Very good presentation. My one nitpick is at 4:58, where you should probably specify that we could pick either orientation of vectors to have positive area: we simply have to pick SOME orientation to be positive, and pick counterclockwise by convention.
Yep, it should be pointed out that it’s precisely a consequence of Rule 1 that certain orientations are considered positive and others are negative - namely, if we decided that A(j, i) = 1 in Rule 1 instead, then A(i, j) would be -1 and all the areas will be negative of what they normally are.
5 days ago I typed into the UA-cam search bar " Why are determinants like that?" but I couldn't find an intuitive enough explanation -- you read my mind and I'm excited to watch this video!
In my experience I've never run out of drawing colors, but I sometimes have trouble choosing alphabet letters. It seems there are not enough (dissimilar) ones that don't already have important uses or meanings. Maths could benefit from a bigger alphabet.
This is the first time I’ve seen a decent visualization of where determinants come from without using the words “geometric algebra” talking about how it’s so much better than vector algebra! Also, I had no idea that the whole “crossing” thing was part of the standard! I thought that was just a weird tangent for talking about multivector rearrangement, I hadn’t figured out how to relate that back to vector algebra. Genius explanation!
I have a PhD in mathematics (granted not algebra) and the 10th minute (plus some wine) caused the increasingly rare epiphany as to the n! terms in the determinate formula. Thank you, sir.
I have a PhD as well, no I have let me check ... yes umm of course my kidergarten diploma here (totally the same as a PhD ;) right?) of course it is about the study of "Super Proportional Vector Fields with a distinct continuus Z Functions ". It's the real deal guys, real deal ...you know that pesky... riemman million dollar problem, the answer is right there. I found out that it stops at 10^363262363 + 2353534532^2. Totally did not invent my field of study and the solution right now. Of course not. Why should I do that. In all seriousness, congrats on having a PhD the only thing I have to brag is that I like math???. I know it is really out of place in a youtube channel called "broke math student" but still.
I always wonder how lives of these people would look like nowadays. Would be they become some big mathematican youtubers? Would they silently work on some research? Or would they get sucked into games and tiktok and wouldn't learn at all?
I am facinated by the fact that you call unit vectors i-hat, j-hat and k-hat, while I am being taught about them as i-cap, j-cap and k-cap. There can be languages inside languages sometimes.
Okay this channel is VERY good. I appreciate math videos more when they help me understand how to discover my formulas. Also, you do a good job pausing in your speech to give me time to process what I saw. This is something I hope to see more mathtubers do well, but you've killed it
I thought of cross product too, but I cannot see the connection. I mean it looks like the determinant of 2 vectors IS the cross product of these vectors? Or?
This is THE BEST explanation of determinants on the internet by far, and I am saying this after days of searching. Thank you so much for this beautiful video. I also started a UA-cam channel explaining math stuff and I actually thought about making one about determinants, but I am sure it would not have been as good as yours. I am so glad that I live in an era where I get to see such beautiful visualizations !!!
With this video, Broke Math Student has surpassed even 3b1b, the master of this genre and the original developer of the animation tools. That is quite an achievement.
I had seen the permutation formula for the determinant years ago and had always wondered what on earth it had to do with an area. This was a great explanation!
This is the most human explanation, for human to understand. I have never found any person on internet that can explain determinant very well. Thank you so much
I only see act I and I want to cry. I have struggled for so long with the relation between the intuition of the determinants and the calculation of them. Thanks for merging them in an elegant way. I cannot describe how grateful I am for content like this.
In Geometry I I learnt the definition of the determinant with the permutations, and it was really odd, as it just poped up, without further explanation. This video's topic should have been that class XD
On the braid crossings, in exterior/geometric algebra one can represent the volume elements by means of the wedge product, and then the crossing numbers can come from NOR operations applied in switching said wedges.
amazing video !! i loved your coverage of the topic, and your way of breaking apart and explaining the subject was very well done and easy to follow !! i cant wait to see more stuff from you :)
Outstanding! Nowhere could I find an explanation of why the sign on a volume in space related to the permutation ordering of matrix columns. This is the only web resource I have found that explains it. Thank you.
