Three-dimensional linear transformations | Chapter 5, Essence of linear algebra
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- Опубліковано 1 жов 2024
- 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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I count myself as one of the luckiest people to have stumbled on this playlist just before taking linear algebra in college
That’s cool
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.
same here
Yes, you are.
I found it many years later :) Still, it's never too late to understand what I memorized back then.
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.
Me too, I sacifice some sleep just to watch these videos!
*decryption
So do I
Same
Same, and now it is a quarter past 1 at midnight.
I think the best way to use his amazing videos are:
1) watch his series on UA-cam 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
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!
I'm going to go through a LibreTexts textbook on linear algebra after this video series.
Yeh. that's actually what I am going to do !
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.
Can you please explain to me what is libretext books?@@anglaismoyen
There s one special spot in heaven saved for you. For helping all those math-lost souls and shedding light on their minds
Omg you deserve my money more than my teachers!! What is your PayPal??
I also want to donate to him.
it's a little boring to create an account. Do you have btc/bch/other crypto address?
@@lucciopreso1 "I want to donate in a way that's easy for me, not easy for you."
@@carterfogelman4613 what do mean easy
@@lucciopreso1 He is a verified Brave creator. You can tip BAT to him if you use Brave browser. I just tipped him :)
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.
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.
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!
AGREE
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)
How
@@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
Lesgo i got it right thanks
@@mdayanali8885 Watch the last vid again
yesss after a failed attempt, i figured it out!
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.
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
omg i had that too!!
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...
Wow!! Never thought matrix was this intuitive!!
and also if a matrix has one or two columns like the identity matrix!
@@padmasaran I'd say everything is intuitive, but the intuition might not always be intuitive. Why is everyting intuitive? Otherwise why bother?
And now I understand identity matrices
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
i agree wow
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
Wait explain? I think it flew over my head
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.
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!
Is that right?
6, 6, 6
33, 44, 55
6, 10, 14
yes
Got the same
i hate arithmetic.
same, i love these videos my god
wrong
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.
I wish I'd stumbled upon these videos earlier than 3 hours before my math final.
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
Uh...yeah...totally meant that 3d pun :)
Yay! Without using any formula! :)
@Trevor Carey It's very helpful to visualize 4D I recommend it
I was going to type the same thing--3 years later :-) !
@Trevor CareyIsnt it a book
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!
Me too!😆😁
@@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
I can't believe I'm actually gonna have any sense of what the heck determinant is
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.
The power of abstraction.
yes
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!
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!
4:15, yes but I don't have a graphic card in my head
For the matrix multiplication at the end i got
6 ; 6 ; 6
33;44;55
6;10;14
Seems about right
can confirm
@Yael Oosthuizen Can you explain the steps please?
@@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
i had an error in there but i was able to find and correct it. I got the same result.
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?
You should talk about Quaternions in a side video
please! I've used them before and I have no idea why they encode a 3d rotation
Yes, please do quaternions.
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.
Usually it's :
x: red
y: green
z: blue
Mentioning handedness here could have been good too
just finished a semester with some linear algebra, but now i want to go back and try more
The waffle house has found its new host
When math gets more exciting than watching movies....
The intro and outro music creates an emotional bond with this channel and grant... It's like a lullaby :')
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.
lol the moment when it only takes a 2 second clip to finally understand the determinant ^^
Nooooo! There is not Russian subtitles more since the video! I'll never become a smart.
Well why not learn English, there's a lot of benifit doing it
Satyam Prakash maybe he doesn’t have the resources
0:12 lisa- smiled awkwardly, cursed that guy in her mind and walks away.
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.
4:27 Not me wondering if the first transformation matrix would have some special meaning to just realise it's literally 0 1 2 3 4 5 6 7 8 lol
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
After seeing this, I can say that I never had real linear algebra in my past three years studying engineering.
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.
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.
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
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.
I attended linear algebra at university and the teacher got through all of it without using a single freaking image AT ALL !
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!
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 😊😁
You should a video about homogenous coordinates. They are very usefull in computer graphics and robotics.
Where were you till now,why aren’t you having billions of subscribers
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.
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.
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.
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 UA-cam comment, but I am truly grateful for your work!
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.
Anybody reading this comment, please do 'LIKE' the video. This is the least we can do for this masterpiece.
Unfortunately, computer graphics requires 4D matrices
oh god oh fuck why
aaaaaaaaAAAAAAAAAHHHHHHHHH
@@defene4966 don't worry, the computer can do the calculations
You should start writing books 📚 you know. You are a genius 👏 😊
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.
Lisa: Well, where's my dad?
Frink: He went out to get milk.
3차원에서의 비선형적인 함수나 그래프등을 선형적으로 표현시키는 것에 대해서 알 수 있었습니다. 미분방정식을 풀 때 비선형적인 항이 많이 나와 힘들었었는데 덕분에 방정식을 풀 수 있었습니다.
