Three-dimensional linear transformations | Chapter 5, Essence of linear algebra

What do 3d linear transformations look like?
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Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
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[@shinequashie393]

I count myself as one of the luckiest people to have stumbled on this playlist just before taking linear algebra in college

[@realbignoob1886]

That’s cool

[@abeestosruinsageneration3725]

Someone recommended this to me in the resources section in my class's study discord server. Already worth it on the first day of school.

[@beytullahberk3632]

same here

[@zhilihuang6609]

Yes, you are.

[@suncho6502]

I found it many years later :) Still, it's never too late to understand what I memorized back then.

[@DekuStickGamer]

I never thought I'd say this but I stay late up at night to watch math videos. These videos are incredibly intresting and rich with decription and fluid animation.

[@jlpsinde]

Me too, I sacifice some sleep just to watch these videos!

[@huverdoose]

*decryption

[@attilabpc542]

So do I

[@stefanoctaviansterea1266]

Same

[@xiangli3322]

Same, and now it is a quarter past 1 at midnight.

[@enricoiraldo9057]

I think the best way to use his amazing videos are: 1) watch his series on KZbin to get a general idea of the mathematical significance of the topic 2) study on manuals/uni the details and the exercises 3) re-watch to wrap up and abstract from details for a last consolidation

[@fahad_hassan_92]

Yup! Unfortunately, I had to deal with all the methods and rules for matrices, but preparing for newer topics like the way you have described is extremely helpful!

[@anglaismoyen]

I'm going to go through a LibreTexts textbook on linear algebra after this video series.

[@LinhNguyen-nh8oq]

Yeh. that's actually what I am going to do !

[@somebodyelse9130]

I ended up doing it the other way around. I have been taking a university course in linear algebra, and watching his videos has been putting everything in perspective for me. I had just been memorizing facts and definitions of jargon without context, and using those facts to solve basically puzzles I didn't know anything about; now, I'm actually getting an intuition for how this relates to real vectors in space.

[@mthibrahim]

Can you please explain to me what is libretext books?​@@anglaismoyen

[@AKSHAYKumar-gd7uq]

The composition of the two 3D matrices he shows near the end is 6-6-6 (top row) 33-44-55 (middle row) 6-10-14 (bottom row)

[@mdayanali8885]

How

[@AKSHAYKumar-gd7uq]

@@mdayanali8885 Check the last video about how to multiply 2 by 2 matrix and try same technique on 3 by 3 matrix For more clarity use online matrix multiplier to recheck

[@mr.noodle6931]

Lesgo i got it right thanks

[@leonaise7546]

@@mdayanali8885 Watch the last vid again

[@AlexAsh25]

yesss after a failed attempt, i figured it out!

[@jmiquelmb]

I had a little eureka moment when I realised that the identity matrix is the only transformation that won't change the position of the basis vectors. This series of videos is amazing

[@ababey1644]

omg i had that too!!

[@scriptkiddie7485]

well yeah, to visualize a matrix is to take each column and say that it represents a basis so the identity matrix would just contain i hat, j hat, k hat, etc...

[@padmasaran]

Wow!! Never thought matrix was this intuitive!!

[@nikhilnagaria2672]

and also if a matrix has one or two columns like the identity matrix!

[@esajpsasipes2822]

@@padmasaran I'd say everything is intuitive, but the intuition might not always be intuitive. Why is everyting intuitive? Otherwise why bother?

[@KirillBezzubkine]

There s one special spot in heaven saved for you. For helping all those math-lost souls and shedding light on their minds

[@dingo4229]

That feel when you've watched all the videos in this series the second they came out and can't wait for the next one to be released. Never been so stoked to hear how a person describes a determinant.

[@MarkCidade]

Me, too! Like wtf is that-you add up only the diagonal numbers or something and somehow that's the area of the parallelogram made by the column vectors?? This, I gotta see.

[@GelidGanef]

Yassss! I watched through all the Khan Academy videos on linear algebra. And I got it, more or less, they're great resources. But now actually I'm having fun doing this!

