see a striking similarity with 6:58 (wich I understood), but something did not click yet in my brain... What could you have done different here ??? Huh I don't know...could you have done the all example geometrically? Would it even help ? I think in my case it probably would help, but you never know 6) Another thing I feel it helps is to show the broader picture on how linear algebra is useful. I know you mention is used in quantum mechanics, but just like that it sounds like me saying linear algebra is used in Mathematics... What I think it helps is trying to extend the new field from something that is already known, and show how much more we will be accomplished with the new knowledge. Sort of like when you teach kids the rational numbers by extending the concept of integers by simply dividing 2 integers, which later you further extend with the introduction of irrational numbers, etc, etc Obviously here you have to assume some degree of knowledge for you audience. Bottom line it is very hard for you to please everyone and for you to put yourself in everyone else's shoes. Also it is very easy when teaching about some subject you know well to skip steps that for you are obvious, but that for the listener are not. I call it the teaching syndrome, because teachers as they repeat year after year the same thing to students they lose the notion of the student side. Holy crap, I wrote too much. Please don't take it wrong cause you're doing a great job. Cheers.
justpaulo Thank you so so so much- this is the most detailed feedback anyone's given me on a video! I'm going to go through them carefully and understand what I should do differently in the future. Thanks for being so nice about it! I'm glad watching a second time helped a bit. I have a plan for a project that might be fun to show everyone how linear algebra is useful, and to help people get familiar with it. Maybe that'll work? We'll see!
Good video with good intuition but I have a point of confusion. When I tried googling how to represent a linear transformation in different bases than the one we are currently working with, I get two equations. The first is the one that you derived in this video: M_B = Q*M_A*Q^(-1) The other is M_B = Q^(-1)*M_A*Q See this wikipedia article on similarity transformation: en.wikipedia.org/wiki/Matrix_similarity Could you clarify the differences in these two equations? Thank you! Is the difference simply due to the fact that we are changing from basis A to basis B in one equation and from basis B to basis A in another equation?
I've seen a few different series on Linear Algebra, and this is definitely the easiest to follow. Thanks so much for making these :)
Thanks very very much, that's beyond kind!
Watch 3Blue1Brown...
Some professors should get fired and replace with your lecture instead for real. Thank you for these videos !
See, this is what I love about this channel - it's just the same attitude I've always had. I can understand something and work out the formula when I need it much more easily than I can memorise a formula, and about half of my professors at uni just didn't get that. I flunked a whole class on calculus because I couldn't internalise the maths until a friend described it in terms I understood and it clicked.
I was doing linear algebra till now mechanically just applying formulae everywhere,but your videos really helped me.
Ah! This makes me so happy!
Isn't that what indian education system is all about?
5:49 _"usually a matrix takes a vector and transforms it to some other vector"_
The matrix takes actually a coordinate vector of some vector v and transforms it to the coordinate vector of f(v) for some linear map f (represented by the matrix). Note that the coordinate vectors of v and f(v) can be with respect to different bases, even if f maps to the same vector space. This becomes important when we want to understand that the matrix Q is a "real" matrix after all, it is the matrix representing the identity map with respect to different bases for domain and target (which are both the same space).
This series is truly amazing, you're doing an awesome job! Thanks to you after more than a year after I was introduced to linear algebra (by a series of 3blue1brown) I've finally understood the change of basis. You've been explaining everything very clearly and you put emphasis on intuition, which is really important in mathematics in my opinion, and not many people are capable of doing it as well as you do. Thank you for making this series!
Awww, thank you very very very much! That’s really kind! I’m glad it helped you.
As someone who teaches mathematics at university level (to math majors), I think you are doing quite fine. I (like to) think that most of my colleagues who teach math majors do actually explain this stuff, but probably in other STEM fields, due to the lack of time to cover it, lots of these intuitive explanations get skipped over. So, please keep making these videos, they are becoming a good, casual, reference to the subject. You are a huge help!
Thanks very much! I appreciate it a lot coming from someone who teaches this properly. I did the easier linear algebra course at university because at the time I thought I hated maths (actually linear algebra was one of the things that convinced me that I don’t and soon after I switched to a maths major). I haven’t got a clue what the harder linear algebra class did, but I think even my class was pretty good with intuition and all mostly. Still, a lot of students seemed to struggle and I think that this was probably because they were too concerned with answering questions correctly to listen to the intuition! I suppose it’d be pretty different in a linear algebra for pure mathematicians class?
Thanks very much!! Actually, even though I enjoyed linear algebra very much, after a few years of not using it I think I had a lot of similar issues which meant I had to relearn it. I’m really really glad these videos help you brush up!
Absolutely! I think it should be a mandatory thing in classes to go back to what you learn a couple of years ago. With your new and nuanced understanding, a lot of ‘basic’ topics are absolutely fascinating. I got interested in QM itself because I decided to go back and learn it again after undergrad.
