Matrix multiplication as composition | Chapter 4, Essence of linear algebra
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- čas přidán 7. 08. 2016
- Multiplying two matrices represents applying one transformation after another.
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Learning matrix in high school was like learning how to construct a sentence but never know it was for communication
So, a standart school second language class?
Except if high schooler taking Linear Algebra course
Wow, that's deep!
Not communication.. It's connection
Wow, that's exactly what is like, very giod analogy
20 years after first being exposed to matrices, and getting a computer engineering degree, I finally understand matrix multiplication.
This video is life changing for me
haaah 6 years since i left my 12years of education. And whenever I feel like something interesting to watch I occasionally come to this channel.
I doubt it took you 20 years to understand matrix multiplication
Then congrats, I must say?
cool
I have a degree in math and actually run a successful mathematics tutoring service. I have never seen anything like these videos. They are incredibly intuitive. Every time - every time! - there is some step where I say “oh, but you’re not paying attention to *this* detail or *that* detail,” within seconds Grant addresses exactly the misgiving I have by saying “now, it may seem like we’re being a bit dishonest here,” or some other welcome mixed dose of humility, honesty, and humor. Thank God for this channel - a rare glimpse into what it is like for mathematics to be considered a subject worthy of human inquiry.
This series is without a doubt the best educational content I've ever come across on the internet. I can't thank you enough for these videos
Absolutely.
There is now "Thanks" option in every youtube video. Donate some amount !
rly? i didnt understand shiet what his talking it was so fast iam still trying toprocess what he said in the beginning like wtf
@@simply6162 This is a great benefit of the video. You can replay it. Pause it. Think. Repeat until you understand. Can't do this in a lection!
Same
So for everyone else that had trouble with his "honest to god proof"... I've taken a number of courses in linear algebra and many proof courses, and found the same hole in his explanation. Here's what he means though, he just left out a crucial intermediary in his proof:
A(BC) means apply the overall effect of BC and then A. Of course, the overall effect of BC is equivalent to applying C then B based on what he explained earlier in the video. So we have just shown that applying the overall effect of BC and then A is the same as applying C, then B, then A. Similarly, (AB)C means apply C then the overall effect of AB. But applying overall effect of AB is equivalent to applying B then A. So we can just apply C then B then A and get the same thing. Since both A(BC) and (AB)C decompose to applying C then B then A, we have that A(BC) = (AB)C
Thanks for adding this, it's a really nice way to phrase something I should have communicated better.
+3Blue1Brown No no, thank you for doing such a good job in making these! I saw someone on my Facebook who just graduated from software engineering link this with the caption "I learnt more linear algebra in 30mins than I did in 5 years university" and decided to give them a watch. Admittedly, I already know this stuff, but these are entertaining to watch in and of themselves just for the gorgeous animations ahah if only you'd surfaced last semester when students in my mechanics class were like "What does a matrix have to do with stress transformations" and our engineering prof was like "EVERYTHING! YOUR MATH PROFS DESERVE TO GO TO JAIL" xD Keep up the awesome work!
I disagree with this explanation. You say, "Of course, the overall effect of BC is equivalent to applying C then B based on what he explained earlier in the video." Unfortunately, I can only assume you mean the section from 3:40 to 3:50, but in that very section, he simply drops some parentheses as if associativity has already been proven. So the argument, as far as I can tell, remains circular, and is not fixed by your comment.
By the way, I'm REALLY enjoying this video series. But this was my first major disappointment, with the claim that "This seems like cheating, but it's not; this is an honest to goodness proof." Actually, I think it is cheating. :( That's one of the dangers of visualization, not noticing the hidden assumptions necessary inherent in making the visualization in the first place. Visualizations are WONDERFUL for intuition, but can be very tricky as a deductive system. If this associativity confusion ends up being a circular proof (as I claim above) I hope that the end of the video can be redone so as not to misinform your (impressive number of) viewers.