Thank you a lot for this video, I have been asking that same question for the last 5 years and yet could not find anything but the simple geometric demostration or the derivation for the simple 2d determinant. Most of people stop there, but for me it feels like proof by induction, it works but you dont learn anything. After watching the video I feel like the awnser was always there! and that I was overcomplicating things, as always the genius of linear algebra is in its simplicity, you made it pretty simple every step of the way!
Also, it's useful to note that a circular permutation of an even sequence (ab, abcd, ....) has alternating sign, given by mod 2 formula, while an odd sequence (a, abc, abcde, ...) has unchanging sign. That's rather useful to rearrange subsequences of symbols in an order more fitting for partitions sorting, that also doesn't affect the total ordering.
hloo sir ,love from india , u explained the topic which even 3b1b failed me to, this shows ur understanding of concept. furthermore i want u to keep making videos on this linear algebra topic its just seems to out of the box to understand this
This video catch my attention so hard that I actually stopped studying just to watch this beauty explanation and visual representation of the Determinant of the Matrix
Fantastic explanation. One thing that you didn't address but I personally always found important, is that in 2 dimensions if you swap the coordinates and flip the sign of one, you get a normal vector: same size but orthogonal to the original. The determinant is the dot product of this one orthogonal vector, and the other original one. Carthesian dot products are independent of the choice of coordinate systems, and thus, are a physical value. That value is the (signed) area.
Oh my god, I loved this. Illustrating the geometric object associated to each algebraic step during dimensions two and three was fantastic, and then the inclusion of braid groups!
Excellent Video! I really like proving all the complex applications, by first starting with simple rules. I think, this is the best way to teach mathematics in schools too. Please try to do more video like this whenever you can, as it will be a big service for humanity.
This was excellent, the determinant always annoyed me but this is so illustrative and makes me want to learn about symmetric groups more. Thank you so much for such a good video!!
It is an amazing concept that in Greece we learn in the 2nd year of high school, but no one gives attention because vectors aren't part of the examination in order to get to the public universities
Oh this is amazing. I'm taking a group theory course, without having taken linear algebra, and i get somewhat lost when they talk about matrix groups. Having recently learned about alternating groups (groups of even permutations), it was really cool to see them crop up in determinants
This is what I taught my students, but you did it better. I think that if we chase the thread deeper though, it appears that the conceptual dependency goes the other way. That is, the only reason we have a concept of area and volume in the first place is because of the algebraic coincidence that the top exterior power of a vector space is one dimensional.
I have been thinking about the determinant lately. Using the most natural interpretation of the determinant, through exterior product, I almost grasped the determinant inside out, except a single detail. I have been especially troubled by the notion of volume, because the area we usually talk about only makes sense in the Euclidean space, where as the volume in the sense of determinant is much more general, it works for all vector spaces over any fields (and even free modules over commutative rings). Furthermore, the usual sense of the volume is extremely complicated (it requires some sort of integral calculus or measure to be defined), using it to explain something as fundamental as determinant feels very wrong to me. This video filled the final piece of the puzzle that I am missing. In mathematics, instead of describing a concept concretely, it might be useful to specify properties that we want the concept to have. So I'll just make up a definition. A (not "the") volume operator f on a n-dimensional vector space V is simply an alternating multilinear map (a function that satisfies rule 2,3 and 4 in the video) that take a list of n vectors from V to another vector space W. As the video demonstrated, if V=ℝ² and W=ℝ, then the ordinary signed area operator is a special case, the justification only relies on simple cut and paste rather than complicated integration. So here's the thing: the determinant is defined on linear maps V→V, not lists of vectors. Although the linear map can be written as matrix, doing so requires an artificial choice of basis, and also obfuscate important insight. When we think of the determinant, we think of the scaling effect of the linear map on volume, not the volume of the linear map. Scaling volume can be formalized as for any volume operator f: (det T)⋅f(v₁,…,vₙ) = f(Tv₁,…,Tvₙ) There is exactly one function det that satisfies this property, and its the determinant. (The proof for this requires the knowledge of the exterior product.) I just wrote a whole essay that probably only 5 people will ever read.