Well, the first course in linear algebra can be arranged in many different ways and still make sense. The topics covered are usually vector spaces, matrices, linear equations, linear operators, spectral theory, diagonalization (all finite dimensional, ofc). In pure mathematicians class you'd cover a lot of theory and are required to learn the proofs, so exercises and formulas actually do make a lot of sense if you are paying attention. As I've said, you are doing a good job of explaining why stuff works the way it does, especially since you immediately established the connection between matrices and linear operators, which is the most important thing in the subject. Mathematicians often say that one can never know too much linear algebra, it appears literally in every other math field.
Glad to see there are no dislikes for this video - anyone who tries to explain anything to do with vectors is a hero of mine :-)
Vectors ❤️
The answer is two.
since for every matrice ([(x,y)]) in A basis we get ([(y,x)]) in B basis
Great Video! I feel so much comfort dealing with this now!
This is the BEST video for change of basis that I have come across! The Alice and bob example was amazing
From the time I'm writing this comment I have yet to do an assignment due in 3 days. I haven't watched lectures (hard to follow), I'm out of hope and keep questioning my existence but this video has given me a slight glimmer, a slight chance. Thank you
This series made me literally understand all the unconnected facts i had from my uni course and it finally made SO MUCH SENSE. It makes so much sense it made me have my own amazing insights over lots of things you haven't covered!!
This is the best explanation of change of basis I've seen. Ever text book I've read talks about changing one vector into another. This is the first time I've seen someone say that it's the same vector described a different way. ( which is what change of basis is really about).
Thank you!
Yay! Thank you so much!!
I think having the multiple choice questions built in the video is an awesome idea, because it gets us to actively think about and apply what is being taught and to gauge how much we really understand. Instead of passively watching and possibly forgetting the content soon afterwards, it helps us retain the information. Thanks for the videos!
Thanks very much!! Do you think this format could work with the QM videos too?
@@LookingGlassUniverse Sorry for the late reply, but it should absolutely work with QM videos, since QM is so famous for being unintuitive, and so it makes sense for viewers to check their understanding while learning it.
Belated reply to the third homework question: I knew some linear algebra before, but this was still informative and also really inspiring. The problems were all straightforward in a good way, and I was utterly delighted when I realized you stuck an information-destroying transform into the initial video on transforms to prepare everyone for when matrices with no left-inverse turned up later on.
Speaking of which: the concept of a left-inverse was incredibly clarifying. Thank you for including that.
This is how I was taught to think about linear algebra by my great prof in undergrad, and ever since, I've felt like I've had a leg up on everyone who didn't take the time to think about the subject as much or didn't have him as a prof. Seriously, I still run into people in grad school (currently a physics grad student) who struggle on certain problem sets because they never were taught to think about linear algebra the "right" way years ago. I just wanted to share this so that other viewers know how lucky they are to have your videos. Really great stuff!
That's so kind of you to say- thank you! I agree, linear algebra makes so much sense this way, but most people don't seem to know this version. Good luck with grad school!
@@LookingGlassUniverse It's also interesting to think about how the change of basis matrix may be interpreted as representing its own abstract linear transformation, in addition to the perspective you take here where it's just a machine that changes basis but leaves the abstract vector the same. I'm still not really sure if this is a widely useful way of thinking about things. I suppose it's the difference between an active and passive transformation?
Wow...no words to explain the joy I now have after understanding 'Change of basis' from you. Wonderfully described. Thank you.
Woman in mathematics.! Really appreciate. Keep up the good work
I absolutely hated matrices and vectors before your series, now I've found myself playing around with them voluntarily. These are great videos, thanks for putting in the effort to make them.
This makes me so happy to hear!! How have you been playing around with them :)?
Keep uploading your extremely interesting videos!!👏🏻
Thank you so much!!
Very helpful.... may god bless you.
Understanding a topic, brings natural happiness
Thanks for making me happy.....
Your very kind comment makes me happy- thank you :)
Linear Algebra is the most fun subject of my first semester in college and your serie of LA is helping me a lot to absorb the subject and have fun with. To understand the concept behind helps a lot and Ive been indicating the series for all my friends. Thank you!
Oh wow! Thank you very very much! I’m so glad to hear it!! And I’m really glad you enjoyed your semester with it!
5:46 There is nothing strange about this matrix. Q_A->B never acts on the original vector but its representation in basis A.
Suppose the vector Alice wants to communicate to Bob is v. Put the basis A into columns of matrix A, do the same for B:
v = A [v]_A = B [v]_B
Q acts on [v]_A and transforms it to [v]_B.
Q doesn't act on v.
Usually a matrix takes a vector and transforms it to some other vector, same thing happening here ([v]_A and [v]_B definitely qualify as vectors, though their nature depends on the underlying scalar fields, whereas v doesn't.)