I can no longer watch these videos at night because I get too excited and forget to sleep
how would you show the transformation where a 3x3 matrix multiplies a 3x2 matrix tho
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!
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.
If I did it right then the answer to that question at the end should be [6, 6, 6]
[33, 44, 55]
[6, 10, 14]
Got the same as you :-)
Very pretty animations indeed.
However my one complain: When you rotate your vectors you shorten them along the way. I'd animate them in such a way, that the length stays the same all the way through. But this is nitpicking at its finest ;)
Pure rotation does not change the length. And he is also showing the "keep length"-version in his 2D examples (0:47). However in his 3D animations (2:25) he now changes the length along the way.
I'm guessing he's doing a linear interpolation (lerp) between some start and end vectors
He could normalize the vectors at each step, but then the speed of rotation would still not be constant.
I read about this sometime ago, and from what I remember, the rotational analog of "lerp" that preserves lengths and angular speed along the way is called "slerp"! (for spherical lerp).
Not sure if there's a way to interpolate a general transformation in a nice way though.
can u animate last 3D animation using grapher ?
I wonder why this has 0 dislikes yet... don't get me wrong, I love these videos, but the internet is full of trolls.
Math video ratings are invariant under Internet trollformations.
There. I disliked it. I actually love these videos. But take it as a lesson not to tempt fate.
>>>>>>>Hugelol
guys lets all dislike the video
Great series!
Btw, in 3d software (cad, modelling), it is convention to map xyz to red, green and blue respectively 🤣
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...
amazing, i wander if you can make a series to explain tasors and tansor calculus
so basically you always and only need n^2 numbers to definite that transformation, with n being the number of dimensions, right? E.g., a 2-D space would require 4 numbers, a 3-D space required 9 numbers, 4-D requires 16, etc.
yes, that's because the matrices are square :)
watching 3Blue1Brown's videos helped me grow my imaginations.
The homework:
[
[6,6,6],
[33,44,55],
[6,10,14],
]
yup awesome.Reminds me of 1960's schools mathematics project (UK: remember that anyone?) .. also set out to explain the 'why' more than the 'how to'. Alas, that collapsed inside the ridiculous UK education football*, that has seen fit to dumb down deeper mathematical understanding; now you don't get that until university maths, and teaching this at school is regrettably all about rote again. Not only are the videos excellent and illuminating, but your entire agenda here is admirable .. every A level (further) maths student should play them out and be enlightened.* not many years ago, it was debated in parliament whether to retain the quadratic equation in GCSE maths - it's (shamefully)true
Agreed
Studying computer science this course/playlist is awesome ❤
I would like to give you all the prosperity in the world.
Did you linearly interpolate those basis vectors to make the rotation animation? :) Because during that transition the vectors cese to be unit vectors, and the grid gets squashed. It would be best to just animate the angles and recompute the rotation matrix for each frame, or use a unit quaternion and slerp.
*sigh*...yeah. It was a bit of laziness in the moment to just reuse the same general format that I had for the previous transformation. Some of it stems from the fact that I was using a different tool from usual that wasn't quite as easy to tweak.
Only that computer graphics mostly use 4D matrices known as homogeneous coordinates to be able to represent projections. Or simply quaternions.
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.
Thank you so much for this footnote, it totally helped visualize 3d space, totally love your series.!!!!
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!
Who the heck thought it was a good idea to make the J and I symbols so similar, and to us J and I in the first place.
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.
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.
Sir, How active and passive transformation are different and how to compute?
You could expand the series if you talk about the fourth row that's used in 3d graphics.
Thanks!
where were you when I was taking these classes????????
what's the mean of 3*1 matrix multiply 1*3 matrix
(6 33 6) for new ihat
(6 44 10) for new jhat
(6 55 14) for new khat
cmiiw
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 !
Will you have a video on the 3D to 2D perspective transformation? I know it isn't linear, but it would be nice to see a transform in action
1:24 "Moving over"? Surely there isn't a unique way for them to move, isn't there?
Surely it is more similar to mapping the original vector to the transformed ones, is it not?
You're right, the movement in between does not matter, all that matters is the starting and ending spots. But for the sake of having something clear in one's head, I like think of the movement in between.
if you apply the same small transformation again and again a path of some conic section seems to connect all the outputs,maybe there is a correct way to move through the first one...
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.
This is the very core of video game development
Your opening quote led me down a rabbit hole! But I'm back after 7 hours..and I have a new favorite TV show.
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.
Simply beautyful work done here. Thanks to 3Blue1Brown
Amazing project. Unrecognizable contribution to science students and even graduaters.❤
Now I need four more of these, because we regularly deal in 7D matrices
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!
I get that it's helpful to think of matrices in terms of linear transformation, but for me to imagine how the basis vectors transform in 2-d or 3-d is still really hard. How do you look at the numbers and understand the general direction of its transformation?
This is a big service you've done for all generations... any way I could do a one time payment to support?