He doesn't drop the parentheses to imply any sort of associativity. In that part of the video, he's actually still trying to define matrix multiplication. Essentially, he says that matrix multiplication should be defined in such a way that if you want to multiply two matrices A and B, then the resulting matrix C should be the one that transforms all vectors v the same way that A(Bv) does. i.e. we look for the matrix C such that A(Bv) = Cv for all v, in which case we say AB = C. By defining matrix multiplication this way, there turns out to be one and only one way to multiply matrices algebraically, and it's that funky little dance that you learn in a linear algebra course (proving this is not so easy). Using this definition, the proof of associativity is as straightforward as you saw in the video.
I have a masters degree in engineering and this series is blowing my mind.
That's because engineers think "what can we use this for" instead of "why does this work or what does it mean".
Well as an engineer, I really, really wanted to know this, but you know, sometimes teachers are not good and when I asked how did anyone found out about matrices, their properties, how they work, why they work, etc. the teacher could not answer anything concrete, and the book on this also was really confusing and vague.
I have A BSEE and I am really loving this. My Linear algebra courses consisted of Appendixes in the back of my text books that summarized linear algebra in 4 pages. I never actually understood any of it but had lots of disconnected factoids about linear algebra.
@@brentlocher5049 true2. I know how the numbers add up. But never why, no fundamentals. Shear and rotation are new to me.
Personally, I find it a bit sad that engineers often learn math for 'practical application' without actually understanding the math. This is coming from a 3rd year engineering student.
This is the future of learning here. Learning through playing. Learning for free. Excellent explanations. Exciting and Relaxing.
❤❤
learning through a personal tutor with an understandable answer to nearly every question is what we'll enter soon with AI
@@orang1921I hope AI will use that video as a basis for teaching
@@orang1921 ChatGPT is already helping me understand math much more than teachers ever did in school or university...
7:20 Also, having the intuitive understanding of it means that when you get outside of high school or undergrad you can actually use it to solve new problems rather than answer exam questions as it isn't just an algorithm but a way of thinking about the relationships between sets of dimensions. This is really useful if you are into data science.
I love the small details like how Composition was colored to look like it is a composition of a rotation and a shear.
Yes....same thoughts. He is a perfectionist.
I came to the comments section after seeing this genius at 4:25, ad hoping to add the comment if I didn't see it.
Also, the attention to detail added that Composition first has the ~teal of Rotation, which is the first transformation and then the pink of Shear. Just genius.
also a rotation and another rotation
It's like Khan on steroids! I love it!
Khan is a shill. This on the other side, is quality.
No doubt 3B1B was far too good to be compared with Khan. 3B1B is quality. Khan is quality.
Calxius those are strong words. Explain yourself.
It says in the videos thiss guy created the calc lectures for khan so lets take it easy. Khan is the man, hands down. Of course there will be people who can explain the information in a more digesable way that allows for deeper understanding, such as 3blue1brown, but Sal Khan has put out mucho content on his own that has gotten hundreds if not thousand of people through the first 2 years of their STEM degree. He even tries to relay a more intuitive undestanding as well.
right, as I say below khan has personally helped hundreds if not thousands of kids through their first 2 years of a stem degree
I spent a few minutes being confused about the associative property of matrix multiplication, but I think the key is to remember that matrices are really transformations, which are really functions, and when we multiply matrices we are really *composing functions*. So, ABC can be thought of as the composition a(b(c(x))). Now we can see that if we were to define some other function, q, as the composition of a and b, i.e., q(x) = a(b(x)), then a(b(c(x))) = q(c(x)). Likewise, we could define a function z that is the composition of b and c, i.e., z(x) = b(c(x)), so a(b(c(x))) = a(z(x)). So, q(c(x)) = a(z(x)), and this is pretty much the same as saying (AB)C = A(BC), I think... Am I right?
This is a great explanation, I was kinda stumped over the way he "proved" the associativity rule, but this way got through to me. No disrespect to 3b1b's explanations, they're great, but the way he said "C... then B then A" when referring to (AB)C felt a little funky.
Great explanation, I was also a little confused by the way it was explained in the video
Apparently this is a great explanation. It definitely doesn’t help me. Composition???
I don't know, I personally think there's still something missing from his proof as there is in this one. Say AB = Q. It seems a bit of a jump to me to assume that applying Q after C is equivalent to applying A after B after C. I feel some proof is missing from this. Similarly in the above proof, it seems like a jump to me when you assume q(c(x)) is equal to a(z(x)). If anyone has any way of explaining these gaps I would love to hear!