So I think this unnecessary distinction might be counterproductive, though it's probably worth mentioning that Q is the matrix of the identity transformation v -> v from its domain using basis A to its range using basis B, this is probably relevant to the point you're trying to make here.
Which led me to think that the necessary distinction is between the case of change of basis for a vector vs change of basis for a linear transformation.
To represent a vector you need one basis, so the process of changing basis for a vector involves a pair of bases (say, A and B), this video mostly deals with this.
To represent a linear transformation you need two bases (the input basis for the domain, the output basis for the range), so the analogous process involves two pairs of basis (so 7:22 is a bit handwavy). (As mentioned above Finding the change of basis matrix for a vector can be thought of as finding representation matrix for the identity transformation using the pair of bases A, B.)
So at this point it kinda annoy me whenever people talk about "changing basis of a matrix", what basis? do you mean changing from one pair of bases (where the input and output bases are usually, conveniently, implicitly assumed to be the same) of a LINEAR TRANSFORMATION to another pair (also usually, conveniently, implicitly assumed to be the same) and find out how the representation matrices are related? The most general case was covered in the last exercise in this video but I think it should be the main point of the video, introducing the special case first confused me 😞 and it took me awhile to realize that by allowing input and output bases to be different any linear transformation can be represented by a diagonal matrix, which should have been where the video ends.
#END RANT 😤😤😤
TLDR Love this video, I would never thought about these stuffs if it weren't for your exposition. Wonderful!
This video literally makes the concept crystal clear! Thank you very much!!
Thank you! I was so confused by this in class bc they just gave a formula and explained in a way that I totally didn't understand, but now it makes sense! yay! I think it goes to show how when people think they suck at maths, it is probably just bc they aren't learning/being taught in a way that works for them.
I'm so happy to hear that! Well done :)
This linear algebra series is very fun, educational and useful for everyone interested. On basic QM theory I hope you get to talk about von Neumann entropy but also entanglement and entanglement entropy. And product spaces, s.a. the tensor product of Hilbert spaces etc.
Now on the video... I have a question that I sort of know the answer to, but I want to put it out there just in case. There is an assumption about the underlying space, as in position space, not the vector space, in which the vectors are embedded. It's sort of assumed to be Euclidean. The norm of a vector is assumed to be the Euclidean norm. What happens if these vectors live on a space with curvature, such as a sphere or high-genus surface (for 2 dimensions) among countless others. Would we not need to introduce a metric? What happens if the space has creases, tears, holes or conical points - or their higher-dimensional analogues.
Oh and I have a second question, that is the vector space itself can be a topological space with a non-trivial topology. And the set of linear transformations also form a topological space. You get the idea. How is it best to exploit that.
I'm so sorry.
_"There is an assumption about the underlying space, as in position space, not the vector space, in which the vectors are embedded."_
In general, there is no "underlying space".
Michael Sommers Yeah there is, in two completely different senses as well. One is implied by the definition of the vector norm, the other from the whole set of vectors.
You are assuming that all vectors are geometrical objects; they are not, at least not the way you mean. I suggest you look up the definition of a vector space. If you do, you will see that there is no "underlying space" required.
Ok, first of all some friendly advice, don't tell people to "look up" basic information, it comes across as rude and patronising.
Secondly, and you seem to not have understood that there are two ways in which vectors can be related to geometry. First of all, normed vector spaces can be construed as metric spaces. So there is a link there. Secondly, vectors can act on objects that exist within a metric space. These are two different ways in which vectors can be related to geometry. (Although it's not exactly geometry, but that's a different matter)
Another thing that is rude and patronizing is unsolicited "friendly advice".
I said to look it up because it is a bit too complicated for a TH-cam comment, and so that you would not have to believe what I said.
What you have not understood is that I am not saying that no vector spaces can be interpreted geometrically. I am saying that not all are or can be. Finding examples of geometric interpretations of particular vector spaces does nothing to disprove my statement.
Again, look up the definition of a vector space. If you do, you will see that there is nothing in it that requires a geometrical interpretation.
For some reason your channel is my favourite math/physics one on yt. And there are a lot of great ones. Guess it's the mix of questions, problems, solutions and your nice voice plus alice and all of the funny characters. Are you in postgrad? How's it going? What's your research field (I know it has something to do with quantum, but could you be more specific)? Could you do a video about this? Bye! I'm getting back to my rabbit hole. I'm late. :)
Thank you so much! I’m very flattered! I am a PhD student in quantum computing- if you’re interested I’d be happy to make some videos about what I do!
I just don't get it. Why can't they just have the same basis? Then they would not need all this. How can there be to different sets of basis vectors? In which real world application do we use this?
The internal questions in the video were exceptional!
Wow. Even after Highschool + Undergraduate education and a lot of physics and maths videos, I couldn't answer one of your questions.. now my curiosity is sparked again!! (y) SUBed.