[@noufalbarakati8854]

AGREE

[@atriagotler]

This series is a true masterpiece. I mean whole 3b1b is awesome, but this playlist just sits in a league of it's own. This should be the gold standard for all linear algebra classes. Also, linear algerba should be introduced much much earlier. It's not that complicated if explained in a common sense manner as you can see!

[@danielrabinovitz478]

I just want to genuinely thank you for saving me in my Linear Algebra class. Your style of teaching is extremely simplistic and fluid. I'm remembering why love math so much even when classes frustrate me immensely.

[@dynamicgecko1213]

I just now realized that the reason the identity matrix is a neutral element in matrix multiplication is because it is the default values for each basis vector (i, j, k, etc). It transforms every vector back to itself. It makes so much more sense now! Man, this series is amazing. Thanks!

[@markostojanovic5930]

Our linear algebra prof. always shares a link for one of your videos in this series at the end of every chapter in his skript because he is a big fan of your videos. I think that‘s great and we have a much better intuition for this topic, thank you!

[@tobibello4160]

At this point in the tutorial, I just want to give you all my money. You've made my mathematics make so much sense, I feel like a child with an endless supply of candy!

[@whimsicalkins5585]

Me too!😆😁

[@frankdimeglio8216]

@@whimsicalkins5585 WHAT IS E=MC2 is taken directly from F=ma, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. Consider TIME AND time dilation ON BALANCE. The stars AND PLANETS are POINTS in the night sky ON BALANCE. The diameter of WHAT IS THE MOON is about one quarter of that of what is THE EARTH. On balance, the density of what is the Sun is believed to be about one quarter of that of what is THE EARTH. Excellent. Consider what is THE EYE ON BALANCE. The TRANSLUCENT AND BLUE sky is CLEARLY (and fully) consistent WITH what is E=MC2. WHAT IS THE EARTH/ground is fully consistent WITH what is E=MC2. CLEAR water comes from what is THE EYE ON BALANCE. Notice what is the fully illuminated (AND setting/WHITE) MOON AND what is the orange (AND setting) Sun. They are the SAME SIZE as what is THE EYE ON BALANCE. Lava IS orange, AND it is even blood red. Yellow is the hottest color of lava. The hottest flame color is blue. What is E=MC2 is dimensionally consistent. WHAT IS E=MC2 is consistent with TIME AND what is gravity. What is gravity is, ON BALANCE, an INTERACTION that cannot be shielded or blocked. Consider what are the tides. The human body has about the same density as water. Lava is about three times as dense as water. The bulk density of WHAT IS THE MOON IS comparable to that of (volcanic) basaltic lavas on what is THE EARTH/ground. Pure water is half as dense as packed sand/wet packed sand. Now, the gravitational force of WHAT IS THE SUN upon WHAT IS THE MOON is about twice that of THE EARTH. Accordingly, ON BALANCE, the crust of the far side of what is the Moon is about twice as thick as the crust of the near side of what is the Moon. The maria (lunar “seas”) occupy one third of the visible near side of what is the Moon. The surface gravity of the Moon is about one sixth of that of what is THE EARTH/ground. The lunar surface is chiefly composed of pumice. The land surface area of what is the Earth is 29 percent. This is exactly between (ON BALANCE) one third AND one quarter. Finally, notice that the density of what is the Sun is believed to be about one quarter of that of what is THE EARTH. One half times one third is one sixth. One fourth times two thirds is one sixth. By Frank Martin DiMeglio

[@marciomaiajr]

There's an old programming book titled "flights of fantasy". The author uses a lot of matrices for the 3d calculations and finally now, 20 years later, I understand why.

[@ExCoSeH]

Omg you deserve my money more than my teachers!! What is your PayPal??

[@sillymesilly]

I also want to donate to him.

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it's a little boring to create an account. Do you have btc/bch/other crypto address?

[@carterfogelman4613]

@@lucciopreso1 "I want to donate in a way that's easy for me, not easy for you."