Great video, thank you! The best explanation of where matrices come from that I've ever seen! The only request is to make the time when the final exercises are shown on the screen to be longer than one second and move them to the center. When I stop the video at the last second, I can't read the bottom lines because they're covered with UA-cam video buttons.
bro this is EXACTLY what I'm trying so hard to read and understand in my textbook rigght now... your explanation is perfectly clear and blowing my mind, kind of :0 thanks for this although I will saay, my book is also throwing differentials at us so to me it still feels just more complex and yet also beautiful :D
If you just keep the V(i,j) instead of saying that it is 1, you'll recover the whole exterior algebra. I consider the formulation there much nicer: no need to introduce an arbitrary rule about the ordering, you just simply compare with one ordering and see how many permutations are needed to convert an ordering to it. Anyway, I opt to teach my kid geometric algebra right away so that formulas like projections, mirrorings and rotations are immediately accessible, and so that one can do things like division.
Such a good video, helped a lot with understanding/recalling what i've studied a year ago :D I was particularly confused with why determinant is defined by permutaions, thank you for your effort!
This is one of the best (if not THE best) video I've seen on the topic! Also, regarding 8:55 - I really hope you're not an Austrian planning on moving to Germany... 😛
I wish my linear algebra professor showed those 5 axioms as the defining property of the determinant. It would have justified the intimidating sum-product definition over the permutation group. We only went over what the determinant truly was once we got to multilinear algebra, but by then, I was burn out by the subject.
YES YES YES, you showed the method of calculation too, oh god I feel like the determinant has been a little neglected as a concept to be explained. If you're searching for such concepts, how about the hessian matrix and its determinant. After all it's pretty important for multivar calculus
This right and left handedness of the det. For me it's a demonstration that higher dimensions are implicit. They're facing the other way in a plane in 3d. Likewise a 3d object can face two ways in 4d and on we go. In fact, the convention of dimensionality is wonky. If a 2x2 matrix is linearly dependent then we say it's 2d (with a rank of 1 admitedly) but the cofactor will never project into the 2nd dimension unless you take the bizarre view that the line is 2d because it's not flat. Anyway, that's only really a debate on terminology, it doesn't change anything fundamental.
I had to slow the video down to see that last frame. Task 4 requires writing a textbook which is uhhh. Anyway amazing video, i loved your manim animations ❤
HELP! Have you learnt the Manin all by yourself or is there a more efficient way? I tried making a video with manin but there were a lot of new functions and arguments to remember.
Congratulations! This a fascinating, amene, as well as relaxing, approach to a usually non well explained algebraic geometry. I love this kind of videos. Thanks for the post. +1 subscriber❤
2:55 what? I don't understand what you did with the triangle at all. Should the areas added together be the area of hypotonus? it seems more like a Pythagoras situation doesn't it? edit: oh its still in 2d
@@greenguo1424 So my problem was that I thought the image was of a 3d triangle, but since its only 2d you can transform it like he does in the video, cutting the triangle created by the top 3 points and seeing that it exactly matches the empty space created on the bottom.