That’s great to hear! Being curious is the best thing :)
Hi, I have a question. At 7:28, why did we have to take a vector from Bob's language, convert it into Alice's language, track the transformation, and then convert it again to Bob's language? Why couldn't we just take any vector from Alice's language, apply the transformation and then covert it to Bob's language? Wouldnt the latter convey the same information i.e., the transformation applied in Alice's language? Thanks in advance.
It's surprising how concepts like this taught stupidly in class room can be taught in such a lucid fashion. Good Job.
Thank you so much!
I really like this series on linear algebra. I have done this stuff during my studies and a lot of the things you make clear - especially the (hidden) role of bases, eg Matrices being basis dependent - caused massive confusion when I was studying. Keep up the great work!
Oh thank you so much!! I really wanted to clarify some of the things I personally found hard in Linear Algebra so it means a lot that you’d say that!
Hi there. I got hooked on your quantum videos and I was waiting until your next video to mention something in hopes you would be able to read this. You have by far provided the best, easiest to follow, and funniest series to explain quantum mechanics and so many details surrounding it. I hope you don’t mind me saying even though that isn’t the focus on this video. I still have to finish this one really. Looking forward to it.
Thank you very much!! I promise I’ll get back to them soon!
Haha, no pressure :)
I really liked this video. It's very fortuitous that you made this video because I have a test next week on change of basis. Thanks for helping me out!
Yay! Good luck. Let us know how it goes :)!
What a lovely and artfully made video. I wish all math were illustrated like this. :)
It's interesting how there's an infinite regress problem in the example scenario, because in order to perform the transformation Alice must know what Bob's basis is, which requires them to communicate vectors from one to the other, which requires them to solve the original problem already before continuing.
Great point!
yeah but like only one time so still quit an improve
I know its too late for a feedback,but still the pace was perfect, the illustration was to the point ,i liked when you make M_a said to co-ordinates of in base B as "no thanks" ,and take the base A. I really dream that one day you will push the series towards advance linear algebra topics ,like complex vector spaces or even upto spectral theorem,who knows !
I love how you say the plural of basis.
Alice and Bob are working with vectors. Alice and Bob speak different languages. Bob needs Alice's help with a problem e.g. he wants to rotate a vector by 30 degrees, but the resultant vector must be in his language. 3 key steps: 1. Translate the problem from Bob to Alice's language (Q); 2. Alice uses her version of 'rotate 30 degrees' matrix to left multiply onto that translated-vector to rotate the vector (A) (yes still legit bcos the vector is in space. Alice and Bob are looking at the same vector in space. It's just that the x/y/z axes are different written in a different 'language'); 3. Translate the vector back into Bob's language. (Q^-1)
By language, I mean the basis. Written in different language means the vector is a coordinate vector of a different basis/the x,y,z axes are different
So basically a change of basis is essentially just a transformation matrix to transform a vector from one coordinate grid to another. Not really into the maths of quantum mechanics, but this is also very crucial for doing any sort of 3D work as you are guaranteed to be transforming coordinates from world space to view space and understanding the difference and why you do it is pretty crucial in order to further understand the tricks you use to reduce the amount of objects/points that are rendered and how objects are given depth despite not really having any.
I really like the multiple choice checks. Complex enough to check understanding but not so complex that I need to go find paper. Also I think you still owe me a promised answer to why gravitational interaction doesn't count as an interaction that would collapse a superposition.
🙈someone remembers I’m supposed to answer that! I worked out the answer at some point- but I think I need to go figure it out again. But yes, I’ll cover it in the measurement video :)!
I think this can be improved by explaining/demonstrating how can this be used, not with just a linear algebra problem but with a physics/"real world" problem that doesn't begin with "bob is using a different basis" but with more context as to why he's doing that and what is it good for..
That is coming (and if you watch the video carefully you will see a few hints of how it is applied). You can't put everything into a ten-minute video. Have patience.
Thanks very much for this feedback, and you’re very right! I think I’ve thought of a good way to show you guys how to apply this stuff.
Thank you very much for the insightful lecture. If it is not too much for asking, I was hoping you could please have a lecture regarding SVD. As I understand (probably incorrectly and hence requesting a lecture) as it closely ties in with the idea of changing the basis vectors which makes the original vector is an Eigenvector such that when the matrix multiplication is performed with the vector in the new space, the direction remains the same. Then we come back to the original space by taking an inverse of "change of basis vector". Thank you.
Your videos have always been good, and they are just getting better and better. This linear algebra series has been especially good.
I have (I think) three linear algebra books. Two are quite abstract, and one is very concrete (the first two chapters are on Gaussian elimination and systems of equations, and linear transformations are relegated to a very short appendix). You are steering a course between those two, which is just right for physics.
Thank you very very much- that’s so kind. I really do appreciate you saying that since you’ve been following my videos for so long.