[@puckry9686]

@@carterfogelman4613 what do mean easy

[@Paul-qn7qv]

@@lucciopreso1 He is a verified Brave creator. You can tip BAT to him if you use Brave browser. I just tipped him :)

[@seanm9306]

And now I understand identity matrices

[@IX0Ahmed0XI]

Oh god! Everything started to click once I read this comment! Like I understood what the video was saying but it never came to my mind how identity matrices make perfect sense now because of this video :D

[@daredo2]

i agree wow

[@cameronfitzpatrick2489]

Yup - so why does finding the inverse using the identity matrix work the way it does? Using rref of [A|I] to making [I|A^-1] - using the identity matrix should make much more sense now too

[@carbon273]

Wait explain? I think it flew over my head

[@lazypunk794]

In and identity matrix, the main diagonal elements are 1 and others are zero. Which means after transformation, the basis vectors remain in the same place.[the first column is 1 0 0 , second is 0 1 0...]. This is why a vector remains unchanged when multiplied by identity matrix.

[@Cleisthenes2]

All I can say is that if I were a mathematician, I would have three hats and call them i-hat, j-hat, and k-hat.

[@bungeruwu]

Is that right? 6, 6, 6 33, 44, 55 6, 10, 14

[@denascite2029]

yes

[@SebastianLopez-nh1rr]

Got the same

[@reinux]

i hate arithmetic.

[@leonardofilippini]

same, i love these videos my god

[@musamwinyi2049]

wrong

[@austinglander1337]

Currently studying for my first linear algebra exam tomorrow. I've been scratching my head on how solving matrices actually works but this video gave me a eureka moment where I just *had* to pull out my tablet and start scribbling. I realized that if A is just an encoded set of transformations that we apply to some vector x to get vector b, then when we solve for x by multiplying b by the inverse of A we are essentially just finding the set of transformations that returns a matrix transformed by A to its original self! My ramblings may not make much sense in the form of KZbin comment, but I am truly grateful for your work!

[@TimCutts]

This is so clear. I remember learning this in A-level Maths when I was about 17, and using it to write a computer program which would display a wireframe globe of the Earth in any orientation. But we were taught the formulae by rote, the words "basis vector" were never used that I can remember. I'm learning this again now, more than 30 years later, so that I can get to grips with the basics of quantum computing, which seems to depend very heavily on linear algebra.

[@djec-eazy1229]

Your ability to create fluid and intuitive animations to represent a topic is incredible and never ceases to add richness and clarity to a topic. Thank you for everything

[@kabeerkumar4334]

The intro and outro music creates an emotional bond with this channel and grant... It's like a lullaby :')

[@romanalaivi6967]

I forgot essentially all math from my school days. The way you explain this makes me realize there are actually many weak analogies to daily life and other tasks I've done like balancing a gimbal on each axis first. Amazing how intuitive this all is for someone whose forgotten just about everything and trying to get into machine learning. Thank you so much for making this so incredibly easy to understand. I feel lucky to have found these videos.

[@scriptkiddie7485]

Wow, I'm speechless. In a matter of hours, these videos caused me to go from hating linear algebra to one of my favorite subjects. At my high school, since not many people are advanced enough to do linear algebra, we have to teach ourselves. There are about 20 or so kids in my class and we each alternate teaching the class, and everyone is new to teaching. So, we ended up doing about 6 numerical theorems for every section without thinking about it, many without proofs. I was desperate for some type of visualization of linear algebra, and I just found the complete gold mine.

[@teddybrow]

Five years after the initial release, I'm watching this series a third time around. It's a great refresher and a compelling watch every time. Thanks, Grant!

[@harishjain2612]

After tirelessly working on my project in robotics, I never really gained a good understanding of the 3d transformations, which is ubiquitous in robotics.Your videos have greatly helped me in visualising the math behind the transformations. Thank You.

[@lexalious47]

I wish I'd stumbled upon these videos earlier than 3 hours before my math final.

[@Supware]

I think there's a typo at 0:46 (peek*) unless there's a pun I'm missing. Loving this series man (: your best videos yet

[@3blue1brown]

Uh...yeah...totally meant that 3d pun :)

[@vicente8749]

Yay! Without using any formula! :)

[@odyseuszkoskiniotis6266]

@Trevor Carey It's very helpful to visualize 4D I recommend it

[@babbetteduboise4284]

I was going to type the same thing--3 years later :-) !