Seeing your videos you seems like a very interesting and smart person , I saw your last video on doing maths alone and when in the last part the quote by Robert pyar from Zen in motorcycle on pursuing highest knowledge highly resonated with me can you give me more recommendation on such books and quotes
6:30 Suddenly anouncing the Determinant was a bit of left field play. You introduced determinant as an equation derived from a matrix, You said nothing about area, now you declare it so. Even if it is that was confusing. 10:58 I am glad that you mentioned the Permutations for that is how I first met and got confused by determinants. I like that you are giving a vector reality to them. Whilst I find you video Brilliant! can I ask: Do you have a similar vectorial interpretation of 'the adjunct matrix'? That would be awesome🙂
Not gonna lie - I got lost along the way, but still watched it and I really appreciate all the visuals and work that was put into this video. I missed though explanation what is actual determinant. From video it looks like it's the area, but is it just an area, or something else? I would watch a video explaining what we use it for. For example how it relates to systems of linear equations? Or how it relates to Jacobians? Matrices is some wild shit for me. I don't remember much from matrices, but I remember determinants refered to some matrix properties. For example, if 2 rows are the same, then the determinant is 0. Is it because they basically "overlap" so that the area is 0 as you said? Or also, when there's column or row of zeros, then the determinant was zero. Is it also because the area basically becomes zero? I remember some time ago I teached one guy how to calculate 2x2, 3x3 and 4x4 determinants and some other matrix operations. I just knew how to do it from the college and he just started and needed it to pass some test. But when he asked what it's used for, I just said I have no idea. All I could tell him that matrices can be used in 2D and 3D graphics and that there are things like rotation matrices. So I knew they are used there. But I had no idea what to tell him about purpose of determinants. And I felt bad, cause I always like to explain something intuivitely, and I lack this intuition for matrices.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
Very good presentation. My one nitpick is at 4:58, where you should probably specify that we could pick either orientation of vectors to have positive area: we simply have to pick SOME orientation to be positive, and pick counterclockwise by convention.
Yep, it should be pointed out that it’s precisely a consequence of Rule 1 that certain orientations are considered positive and others are negative - namely, if we decided that A(j, i) = 1 in Rule 1 instead, then A(i, j) would be -1 and all the areas will be negative of what they normally are.
lol it took me a few minutes paused to figure out where the “orientation” came from before I saw rule 1
@@brokemathstudentnegative is just the opposite of whatever the reference orientation is. It gets more difficult to calculate at A^N dimensions!
What is defination of area and why it is a into b in case of rectangle
wonderful presentation! How do you make these animations? Is this Manim?
So humble of you to shoutout the small mathematicians like Leibniz.
😢Yep, leibniz is so underrated.
small? are you joking?
@@jimpim6454 yes, they are
"small" ?
@@jimpim6454 intended as a joke
as shoutouts are given to high profiles of lesser prestigue as a free hand up
5 days ago I typed into the UA-cam search bar " Why are determinants like that?" but I couldn't find an intuitive enough explanation -- you read my mind and I'm excited to watch this video!
You are now a determinator 😎
@@Mr0rris0im convinced most mathematicians have a horrible sense of humor because of this comment.
@@macchiato_1881 well I ain no mathitician if that helps
You didnt find 3b1b's video on this topic??
Literally typed "Why are 3D determinants like that" and got this
11:16 "we dont have enough colors" yes thus appears the biggest problem for going into higher dimensions
Bro the irony of you representing the problem of a lack of representations in another form is so funny lmao
Imagining a world where we only have two colors and hence can't imagine the three dimensions we live in
In my experience I've never run out of drawing colors, but I sometimes have trouble choosing alphabet letters. It seems there are not enough (dissimilar) ones that don't already have important uses or meanings. Maths could benefit from a bigger alphabet.
This is the first time I’ve seen a decent visualization of where determinants come from without using the words “geometric algebra” talking about how it’s so much better than vector algebra! Also, I had no idea that the whole “crossing” thing was part of the standard! I thought that was just a weird tangent for talking about multivector rearrangement, I hadn’t figured out how to relate that back to vector algebra. Genius explanation!
I have a PhD in mathematics (granted not algebra) and the 10th minute (plus some wine) caused the increasingly rare epiphany as to the n! terms in the determinate formula.
Thank you, sir.
I have a PhD as well, no I have let me check ... yes umm of course my kidergarten diploma here (totally the same as a PhD ;) right?) of course it is about the study of "Super Proportional Vector Fields with a distinct continuus Z Functions ". It's the real deal guys, real deal ...you know that pesky... riemman million dollar problem, the answer is right there. I found out that it stops at 10^363262363 + 2353534532^2. Totally did not invent my field of study and the solution right now. Of course not. Why should I do that. In all seriousness, congrats on having a PhD the only thing I have to brag is that I like math???. I know it is really out of place in a youtube channel called "broke math student" but still.
Leibniz really needed that shoutout, it's always good to see big creators help smaller ones. (great video)
I always wonder how lives of these people would look like nowadays. Would be they become some big mathematican youtubers? Would they silently work on some research? Or would they get sucked into games and tiktok and wouldn't learn at all?