That’s the way I was hoping to go! Add enough of the pure side to clarify but still have the intuition from examples. Glad it’s striking that tone!
how ru saving my grade rn 😭😭😭 thank u so much
I believe in you!
wow thank you really much I'm a physic student and i really only plugged in the formula for basis chance until now, telling myself that i already understood it and just don't want to go through another time. really thank you verry much
sorry for bad English
I'm so happy that this helped! Thanks very much for commenting :)
So one and the same Matrix can have different meanings depending on context ? Either its a linear transformation and transforms a vector or it tells you what the same vector would be if it were expressed in the new basis vectors had the old ones been transformed ?
Some really carefully considered pedagogical notation, illustration, disguised as whimsical fun! Great stuff! A lot of work hidden in these videos, I think. ( BTW basis, plural > bases. 'basees')
This series is just superb! I'd appreciate a lot if you put in the link to the answer of your "homework" in the description of the video.
Oh I should! Ok, I'll do that at some point. Thanks for watching!
Excellent and so sweet way to explain
I've come along quite late, but this is excellent, since you asked!
I really love this series; I'm not sure if there's too much more to say. I'm sure you're getting lots of comparison to 3B1B's linear algebra series, but it's always good to see things explained multiple times in multiple ways, especially since this series has an underlying quantum mechanical slant to it that makes it different from that series (and I can't place my finger on exactly why I feel that way - maybe it's just a psychological bias that I _know_ you're going to be linking back to quantum mechanics so I'm more alert for references, or maybe there's something really there, like the focus on vectors "really" being just a linear combination of basis vectors, or something). Either way, always looking forward to future videos!
Thank you so much for the kind words! I’m glad you think that there’s a quantum slant in these videos. I think QM has quite a different emphasis in linear algebra than the usual one (like you were saying with the matrices being linear operators etc) so that’s why I wanted to make them, even though 3b1b’s series is already excellent. Hopefully the quantumness will start to become more apparent soon!
Simply amazing. God bless you
I wish I had found this last quarter!
I was listening to Information by James Gleick and suddenly when talking about quantum information there appears Alice and Bob. I thought "I heard this somewhere...". Maybe these are common names in examples in English language, but it was funny to remember this channel.
This series is amazing!! Please keep going with your great job about Linear Algebra. If you give us more exemples in physics would be awesome. Thanks anyway 👍
I definitely need Physics examples! You’re right :) thanks!
Just a little observation. The transformed vectors used in minute 3 a1, a2 should have their name switched, right? As it is, it is saying b1 is getting mapped to a2 and b2 to a1.
This series is great . I felt like I understood change of basis before but the explanation was so clear, I definitely would have taken WAY more time answering homework number 2 numerically instead of with the intuition you showed in the video . Keep it up ^u^
Yay! That’s fantastic! Thank you so much
I'm surprised you didn't talk about a transposed matrix. In 3D, if you have a matrix of the three bases in global space (forward, up, right) and you want to transform a vector to be relative to that object, you just need to transpose the orientation matrix to get the inverse (flip it row to column). In plain english, in a flight simulator, if I want to make a cockpit gauge to show where the horizon is, I would use the transposed matrix to transform the gravity vector from world space to relative to the instrument panel and use that to display the artificial horizon. It's super simple and way faster than calculating the full inverse (which is a huge deal in a 3D game). After watching the next video, those would be called "orthonormal basis vectors".
Fantastic explanation!
this video is epic...i like ur teaching..
I finally understand what is in my tekstbook thank you
Most intuitive video thanks!
Your videos are absolutely amazing and very interesting to follow, they ver really really good from the beginning and they get better and better every time. please do not change anything... I cannot wait till you get us to tensors! Or won't you?
I really appreciate the perspective you give and the videos have fantastic content
Thank you so so much!
this is very informative
I don't know what could make this series better. Possibly harder questions since almost everyone who answers them knows the answer easily? But that could just be a sign that we have internalized the lesson easily. I don't suggest harder questions for there own sake.
Thanks for the suggestion! I was wondering about that too... maybe harder questions will help it stick?
I really like this series. In fact, you inspired me to start the linear algebra course in brilliant.org!
I have just one question about this video, and that is: what happens with diagonalization? I thought the point of all these change-of-basis stuff was to find the diagonal matrix.
But that's all for me, I love your videos and I really appreciate your effort to give us some intuition on it!
Oh really- yay! Do you like the Brilliant course? If I do sponsored videos in the future I only want them to be ones you guys like, so I’m curious what your thoughts are.
Good question about diagonalisation! That’s changing the basis of a matrix so that it is now written in its ‘eigenbasis’. Have you come across that idea in the Brilliant course?
honestly i ve never quite understood why people insist on distinguishing changes of basis from other linear transformations quite so rigidly-it is just another one of many perspectives on the same thing.
as for the speed your going, i think the only people that can actually answer that adequately are those who have never learned about linear algebra before. as for me i really just learned this from the not dissimilar 3b1b series and i found the videos on basis matricies and linear transformations to be by far the easiest to follow throughout that series as they require much less intense geometric thinking which is often the most difficult part.