[@emilnavod67]

@Trevor CareyIsnt it a book

[@shaon56]

If you study solid mechanics or finite element anlaysis, then this transformation using matrices gets pretty clear. You break down a structure into points and apply the stress and shear equations between the points and convert them to matrices to get the answer. But even though I did not realize that this is what it actually means when we do matrice multiplication in mathematics. Thanks for the videos. You have explained very nicely!

[@Math_oma]

The power of abstraction.

[@adultishgambino8713]

yes

[@kjekelle96]

0:00 intro 0:13 background 0:52 3D linear transformations 1:56 3x3 matrices, and example 2:49 correspondence 3:29 composition, and applications 4:03 homework

[@dhidhi1000]

I can't believe I'm actually gonna have any sense of what the heck determinant is

[@ljzilchz6548]

I once asked my teacher "what is the real purpose of learning matrices and it's applications in real life and how should I visualise it? " But All he ever said was "it is not really applicable in real life and he added that I should just practice hard to be good at it". After all these years, I finally got all my questions answered. Thank you soo much bro.

[@mrlaul7130]

4:15, yes but I don't have a graphic card in my head

[@sealol5209]

Thank you so much for this video series, feels like i've been cheated by my teachers for only learning how to memorise formulas and not have any understanding further than that.

[@TheHippyhopp]

just finished a semester with some linear algebra, but now i want to go back and try more

[@taprusthe3rd496]

For the matrix multiplication at the end i got 6 ; 6 ; 6 33;44;55 6;10;14 Seems about right

[@yaeloosthuizen725]

can confirm

[@Test-ri2kr]

@Yael Oosthuizen Can you explain the steps please?

[@yaeloosthuizen725]

@@Test-ri2kr you have a b c |0 -2 2| |0|. | 6 | |5 1 5| |3| = 0xa +3xb + 6xc=|33| |1 4 -1| |6|. | 6 | That's for row one You also do: For row 2 1xa + 4b + 7c For row 3 2xa + 5xb + 8xc

[@esajpsasipes2822]

i had an error in there but i was able to find and correct it. I got the same result.

[@guywhoistooparanoidtousere3758]

Would it be correct to say that i-hat after the first transformation is [0,3,6] and after the second transformation it is [0,5,1]? Am I understanding that right?

[@shiyi9993]

This content is literally GOLD! I always don't understand the point of exam just for the sake of exam, but I absolutely love digging into reasons of "why"! Haven't enjoyed this kind of joy of pure learning for soooo long since college!

[@RandomTot]

Thank you soo much 3blueonebrown, i have for five months tried to come up with a formula that does exactly this. I got pretty close even though i did not not anything about linear algebra just some trigonomerty i have been studing by my own. Thank you so so much this really helped me with my boid experiment i am cunducting in unity.

[@fancyphysics8458]

Two years ago, I had to learn Linear Algebra as a core subject in my university. I hated it because neither did I know whats going on nor my professor put any effort to teach us properly and thats happened all it was online classes because of Covid- 19. I surely had to memorize tons of formula without understanding to pass this exam. Now I am glad I am finally understanding this concepts little by little. Thank you so much for this 😊😁

[@siddhantdeshmukh7120]

ok so when you turn in a video game you are essentially rotating the whole environment around you in the opposite direction. To do that you are multiplying every vector of every object in your environment by a matrix which is the transformed version of the original view's basis (now I understand why do some graphic developers use the term rotation matrix ).And if you turn again, you transform the transformed vectors and keep on doing it every time you turn. Oh my god this is really EXHILERATING. Much appreciations for the content, very good content.

[@vivekjain8181]

I never really understood what matrices represented or what linear transformation actually meant. It was my lucky day when I randomly just started watching these videos. These are absolute gold! Anybody who thinks mathematics is all about arbitrary number crunching will have a change of heart. The meaning of an identity matrix (diagonal matrix with 1s) is so clear after watching this. Basically the basis vectors remain at the same place before and after transformation and hence the multiplication with another vector/matrix results in that same vector/matrix!

[@romance177]

i enjoyed every second in this series. you explained the material better than any professor I have encountered during my bachelor and master degrees. well done.