The transformation is always happening right to left , so the way I understand is (AB)C is apply Transformation C then B then A; even A(BC) says apply C then B then A; the brackets only change the order of multiplication but we are not changing the order of transformation
I've only seen the first 4 videos in the series, and I've gained more valuable intuition than my semester long engineering linear algebra course. Thank you!
Sir you have no match. I have many books of mathematics none of them explains the basic concepts.They just explain both basic and advanced concept in a way that we would memorize them, without understanding the essence.You make mathematics real and alive and make us get the real feeling of it. Keep it up and thanks
real and alive
Something interesting: I watched this series before learning about matrices in school, and it was extremely helpful to have this conceptual grounding.
you are lucky!
Some really do have all the fun
Doing the same
I am doing it right now and I hope this will be of much help when I start my linear algebra course in a few weeks :)
@@3drws314 how did that go? I am doing the same thing.
I have tears in my eyes I have never been able to visualize math so easily ever before this is a life-changing channel what you are doing is a gr8 work keep doing
Our professor at our university in Germany suggested your CZcams channel to us because he couldn't properly represent the 3 dimensions on the board, and it has been very helpful to me. Thank you for your videos.
Why is this channel not more well-known? These are probably the best math videos I have ever seen, in terms of their potential to make advanced topics easily understandable.
maybe math is not so wanted content these days :)
Because the majority of the 7 billion people dont care for understanding the universe(through math in this case) and care about primal instincts like sex, food and money more. They are on a lower level of Maslow hierarchy. If at least 20% of people really cared about science we'd be on Mars and Titan already.
It is quite well known among math majors and math grad students, as far as I can tell.
Jo Kah he has 2.42 million subscribers, that’s a lot compared to other CZcams channels. I’d say he is doing very well
It has 1.5 million views.
"Good explanation > Symbolic proof "
Exactly. I've been screaming this in my mind every time I see math. If my middle school teachers taught like this I wouldn't have hated math.
Thank you so much :D
I had to disagree with that part actually. I came back to this video to see how he proves associativity again and realized he just said the translations are in the same order. That's just explaining what associativity is. lol That's what we're questioning and seeking to actually prove. Obviously you can imagine translations in the same order. That's not what we're asking. We're asking whether it's associative. Which is about different orders.
Only A(BC) is "C then B then A"... (AB)C does "B then A" first (producing a whole new translation) which means we're doing the math in a different order, hence the word "associative" exists. Because we're smart enough to realize some things may not be. Or we can all act like everything's associative just cause we can imagine them being the same order we want them to be. lol
With something like associativity, this is the pure example for symbolic proofs. You can't rely on your "good explanation". You wouldn't even begin to start explaining something until you've actually proven it.
If you're someone who likes to actually understand, you want solid proof. You don't want some simple "it makes sense so just accept it" so called "proof". And then you look back with hindsight and tell others it just makes sense and ask if they can see why. lol
A(BC) is "C then B then A" is the same as (AB) C because you're doing "B then A" then add C to the front ending with "C then B then A"
Exactly at 7.37 you told that ""Take a shear which fixes i-hat and smooshes j-hat over to the right and then
Rotate 90 degrees""
At 7.45 you first did the shear fixing i hat and rotated 90 degree.
clear and perfect...
But at 7.55 you first rotated 90 degree and took a sheer ""Fixing J-HAT"" instead of fixing i-hat.
So,you end up having different results.
I want a clarification whether that's right or wrong???
Thank You..
By the way you are the best in the business for explaining mathematics.
@@DlcEnergy there is another comment regarding this which actually makes this proof quite rigourous. You may like to check that.
@@nikhilnagaria2672 i'm interested. can you link me to it?
I don't know how much blessed I felt myself after understanding matrix as linear transformation column vector representation. This concept really changed the way i imagine about matrices
the amount of effort he puts into these videos is incredible , I really appreciate his work!
Good lords! I've never been thought what matrices represented... this changes everyhing! and makes so much more sense! thanks!!