Oh I see! So what do you suggest to make the geometric parts simpler?
i do not really have a good answer to that.once you kind of understood what is going on the geometry is an asset, but getting to that point is much more difficult. i suppose good visualizations go a long way already. though context can be immensely helpful too. a great example of this is chapter 11 of 3b1bs series i mentioned before, by giving more context it makes things a little more complex but simultaneously allows for a deeper understanding which helps provide confidence in your understanding
Very helpful!! Thank you!
This is so beautiful 😍
Okay I checked this channel after a while....and wow...u had videos on matrices....the stuff going on in college....wow
i didn't studied in college and skipped through book pdf...i remember there were words like span..etc...i thought not important ..but wow it was really surprising to me
Haha! Turns out linear algebra is really important for other physics topics.
Okay I dunno if it was god Playing Dice or stuff but i was lucky to have these videos here.....i want you to continue making such videos.... definately i can pass my semesters....by your videos :)
Interesting, try to make a video on Hilbert Space and some other spaces in this series, because you may now understood formalism
Yup! The next couple of videos are leading to that :)
@@LookingGlassUniverse are they ?
You are so good I don't think that you need some linear transformation to your "video-making" vector. 😂
Haha! Thank you!
Typo: The 1:44 basis is reflected, not "rotated by 45 degrees".
Thanks for the videos.
Yeah, I wasn’t too sure about my wording there. Thank you!
Amazing! Thank you❤
7:40 It must be Q_{A->B}^{-1}, or Q_{B->A}.
Ah! This is bad notation on my part- the matrix is Q, but the subscript is just to help you remember which way it goes.
Ok, got it. It's more an annotation and not a subscript of the matrix.
This worth more than 4 years of college ♥️
That is far too kind!!
tbh, this was easier to understand stand than 3blue1brown's course
😮
That's true, he went too far with his "I won't show you calculations and exercises" approach, glad we get those here as they are so crucial to understanding..
@@borg972 in the end, it's practice and making mistakes that are the greatest teachers. although he does encourage us to practice on our own every chance gets
amazing video , thanks a lot
When you say "vector" (or "vec-ttah"), what accent is this?
Plz can u say what courses need to be done to pursue PhD in quantum chem ??
If I had to guess, I would say chemistry 1 & 2 (I don't think there's a 3 in most colleges), physics 1,2, and 3, calculus 1,2, and 3, differential equations, multivariable calculus, linear algebra, and perhaps some complex analysis. But a lot of it will be working with quantum mechanics, which will be either its own course or part of physics, depending on which college you go to. All this being said, if your college or any future colleges you're thinking of have quantum chem as a major, then you could just check the course requirements. A lot of these courses can be skipped if you take/took the AP versions in high school, so it shouldn't be as crazy as it seems, at least in the beginning.
Wow, thank you so much for answering this question so thoroughly!
@@LookingGlassUniverse No problem!
Thanks 46 & pi.thnx a lot
@@dipanitajana6213 You're welcome! And like I said, that's just me guessing. And if you do get a hold of the course load and it looks intimidating, don't worry - all of the subject matter these courses is pretty fun amd interesting and it isn't too hard to understand topics like these on this channel.
What about converting between matrices of different bases that include rotations? Can you make a video regarding that?
Thanks so much .ur videos r really helpful
Thank you!!
Excellent video!
Thank you!
I feel that your Linear Algebra (LA) series is slightly better than 3blue1brown's, because not only do you cover the "intuitiveness" of LA, you also show much clearer examples to explain how the transformation (or anything else) is evaluated. 3b1b does a "pause and ponder" moment as well, but your mid-video polls are much more helpful in making the target audience understand LA, especially those who are weaker in it. Although your videos are much heavier in content and tend to be pretty long, someone who's determined to pick up LA is going to learn quite a lot from your videos. 3b1b attempts to do the same but sometimes he gets too far away from the theoretical aspects and doesn't return to the solving problems (mainly because he uses cool animations to explain which looks way better but when you think back about it you can't really remember much, whereas you use caveman methods like whiteboard marker and toothpicks to show your basis vectors - presentation is not that good but much more impactful). Though I've taken an LA course and hence not very qualified to judge your videos and his, your videos have made me learn a little more than his, and that's what I have to say. I've tried teaching LA to some people and I personally feel that watching your videos would have been way more effective than my own way of teaching. Anyway, all the best in your course and keep these videos coming!