[@wackywally69420]

my linear algebra course has been taught in the reverse order - matrix properties first, then transformations. its like a bunch of random theoretical stuff that doesnt seem to be related thrown at us but watching your videos puts it all together so simply.

[@noufalbarakati8854]

I've been studying, working with and even teaching linear algebra for years. This series blown up my mind. Thank you so very much.

[@maxscribner1743]

You should talk about Quaternions in a side video

[@Carutsu]

please! I've used them before and I have no idea why they encode a 3d rotation

[@OttoFazzl]

Yes, please do quaternions.

[@ChethanSrinivas]

Excellent, this should be the way that is to be taught at engg schools, tensors is one of the most important topics for engineers. loved the whole linear algebra series ( I liked all of 'em). Typically used for 3d stress transformations and even mass inertia matrices, however this can be used for 2d stress transformation as well, but usually also done by trigonometry. Man you are a boon to lot of people like us, thanks a ton x (1/0). Have a great day.

[@rajaparameswaran1119]

How do you make the vectors and the coordinate system dance so well to your music? It is such an impressive way to learn Liner Algebra.

[@AltairCrysis]

Note: For anyone else who's confused, at 1:38, each tick on the graph represents 0.5, not 1! That's why each of the "unit" vectors (i, j, k) are shown with a length of 2 ticks.

[@viratzz]

Oddly enough this video on 3d transformations was what finally enabled me to have Linear Transformation as a concept click into the place! Thanks for your work.

[@spelunkerd]

I love the way this dissects each of the original basis vectors into three elements. Then the product of those with the new transformation determine where the new basis vectors will lie. Even so, for juggling all the relevant details, it is still helpful as an accounting aid to remember conventional multiplication patterns. I guess that says something about human memory.

[@aminpakshir5528]

it all started with a click on a video on the youtube and now I have got to the 6th one!!! amazing!!! great job!!! probably I will not leave until I cover ALL OF YOUR VIDEOSS !!! THANK YOU!!! I make robots and this was exactly what I wanted...

[@davidm.johnston8994]

I was gonna say that the rotation wasn't about y but then I realized y is represented by the red vector and x by the green vector. It's very misleading for someone who spends his life working in 3D software like I do haha.

[@davidm.johnston8994]

Usually it's : x: red y: green z: blue

[@totallycarbon2106]

Mentioning handedness here could have been good too

[@김현지-p5m]

This is the best linear algebra lecture I've ever watched

[@kingNikki]

Studying computer science this course/playlist is awesome ❤

[@Daddy_Skeletor]

I am baffled at the fact how understandable these videos make linear algebra, if only I had discovered them sooner. Thank You!

[@JerikRiza]

linear algebra has taken its toll on me this semester, I'm going through this entire playlist during my readers week right now.

[@swastiksanmay2315]

I realise every time your video ends, there is a soft smile on my face and feeling of content. Your videos are SOUL FOOD

[@motazart5961]

Your opening quote led me down a rabbit hole! But I'm back after 7 hours..and I have a new favorite TV show.

[@humanbot4]

I always had a problem in multiply 3 by 3 matrix in my high school. And now I stumbled into your videos about matrix, and I finally understood what it actually means. Thanks you very much.

[@scottdotjazzman]

This just made sense of the hours of struggling I experienced with the XNA game studio way back in 2010. _MAN_ I wish I'd had this so clearly demonstrated back then!

[@sinasafizadeh1164]

When math gets more exciting than watching movies....

[@hu-tv]

0:12 lisa- smiled awkwardly, cursed that guy in her mind and walks away.

[@anonymoususer4356]

Thank you very much. Had zero intuition about the subject until these videos.

[@Jame5Mead]

Thank you for making these videos. They helped me to properly visualise what's really going on.

[@gnuPirate]

Just going through the play list, leaving repetitive thank yous for this AMAZING CHANNEL! I really appreciate these. Thank you.

[@user-dw8lv6sy2y]

3차원에서의 비선형적인 함수나 그래프등을 선형적으로 표현시키는 것에 대해서 알 수 있었습니다. 미분방정식을 풀 때 비선형적인 항이 많이 나와 힘들었었는데 덕분에 방정식을 풀 수 있었습니다.