I know right!!!! This gave me the intuition behind all of matrices.
This is a million times more interesting than just learning the formula.
"I've never been thought" What an ironic typo.
+Zardo Dhieldor ooops! thanks for spotting that one out. =)
Jérôme J.
This should become a new figure of speech: "being thought sth." Only, I don't know what it would mean.
Non-square matrices laughing maniacally in the background.
"Muuuahahah... Try *that* with us, you fools!!!"
+SafetySkull Atleast when the right matrix is non-squared, you can consider it as just a (linear transform) x (vector) multiplication. Giving you a transformed vector.
THIS, I should've scrolled down earlier, I was so confused about vectors losing dimensions, unexplained by any simple geometric transformations.
non-square matrices can be represented as square matrices by substituting 0's on missing dimensions - the video still makes sense on that count
AB,BA are only possible only when B,A are square matrices
I can't make intuitive sense of why M2 applied to M1 at lands where it lands. ( 4:30 ) I can do the math, but how do I visualize it? Applying a transformation to the [1,0] [0,1] basis vectors moves them to the coordinates specified in the transformation matrix, but how is this done once the basis vectors are no longer that simple, and space has changed?
When you get it for the Basic vectors [1 0] and [0 1] it should be easy for every other vectors.
Its a rotation of 90° and then a vertical flip.
Two visualize use your fingers: Raise left hand.
Index finger up.
Middle finger to the right.
The transformation M2:
Middle finger must be where the index finger is now.
Index finger must be where the middle finger is now (and must double - but let's just imagine that).
We can do this by lifting the elbow to the left.
This was the M2 transformation (for the base vectors)
No matter where your fingers are now - if you make this rotation (and imagine that your index finger becomes twice as long) then you perform the M2 transformation.
Bruh !!! ... I watch you videos. Glad to see your interest in maths.
5m subs :0
MAGMA?!?!?!?! WHATTTTTTT
Lm
Thank you so much for this playlist. No one's been able to make me understand vectors as much as you have. Can't thank you enough!
Be careful using red & green for color-coding, it's a common type of color-blindness.
Very good point, I'll try to keep that in mind in the future. For many of the videos already made in this series, though, the green/red is already kind of locked in, and I wouldn't want to be inconsistent.
Additionally, It's worth noting that you've reversed the standard color conventions for X and Y. X is almost represented by red and Y is almost always represented by green. When in 3d, Z is usually blue. In fact, between various 3d applications, there's more agreement over this color convention than there is over whether Y or Z is the vertical axis. (And I maintain Z should always be vertical)
Z going into or out of the board/screen also makes more sense for Z-indexing and Z-buffering.
***** I'm not a board or paper person. I'm a 3d modeling and graphics person. Putting Z up makes the most sense to me and it's the convention used for aerospace engineering and 3d printing. This REALLY wasn't the point of my comment though. I was just trying to point out the standard color convention, which I feel is fairly important. The axis orientation comment was more of an aside.
I will say that I am red-green colorblind and I don't have any trouble with the colors in these videos. Of course, colorblindness has a lot of variation between people, so I'm not representative of everyone.
“Reading from right to left is strange”
Being a weeb is paying off
YES FINALLY
lmao
Knew I would find this comment
Yeah! But it has its own disadvantages. Like I always read comic styled meme in the same way it gets spoiled.
@@AnuragKumar-xh3wc a small price to pay for salvation
I have watched this series before without ever having learned linear algebra. Now I'm watching this again while taking linear algebra in university, and I highly recommend watching it this way. Definitely getting something new this time around.
This is truly eye-opening. Thank you very much!
Each day, for the past few days, I have legitimately looked forward to each release of the videos in this series.
you have some of the best quality math videos I've seen.
On an unrelated note, what song are you using at the beginning and end of these?
Thanks Noah! The song is just a short little made up thing. There's not really a full song to it, just enough to sandwich the videos.
Acttually, it's not a song at all, since those actually have words.
+John David moonlight sonata doesn't have words either
+Yunis Yilmaz We don't call it a song for that reason.