@@PinkeySuavodepends person to person
I am facinated by the fact that you call unit vectors i-hat, j-hat and k-hat, while I am being taught about them as i-cap, j-cap and k-cap. There can be languages inside languages sometimes.
I‘ve only ever heard the i-hat version. Good to know the i-cap version but I think it is more exotic.
People saying "cap" are the cool kids.
Are you from the US? I’ve never heard i-cap in my life. Cap feels like such a British word. 😂
That's how I was taught too. Î ĵ ƙ are all hats.
@@mgsquared5204 Naa, i'm not from US, I'm from India.
I finally understood what the determinant is. Thank you for making this video
Okay this channel is VERY good. I appreciate math videos more when they help me understand how to discover my formulas.
Also, you do a good job pausing in your speech to give me time to process what I saw. This is something I hope to see more mathtubers do well, but you've killed it
Yeah he had perfect pace, not too fast and not too slow.
5:54 This part made me realize the connection between the cross product and the determinant
I thought of cross product too, but I cannot see the connection. I mean it looks like the determinant of 2 vectors IS the cross product of these vectors? Or?
This is THE BEST explanation of determinants on the internet by far, and I am saying this after days of searching. Thank you so much for this beautiful video. I also started a UA-cam channel explaining math stuff and I actually thought about making one about determinants, but I am sure it would not have been as good as yours.
I am so glad that I live in an era where I get to see such beautiful visualizations !!!
With this video, Broke Math Student has surpassed even 3b1b, the master of this genre and the original developer of the animation tools. That is quite an achievement.
I had seen the permutation formula for the determinant years ago and had always wondered what on earth it had to do with an area. This was a great explanation!
This is the most human explanation, for human to understand. I have never found any person on internet that can explain determinant very well. Thank you so much
I only see act I and I want to cry. I have struggled for so long with the relation between the intuition of the determinants and the calculation of them. Thanks for merging them in an elegant way. I cannot describe how grateful I am for content like this.
In Geometry I I learnt the definition of the determinant with the permutations, and it was really odd, as it just poped up, without further explanation. This video's topic should have been that class XD
12:07 > _"how many swaps required? use braid diagrams"_
niice
lol I only just got 'braid', it's like doing your daughters hair before school.
On the braid crossings, in exterior/geometric algebra one can represent the volume elements by means of the wedge product, and then the crossing numbers can come from NOR operations applied in switching said wedges.
What a great presentation vectors are such a rich area
Most intuitive presentation of the determinant i have seen!
amazing video !!
i loved your coverage of the topic, and your way of breaking apart and explaining the subject was very well done and easy to follow !!
i cant wait to see more stuff from you :)
AZALI!? Love your music!
6 years using matrix and vectors, and now I have came up why we do multiply that parameter by this and so on. Cool video!!
Outstanding! Nowhere could I find an explanation of why the sign on a volume in space related to the permutation ordering of matrix columns. This is the only web resource I have found that explains it. Thank you.
this is so cool, u know what are the general questions in the mind of a math student, thanks a lot for making this video
I can already see your channel crossing 1M if this level of content is maintained. Cheers mate. 👏👏
Thank you a lot for this video, I have been asking that same question for the last 5 years and yet could not find anything but the simple geometric demostration or the derivation for the simple 2d determinant. Most of people stop there, but for me it feels like proof by induction, it works but you dont learn anything.
After watching the video I feel like the awnser was always there! and that I was overcomplicating things, as always the genius of linear algebra is in its simplicity, you made it pretty simple every step of the way!
Brilliant video, it's the first time I deeply understand why the determinant is a volume and also a really elegant derivation of Leibniz formula !
Also, it's useful to note that a circular permutation of an even sequence (ab, abcd, ....) has alternating sign, given by mod 2 formula, while an odd sequence (a, abc, abcde, ...) has unchanging sign. That's rather useful to rearrange subsequences of symbols in an order more fitting for partitions sorting, that also doesn't affect the total ordering.
That is the clearest explanation of determinants I have seen!