[@adityabusa2788]

Where were you till now,why aren’t you having billions of subscribers

[@estefaniakiara-elizabeth8538]

Thank you so much for this footnote, it totally helped visualize 3d space, totally love your series.!!!!

[@adolfocarrillo248]

Simply beautyful work done here. Thanks to 3Blue1Brown

[@wither8]

Man, I wish KZbin would have an edu-YT subset, structured somewhat like a wiki. That way, when someone needs to learn, say, to control a robot arm (as you poignantly mentioned), they'll be able made aware of this resource. Half the problem is people don't have a concrete idea as to the appropriate search phrase -- (e.g. for robotic arm control, "inverse kinematics"). Anyways, this is one of those resources that is just absolutely top notch. 20 years too late for me, but I bet it's shaping the minds of children to just want to learn for the sake of learning. Since my parents were richies, we were playing around with matrices as methods to represent co-efficients during the standard linear-systems-of-equations courses that middle-schoolers are taught. Having a resource like this and Geogebra would have totally led me down the road of differential geometry while I still had some neuroplasticity, ah well. Good on you. Again, A+.

[@MeriaDuck]

Great series! Btw, in 3d software (cad, modelling), it is convention to map xyz to red, green and blue respectively 🤣

[@HaraldZlattinger]

Really like your videos. I envy students nowadays for having such a good visual explanation for these concepts. Thanks for making these videos :) I'd just like to point out that in most 3D software packages you would correspond axis x, y and z with colors r, g, b --> x = red, y = green, z = blue. Working with Maya, Motionbuilder, Max, Modo and other packages for so long I have a hard time not automatically interpreting a red axis as the x/i axis and the green axis as the y/j axis. Also it's probably a good idea to mention in your videos, that depending on the software the matrices might be row matrices and not column matrices. And row vectors vs. column vectors.

[@soundsbeard]

thats really hard looking at green vector and think its x axis

[@ahmedheaba3180]

After seeing this, I can say that I never had real linear algebra in my past three years studying engineering.

[@abdallahhashem5135]

Those videos are really inspiring. I've never thought of Linear algebra like that before. I'm just imagining how that was essence. for programming like for example when you search something using your name or ID number it does some sort of matrix manipulation(based on your program) and points to SSD where your information is.

[@nguyenvominhkhoi9877]

I feel so lucky to find this playlist before I study in university , I think I won't be able to understand anything about linear algebra with the traditional way of teaching ( the conceptual and theoretical way of teaching )

[@mnada72]

I wish I have knew these and other concepts earlier in time , may be I would see the world differently. Thanks for the most appreciated effort spent in theses videos. Hoping to see more

[@movahediacademy9017]

Fantastic videos. I'm a detailed oriented person that needs to know why things work the way they do, and not just brute force through calculations blindly. This helped a lot. Looking forward to newer videos.

[@odyseuszkoskiniotis6266]

Hi Grant. I think the visualisation of which transformation is first algebraically by representing the multiplication of matrices as A*B should be A matrix is applied first and then B is applied second. I know it sounds crazy, but it works out right. The thing is that if we were to transform the standard [ 1 , 0 , 0 ; 0 , 1 , 0 ; 0 , 0 , 1 ] matrix re-defining it's basis vectors we would take the Matrix B as the function (a matrix containing 3 vertical vectors - according to Computer Scientist Convention), and use each values of those vectors x,y,z to scale the basis vectors of the system A (standard system). Taking that analogy to matrix multiplication, we would have already transformed standard matrix from the previous example (Matrix B), and we would like to re-define it again, by redefining the basis vectors, but this time the vectors of Matrix B. We would then take 3 vectors put together in a matrix C, to use the Bassi vectors of matrix B to create a composition final matrix. So if we look at the order, the way the transformations were applied, would be first to the standard matrix A we would apply B and then C, which should be written: A * B * C, and not the other way around, as we are basing the new system on the basis vectors of the previous one. While in the video you showed that operation for the inside is applied first, which would make the whole scaling of the basis vectors of the matrix on the left invalid, as the matrix on the left should be the one that we are applying the transformation to. I was figuring it out watching the videos meticulously from the beginning of the series, and understanding each component. And then the analogy of a vector scaling a matrix simply being - scaling each of the basis vectors of the matrix on the left - and trying to apply that analogy to redefining (transforming) the already transformed matrix , just didn't work out in algebraic calculations

[@arezkiaitmouheb7368]

This Video Series just acted on me like a matrix ! I've been Computing matrix operations for years now as Engineering student and this is the first time i understand what the hell It is.... Thank you for transforming my view !