I second this +1. Kudus to the author(s) of these videos and I wish them the best
My ears feeling educated
This series is simply awesome. I got engineering master, and working in quantitative finance now. Lots of matrices during my study and work. However, I never understood how people came up with this kind of method. When I was in college, I had no idea why we need to learn this. After this series, every dot are finally connected. Thank you, and, hats off to all those great great mathematicians. What we are enjoying now are all based on their great and genius work hundreds of years ago.
The best and most logical explanation of vectors and matrices. Have been dealing with these things for years, and just now things clicked in properly. Thanks a lot for your time to make these series.
I just have to say this series (and the rest of your work) has been an inspiration. Getting a solid spatial understanding of these concepts has made diving into the hard mathematics of it not just easier, but FUN. Even working in spaces where these rules are broken, having the visual understanding helps me understand why they don't apply - intuitively.
Good explanations > Symbolic proof
THIS
Yes but you need symbolic proof to ensure correctness.
@@purefatdude2 no you don't, symbolic proof can only be proven on case by case basis (2D), here is a better proof. I will use A' for the inverse of A (easier to type, A(BC)=(AB)C, apply A' on both sides, A'A(BC)=A'(AB)C, A'A=I, BC=A'(AB)C, apply B' on both sides, B'BC=B'(AB)C, B'B=I, C=B'A'(AB)C, but B'A'=(AB)', C=(AB)'(AB)C, (AB)'(AB)=I, for this to be true, the associativity rule must be true. It is consistent with 3b1b's transformation rule
@@tehyonglip9203 I wasn't referring to this specific problem. I was referring to mathematics in general.
Teh Yong Lip The symbolic proof isn't just taking 3 2x2 matrixes and doing the multiplication. You just take 3 generic matrixes for which the product is defined (the number of columns of the first must be equal to the number of lines of the second) and then apply the definition
@@tehyonglip9203 lol this proof even without thinking about not square matrices, existence of square matrices, that have'nt any inverse matrice is wrong
Mistake is in first step apply A' on both sides and A'A(BC)=A'(AB)C is already wrong and the biggest mistake is in not wanting to write * operation symbol(or other symbol for multiplication of matrices)
Because it should be like this A'*(A*(B*C))=A'*(A*B)*C ,but when you are aware of importance of order of operations you could write it like this A'(A(BC))=A'((AB)C) and if you don't consider this as mistake then next step could bring you this mistake even more A'(A(BC))!=(A'A)(BC) before you prove associativity of this operation, which you want to prove using this property
And actually it is often bad to prove something with something you learn in future, because this smth could be proved or invented only because of true of this thing that you want to prove
And I know that you wrote this 8 months ago for me, but what if someone would see your prove as most simple, when it is wrong in a lot of different aspects
Great visualization!!! This series helped me a lot and I’m so lucky that I can find this 2 weeks before the final week!!! Thank you so much!
This is exactly what I've been looking for for so long! I've looked through a bunch of lectures and textbooks on linear algebra, but I haven't found even a close explanation to this! It feels like the authors themselves didn't have an understanding of this process.
4:26
As a Hebrew reader, I find this good news indeed
“It’s horrible ,just horrible” 8:52
I feel you man
0:00 intro
0:10 recap
1:59 order and composition of the transformations
3:42 multiplication of two matrices
5:56 generalized
7:00 what this really represents
8:21 associativity
I can be at peace in 2021 knowing that I learnt something, that I tried to understand for 4 months, in just 2 days. All Thanks to this guy. Thank you so Much sir💟
I was never taught the reason behind matrix multiplication and how all of it is how it is. Thank you so much for posting high quality content, absolutely love it!
I really come back to this Series so darn often! I guess I watched the whole series at least 3 times, not considering rewatching each episode when I need a refresher on each particular part. You are my hero Grant.
Years since i've taken linalg and still find myself coming back to these videos when I get confused about topics in my ML, graphics, robotics courses, get a new sense of understanding everytime. Thank you for these.
Your videos are phenomenal! I am self learning linear algebra from a few books and MIT lectures but I was struggling greatly until you cleared all this up! Your diff.eq videos are amazing as well. Thank you for all you do.
I'm so happy I found this channel!!!
That's the way it should have been taught.