Hands down best video on determinant. Further I have yet to see this so clearly laid out in any text book.
hloo sir ,love from india , u explained the topic which even 3b1b failed me to, this shows ur understanding of concept.
furthermore i want u to keep making videos on this linear algebra topic its just seems to out of the box to understand this
this is one of the most awesome videos on determinants
I love that oriented areas with a + and - sign give complex number a geometric intuition: x^2=-1 is just a square of side 1 oriented counterclockwise.
Best video explanation for determinant! Should share to all students who are studying on linear algebra. Thank you so much for making this
This video catch my attention so hard that I actually stopped studying just to watch this beauty explanation and visual representation of the Determinant of the Matrix
im taking linear algebra right now, this video explained in the terms and notation the professor used!! I LOVED THIS TYYSM
Fantastic explanation.
One thing that you didn't address but I personally always found important, is that in 2 dimensions if you swap the coordinates and flip the sign of one, you get a normal vector: same size but orthogonal to the original.
The determinant is the dot product of this one orthogonal vector, and the other original one.
Carthesian dot products are independent of the choice of coordinate systems, and thus, are a physical value.
That value is the (signed) area.
Oh my god, I loved this. Illustrating the geometric object associated to each algebraic step during dimensions two and three was fantastic, and then the inclusion of braid groups!
Excellent treatment!! I will share this everywhere. Also, I almost fell out of my chair with your closing statement. So funny.
Excellent Video!
I really like proving all the complex applications, by first starting with simple rules. I think, this is the best way to teach mathematics in schools too.
Please try to do more video like this whenever you can, as it will be a big service for humanity.
In a few hours i have an Algebra exam, and this video was quite helpful, if one exercise is determinant based, i will update :)
I passed 😃. Sadly, there was no determinant based exercise, great video tho
please keep making educational math content, this is magical
This was excellent, the determinant always annoyed me but this is so illustrative and makes me want to learn about symmetric groups more. Thank you so much for such a good video!!
Sheer brilliance! All math videos should be this good. 🎉😊
More than a genius. I had never been taught in this way. WOW physical interpretation is awesome
I have done my highschool (cal.maths or add maths) and I never knew determinants can be visualized this way and that it was this simple!!
It is an amazing concept that in Greece we learn in the 2nd year of high school, but no one gives attention because vectors aren't part of the examination in order to get to the public universities
you're amazing dude, never seen math that way, you blew my mind, keep going with your videos, cheers from Brazil
Great video! Anyone else get this recommended to them after the recent 3b1b video where he hinted at a future video explaining this exact topic?
was about to mention that lol
Best outro in a math video ever.
Oh this is amazing. I'm taking a group theory course, without having taken linear algebra, and i get somewhat lost when they talk about matrix groups. Having recently learned about alternating groups (groups of even permutations), it was really cool to see them crop up in determinants
This video is a real treasure, it is one of the best and most important videos/lessons regarding the topic of linear algebra! Thank you so much! 😊
It's also interesting to point out, that determinant in 2d is a bivector, in 3d it's a trivector and etc. In other words it's a pseudoscalar.
As someone going into their first year of a mathematics degree THANK YOU SO MUCH!!!!!!!
This is what I taught my students, but you did it better. I think that if we chase the thread deeper though, it appears that the conceptual dependency goes the other way. That is, the only reason we have a concept of area and volume in the first place is because of the algebraic coincidence that the top exterior power of a vector space is one dimensional.
I have been thinking about the determinant lately. Using the most natural interpretation of the determinant, through exterior product, I almost grasped the determinant inside out, except a single detail. I have been especially troubled by the notion of volume, because the area we usually talk about only makes sense in the Euclidean space, where as the volume in the sense of determinant is much more general, it works for all vector spaces over any fields (and even free modules over commutative rings). Furthermore, the usual sense of the volume is extremely complicated (it requires some sort of integral calculus or measure to be defined), using it to explain something as fundamental as determinant feels very wrong to me. This video filled the final piece of the puzzle that I am missing.