Thanks so much for your detailed comment! I’m really flattered that you could compare my videos to 3b1b’s favourably at all so thank you very much! But as you pointed out, our videos have different purposes. I think his videos are aimed at people who get it at the level school taught them, but what to get it deeper and see the real beauty of the subject. I dig that. But I was aiming at a very different crowd (ironically, not my usual audience). I love teaching people who ‘hate maths’ and don’t get it and these videos are more for them- while at the same time I’m trying to give a perspective that isn’t usually taught so that even more advanced students can learn something small.
I made these videos with the intention that I have to give clear simple take home messages and I needed to make them stick ....which I still need to work but I’ve been experimenting with things. The midroll polls where the main one- so I’m glad you think those help! But yeah, another was specifically the crappy ‘graphics’. There’s some research that suggests that when notes are given in hard-to-read fonts students retain more because they have to engage more. Similarly, if you give students some still images versus an animation of a process, they learn more from the images because they need to figure out themselves what it would look like animated. So while I don’t animate in (large) part because I can’t, the toothpicks that are hard to see and visualise were kind of a conscious choice. Again, that’s no disparagement of 3b1b, because his main aim probably isn’t to have it stick in student’s minds. It was to show how beautiful and visual these things are.
Anyway, you seem to be someone who is able to see and judge how effective various teaching techniques are. Do you have any advice for me? I’m really keen to experiment
Hello, and thank you for this great video! I 've been strugling a little with the notations of the change basis matrix, can we say that (Q_A->B)^(-1) = (Q_B->A)^(-1), meaning that the index _A->B is just an indicator but the matrices are the same finaly, it's just Q?
Yeah, sorry that was confusing! The matrix is Q, the subscript is just to help you remember which way it goes. Thanks :)
Hey, can you do a video on molten salt cooled reactors like pebble bed FHRs, IMSRs, MCFRs, or LFTRs?
Just before I watched this video, I was in the tram and apparently there was a girl who was nervous because she was about to write an exam in linear algebra. Maybe I should have calmed her down by saying that linear algebra is so fun that there's nothing to be afraid about.
Haha! Well I hope she did ok. Are you in Melbourne by any chance?
I haven't been a tutor in linear algebra this semester, so I probably won't find out. I'm in Germany, so I'm far away from Melbourne, sorry. Why do you ask?
Damn, I was off the mark! I didn’t realise they’d be exams in Europe right now. In the UK where I am it’s holidays. Do you supervise maths courses?
We have holidays here, too. That's when exams are usually written. (Officially, it's not called holidays, but instead lecture-free time, but I don't think holidays exclude doing maths.) I don't know how they do it in other countries though. I'm not a supervisor either, I'm just a university student. In the last winter term, I was one of the tutors for linear algebra. Would you believe that only a fraction of all students was able to solve a system of linear equations in the first exam?
Ah that’s just cruel to have exams during ‘holidays’ haha!
It seems sadly to be the case everywhere that people don’t really like or try to understand linear algebra in university :/
Wow, great sweet channel you have here. c: I wonder why you sometimes have a german Title. Bist du Deutsche? However, i guess i ll have to work on my vektor skills, if i m going to study soon. I also have a mathematical quastion. I just had a weird coincidence, when i was calculating something and it blew my mind. I wonder, if you can picture a pattern, why it happens how it happens, or if it actually stops.
I calculated 20 : 19.6=1.020408[...] so you get out, what seems like an infinite 2^x sequence in the decimal place, which occurs every 10^-2x. You can also do this with 19,96 or add more 9s as you like infront of the 6, which just changes 10^-(2+y)x where y would be the number of 9s. I wonder, if there is an end to it, or (what would be much more exciting) if there isnt one, which would mean it would go infinite. Such a weird infinite thougth, because it would be infinite, but not really infinite, that it goes beyond the decimal place. I guess because 10^-2x gets smaller faster, then 2^x can grow. Also there seems to be a weird pattern behind this, which probably has to do with some diversibility rules, but i couldnt really figure out which, because what has all this 20 : 19.6 stuff to do with 2^x and 10^2x??
It just blew my mind, and i wonder, if you can help me out.
Have a nice day. ^_^
The fraction 20/19.6 can be rewritten as 20/(20-0.4), then as 1/((20-0.4)/20), then as 1/(1-0.02). This form corresponds to the sum of a geometric series 1+r+r^2+r^3+... =1/(1-r) . The first terms will be far away enough so as to not overlap when you sum them, so you see them in the result of the division. But as you raise to higher powers, they begin to "collide".
Btw, where have you seen german titles? Only some names appear.
Thanks for that nice explanation. :) Oh and these "german titles" are just a weird youtube translation. So dont mind that. :D
Btw. i noticed, that when you take all "9s" away, like 20 : 16 or 20/20-4 or 1/(1-0.2) you ll just get 1,25. Which means all these collisions actually add up to 0,01, which is somehow really cool i think.
Nice observation. It would take an infinite number of additions to get the round number though.