[@alistairbaker4282]

Thank you so much for your videos. They are incredibly well done. I've shared the Essence of LA with other software developers in my company. If you are able to produce a video about how homogeneous coordinates can be used to represent both position and orientation of an object and transformations of that object, that'd be totally awesome.

[@anderesten]

I´m actually studying tensor algebra, but I needed to remember everything about matrices. Now I can see it all, everything makes sense, it is all connected.

[@rishisharma5827]

I watched these when they first came, and now revisiting for uni mathematics. Your explanations are much more intuitive than my textbook.

[@Alithium0]

I could kiss this man right now because he has made the world make sense, and I love him for it. Jokes aside tho this is the best explanation I've ever heard, and I'm finding great use from it.

[@alonshamay5163]

Amazing project. Unrecognizable contribution to science students and even graduaters.❤

[@mohammedkhalid9317]

I was stuck trying to understand Linear Algebra without going into too many details until thankfully I found these videos, THANKS ALOT

[@C2H6Cd]

These videos teach me far more under a couple of hours than my utterly boring profs at university under months. I think it's ridiculous that the majority of teachers simply can't teach properly at all.

[@jujumx1716]

I do really really thank you. You are the best teacher I've met so far. Your videos are dramatically helpful that I understand Linear algebra.

[@danielm7755]

I love that moment when something *clicks* in my head. Now I understand how rotation in 3D graphics work! Next thing to learn about rotations - Quaternions!

[@ANabih-uo3zy]

I couldn't grasp linear algebra fully in my robotics courses, so here I'm . very glad to find this playlist

[@thomuzmphela]

amazing, i wander if you can make a series to explain tasors and tansor calculus

[@mkb6418]

You should a video about homogenous coordinates. They are very usefull in computer graphics and robotics.

[@hopefullysoonaweldingengineer]

Trying to visualise this: Transform the unit vektors i, j, k by putting a triangular based pyramid on the posive octant. Then mirror to other octants. My brain melted. So sick!

[@narunaram]

This is a big service you've done for all generations... any way I could do a one time payment to support?

[@anahitaabdollahi4584]

Watching this for the 4th time! I cannot begin to explain how much has this series helped me❤

[@anonyone8834]

4:25 1st we multiply nornally the green numbers with all the pink numbers then we do the same for red and blue, we get a new matrix and that is the composition of these 2 transformations

[@psiphisapiens]

For me, your channel reopened the door to mathematics. I love maths! But I find it so difficult to ground. The difference between convention and fact was never taught to me, and so the grammar of mathematics overwhelmed me. I always found myself asking why this and not that. Your videos don’t just offer animated diagrams, they use animation as visual proofs, you clearly state the convention of a system, and then display the facts visually within that convention. As a scientist, thinking about change geometrically allowed me to see graphical representation in wholly new ways. This way of thinking helped me through my undergrad and masters, and will surely help me during my upcoming PhD, and Im currently reading mathematics books to push my learning further. But it all started here with you. Thank you. Ps The first time I watched this, the 3D space you use at 1:04 really confused me, as we are viewing the axis from below, compared to the Y-axis conventionally pointing upwards 😂

[@HerLos-v9r]

very good for a deep research throughout the linear algebra world after studied it in college

[@helo3827]

watching 3Blue1Brown's videos helped me grow my imaginations.

[@srinivasams9323]

We might be born late to explore the earth and born too early to explore the space but we were born in the right time to understand maths like never before.

[@stormer7502]

These are amazing! They clarify so much of what ive observed and figured out. I hated math in school because the derivations were never taught, but out of my own interest ive invested myself into things that require it and now i even find math itself fun and fascinating. So many years shamed for not getting any of it. Im sorry that i couldnt get my younger, blissful self to remember or even try to remember what might as well be arbitrary bs.