In mathematics, instead of describing a concept concretely, it might be useful to specify properties that we want the concept to have. So I'll just make up a definition. A (not "the") volume operator f on a n-dimensional vector space V is simply an alternating multilinear map (a function that satisfies rule 2,3 and 4 in the video) that take a list of n vectors from V to another vector space W. As the video demonstrated, if V=ℝ² and W=ℝ, then the ordinary signed area operator is a special case, the justification only relies on simple cut and paste rather than complicated integration.
So here's the thing: the determinant is defined on linear maps V→V, not lists of vectors. Although the linear map can be written as matrix, doing so requires an artificial choice of basis, and also obfuscate important insight. When we think of the determinant, we think of the scaling effect of the linear map on volume, not the volume of the linear map. Scaling volume can be formalized as for any volume operator f:
(det T)⋅f(v₁,…,vₙ) = f(Tv₁,…,Tvₙ)
There is exactly one function det that satisfies this property, and its the determinant. (The proof for this requires the knowledge of the exterior product.)
I just wrote a whole essay that probably only 5 people will ever read.
This is a really excellent explanation for one of the more arcane formulas from my linear algebra course.
A comment for the algoirthm. I have no words on how great of a video this is, hope more are on the way!
Incredible! Even though this channel is still small, it's amazing!
Not sure if this was mentioned, but orientation of spaces is intrinsic to geometric algebra. Love this demonstration! Thank you.
Great video, thank you! The best explanation of where matrices come from that I've ever seen! The only request is to make the time when the final exercises are shown on the screen to be longer than one second and move them to the center. When I stop the video at the last second, I can't read the bottom lines because they're covered with UA-cam video buttons.
bro this is EXACTLY what I'm trying so hard to read and understand in my textbook rigght now... your explanation is perfectly clear and blowing my mind, kind of :0 thanks for this
although I will saay, my book is also throwing differentials at us so to me it still feels just more complex and yet also beautiful :D
If you just keep the V(i,j) instead of saying that it is 1, you'll recover the whole exterior algebra. I consider the formulation there much nicer: no need to introduce an arbitrary rule about the ordering, you just simply compare with one ordering and see how many permutations are needed to convert an ordering to it. Anyway, I opt to teach my kid geometric algebra right away so that formulas like projections, mirrorings and rotations are immediately accessible, and so that one can do things like division.
"Some of the most interesting mathematics happens when we see where 'nonsense' leads us" . Dang what a bar. Definitely stealing this one.🤣
such a well animated video! i never thought of the determinant like this
masterpiece explanation video.
Man, this is soo cool seeing your videos. Can you share the source code for the scenes?
Amazing animation with great explanations. Thanks! Btw, did you use Manim for animations?
I love the fact u named the first part "flatland"
a mathematic-culturally significant piece of art
@@Yaseenicus yep
I've been trapped in this for years. The moment of understanding make you my father.
Such a good video, helped a lot with understanding/recalling what i've studied a year ago :D
I was particularly confused with why determinant is defined by permutaions, thank you for your effort!
This is one of the best (if not THE best) video I've seen on the topic!
Also, regarding 8:55 - I really hope you're not an Austrian planning on moving to Germany... 😛
Best outro for a math video
I wish my linear algebra professor showed those 5 axioms as the defining property of the determinant. It would have justified the intimidating sum-product definition over the permutation group. We only went over what the determinant truly was once we got to multilinear algebra, but by then, I was burn out by the subject.
YES YES YES, you showed the method of calculation too, oh god
I feel like the determinant has been a little neglected as a concept to be explained. If you're searching for such concepts, how about the hessian matrix and its determinant. After all it's pretty important for multivar calculus
I'm right now just studying differential forms and multilinear functions, I guess this video will be really good to it
Hello from Russian university. Your video is really usefull
I've been looking for a video on this topic for months, thanks
This right and left handedness of the det. For me it's a demonstration that higher dimensions are implicit. They're facing the other way in a plane in 3d. Likewise a 3d object can face two ways in 4d and on we go.