@@Transyst |r|
Switching between different co-ordinate systems
Beautiful
Thanks a lot
- «And now it's perfect»
Awesome video
Hey, could you please increase the video framerate a little? My eyes hurt.
crystal and clear... no1
Homework:
Well, I understood this 3rd video and I did OK on your 2nd (about inverses)... maybe with a small gap here and there. Why? Because surprisingly I struggled (don't know if I still do) in your 1st introductory video, which theoretically should have been the easiest (no?).
Linear Algebra was always a subject that confused me, and, not to blame you, but your 1st video brought up those feelings buried long time ago, to the point that I was a bit afraid to watch your last 2 videos (but I'm glad I did it) and I'm still postponing re-watching video #1.
In summary, to me, when it comes to linear algebra the slower and the more examples the better.
Other than that where did the red nails go ? :)
Thanks very much for the feedback!! That’s very useful! Would you be able to explain what made that video difficult in your opinion? Then I can work on improving!
So, let's see, how much time do you have? :)
Ok, 1st let me say sort of what "Halberdier" told before: have fun doing your videos and don't be so hard on yourself.
I mean, it can be, probably is, totally my (others) fault not being able to understand this subject, and maybe you just can't do nothing about it. I know it can be really frustrating for you, but such is life. Imagine you'll be teaching monkeys linear algebra ? Yeah, right ! No matter how hard you try I doubt that any monkey would get it... Maybe I (we) are just not smart enough like monkeys.
Having said that, and after overcoming my fear I re-watched the AB=/=BA video again. And the good news are that I am not a monkey! I feel much better now :). But it is still sinking in...
What made the video difficult:
1) In college I ended hating the subject. Didn't understand the purpose, so I just memorized enough to pass, and even without studying I passed the exam, and after that I quickly wiped out the all (almost) subject.
2) In college I think their approach was introducing matrices as systems of equations. Your approach is radically different, and so their was maybe some resistance / brain rigidity on my side.
3) On your side maybe your speed was too fast, at least for me, but I understand this is youtube (max 15min videos ) and I can always rewind as many times as I wish. On the other hand you may consider breaking the subject into more videos, each covering less, but with more examples (not just breaking the actual video into 2 shorter videos)
4) The 1st time I saw the AB=/=BA video I lost you completely at the 1st pop quiz at 3:20. What confused me then was that you all the suddenly introduced a V2. And I remember I was already confused cause I was not grasping what did you mean by applying a matrix (due to #2 above), such that I thought the rotation pop quiz was not linear. I guess I was interpreting it more like a change in coordinates instead...
5) At 11:46 I still struggle a bit. Why A(b11 b21) = b11(a11 a21) + b21(a12 a22) ? (this is a rhetorical question only)
I see a striking similarity with 6:58 (wich I understood), but something did not click yet in my brain...
What could you have done different here ??? Huh I don't know...could you have done the all example geometrically? Would it even help ? I think in my case it probably would help, but you never know
6) Another thing I feel it helps is to show the broader picture on how linear algebra is useful. I know you mention is used in quantum mechanics, but just like that it sounds like me saying linear algebra is used in Mathematics... What I think it helps is trying to extend the new field from something that is already known, and show how much more we will be accomplished with the new knowledge. Sort of like when you teach kids the rational numbers by extending the concept of integers by simply dividing 2 integers, which later you further extend with the introduction of irrational numbers, etc, etc
Obviously here you have to assume some degree of knowledge for you audience.
Bottom line it is very hard for you to please everyone and for you to put yourself in everyone else's shoes.
Also it is very easy when teaching about some subject you know well to skip steps that for you are obvious, but that for the listener are not. I call it the teaching syndrome, because teachers as they repeat year after year the same thing to students they lose the notion of the student side.
Holy crap, I wrote too much. Please don't take it wrong cause you're doing a great job. Cheers.
justpaulo Thank you so so so much- this is the most detailed feedback anyone's given me on a video! I'm going to go through them carefully and understand what I should do differently in the future.
Thanks for being so nice about it! I'm glad watching a second time helped a bit. I have a plan for a project that might be fun to show everyone how linear algebra is useful, and to help people get familiar with it. Maybe that'll work? We'll see!
Wow, How great
Good video with good intuition but I have a point of confusion. When I tried googling how to represent a linear transformation in different bases than the one we are currently working with, I get two equations. The first is the one that you derived in this video: M_B = Q*M_A*Q^(-1)
The other is M_B = Q^(-1)*M_A*Q
See this wikipedia article on similarity transformation: en.wikipedia.org/wiki/Matrix_similarity
Could you clarify the differences in these two equations? Thank you!
Is the difference simply due to the fact that we are changing from basis A to basis B in one equation and from basis B to basis A in another equation?
Hey! Good question. Yeah, it depends which direction you define Q- from A to B or B to A
The only problem i found with this series is that come one semester late. At least, inner product come just in time for my algebra ii class!
Haha! I’m sorry! What do you cover in algebra ii?