In fact, the convention of dimensionality is wonky. If a 2x2 matrix is linearly dependent then we say it's 2d (with a rank of 1 admitedly) but the cofactor will never project into the 2nd dimension unless you take the bizarre view that the line is 2d because it's not flat. Anyway, that's only really a debate on terminology, it doesn't change anything fundamental.
this is the best video of mathematics taht i saw
I've been looking for video like this for a loooooooong time...
thank you so much, this Is such a great video, i understood more the determinant thanks to you!
Beautiful presentation.
That is an very interesting visualization of determinant . It has benefited me a lot
I had to slow the video down to see that last frame. Task 4 requires writing a textbook which is uhhh. Anyway amazing video, i loved your manim animations ❤
HELP! Have you learnt the Manin all by yourself or is there a more efficient way? I tried making a video with manin but there were a lot of new functions and arguments to remember.
Yeah these concepts are not even in standard books
@@pauldirc.. please do share if you got any other resources or any other idea.
Congratulations! This a fascinating, amene, as well as relaxing, approach to a usually non well explained algebraic geometry. I love this kind of videos. Thanks for the post. +1 subscriber❤
The
First
Time
Someone
Explained
To me
The
DETERMINANT
:-o
2:55 what? I don't understand what you did with the triangle at all. Should the areas added together be the area of hypotonus? it seems more like a Pythagoras situation doesn't it?
edit: oh its still in 2d
I still don't understand why "a(u+v,w)=a(u,w)+a(v,w)", could you help me a bit out here 😇
@@greenguo1424 So my problem was that I thought the image was of a 3d triangle, but since its only 2d you can transform it like he does in the video, cutting the triangle created by the top 3 points and seeing that it exactly matches the empty space created on the bottom.
@@bakersbread104 thanks!! I guess my problem was not seeing why the triangle has to do with the area of (u+v, w) which is a parallelogram
@@bakersbread104thanks for asking and answering this question! I was too thinking in 3D and assuming that I’m just not smart enough to follow.
I love the content, how did u do this amazing video,what video-rappresentation tools did u use?
Thank you for your precious time.
Stand proud. You can cook.
This is very good, time to make the broke math student rich
oh how i wish these videos existed on youtube 10 years ago
wow, great explanation. math at universities really has a pedagogy problem
Seeing your videos you seems like a very interesting and smart person , I saw your last video on doing maths alone and when in the last part the quote by Robert pyar from Zen in motorcycle on pursuing highest knowledge highly resonated with me can you give me more recommendation on such books and quotes
6:02 Rule 4. can be seen as the distributivity of addition.
6:30 Suddenly anouncing the Determinant was a bit of left field play. You introduced determinant as an equation derived from a matrix, You said nothing about area, now you declare it so. Even if it is that was confusing. 10:58 I am glad that you mentioned the Permutations for that is how I first met and got confused by determinants. I like that you are giving a vector reality to them. Whilst I find you video Brilliant! can I ask: Do you have a similar vectorial interpretation of 'the adjunct matrix'? That would be awesome🙂
Very interesting video of the topic, will come back to watch it again.
Not gonna lie - I got lost along the way, but still watched it and I really appreciate all the visuals and work that was put into this video. I missed though explanation what is actual determinant. From video it looks like it's the area, but is it just an area, or something else? I would watch a video explaining what we use it for. For example how it relates to systems of linear equations? Or how it relates to Jacobians? Matrices is some wild shit for me.
I don't remember much from matrices, but I remember determinants refered to some matrix properties. For example, if 2 rows are the same, then the determinant is 0. Is it because they basically "overlap" so that the area is 0 as you said? Or also, when there's column or row of zeros, then the determinant was zero. Is it also because the area basically becomes zero?
I remember some time ago I teached one guy how to calculate 2x2, 3x3 and 4x4 determinants and some other matrix operations. I just knew how to do it from the college and he just started and needed it to pass some test. But when he asked what it's used for, I just said I have no idea. All I could tell him that matrices can be used in 2D and 3D graphics and that there are things like rotation matrices. So I knew they are used there. But I had no idea what to tell him about purpose of determinants. And I felt bad, cause I always like to explain something intuivitely, and I lack this intuition for matrices.
There are also the series of video made by 3blue1brown deals with linear algebra and matrices