I started learning linear algebra using this series. And I wonder what is going to happen when I see traditional ways in college. Thanks for this great series and videos!
I can't be more thankful for the creators of this series. I am truly grateful.
I am actually crying of how beautiful this playlist is. Sir, you truly are exceptionally inspirational.
These are some pretty godamnded good videos, man. This way of thinking about matrix transformation also makes the connections between zero-determinant, invertibility, eigenvalues, linear equations and a whole ton of other things pretty obvious.
5:23 the heart of matrix multiplication as composition, the thing that took me a while to understand deeply, a huge thanks!
Best Linear Algebra course on the internet. I wasn't getting this basic intuition anywhere else. The content should be put in a "book" to reference at any time.
I'm just sharing this playlist with everybody I know! this series of videos is amazing, it's teaching me more than I learnt in whole semester in college
Binge watching this series before my linear final tomorrow. I gotta say, your channel is amazing! You clearly have passion for this subject.
This channel and series released by them are making my quarantine holidays productive
Surely this channel has a lot of potential in explaining things very easier and thank u a lot fot this beautiful stuffs
And i am lucky i found this channel during my first year of engineering 😁😁
What you just taught here, it just blew me away. Never had I ever given thought to liner algebra in such a light. I am glad that i found this channel.
Thank you so much... Got my math exam's result today and it was all about matrices. The moment I somewhat internalised everything taught here, matrices concepts became so intuitive and I was so confident when I completed the paper. From what I was worried about the most it became my most confident topic. Really appreciate the visual understanding taught here! I've been spreading this video to everyone else since I watched it myself before the examinations and I'll continue to do so for everyone that needs to learn matrices!!!
I always crave true insight into what I study. It's just sad that there's often too much material to cover during the course of a semester to ever delve deep enough for a satisfying understanding. Thanks for making these videos!
These tutorials are better than college level lectures, I think it's a combination of your style and the visualizations that make them so effective. Wanted to thank you for your effort and contribution.
I was stuck in my machine learning preliminary lessons but this series helped me to think of matrices in a different and intuitive way. This series might change my life.
seems to be the most informative math series I've watched. The approach to making the math appear like art and we can figuratively imagine what all the functions and components are doing.
Well done @3Blue1Brown
I like to visualize numbers in matrix multiplication by tilting first row of second matrix counter clockwise and dropping it on top of first matrix, it falls through and multiplies everything it touches on the fall. And then doing the same with next row and so on.
It makes memorizing it very easy.
Agradezco profundamente que tengas los vídeos subtitulados a varios idiomas. Para que así, personas no anglo-parlantes podamos entender el maravilloso contenido que compartes.
Saludos desde el Perú.
You're a teaching genius my man, that is so incredibly instructive and just the right approach for learning linear algebra
Thoroughly enjoying this series, you sir are a savior for many people like me who just crammed the concepts in the beginning and later struggle to understand why something happens the way it does.
I am really enjoying this series : I love how he SHOWS what is happening rather than just doing symbolic manipulation ---so,so much easier to understand. (and remember). I took matrices in Math 1, but never understood what they were, just how to apply the rules. Thank you so much - I'm now having to relearn (or maybe learn properly for the first time!) LA for data science. (ps: like the music too, and great voice)
Same here bro
If I had 3blue1brown when I was in high school I would have gone on to get a PhD in Math and then become a mathematician. Now I’m just a vocational math enthusiast. Thanks so much for your service to math education. I hope your work get immortalized.
Watching this series yet after 6 years of its release, really helpful it is! Thankful to the creator!
You're videos are amazing! You manage to teach the essence of it in a really fascinating way. Thank you!
what the hell! i have been all learned the matrices without knowing what it is practical....thanks a lot man!
Yes CZcams serve me lot of ads , afterall it's for the great tutor.
Love these video courses emphasizing understanding rather than rote learning. Simply brilliant! Learning algorithms has a limited mental lifespan, but understanding is forever. Kudos.
I love how fancy and difficult these things look, then you just explain it a bit by bit and it all makes sense.
this guy's videos are going to make me cry, I always liked math but what I learned in school always did not make sense for me. Initially, I liked solving problems afterward I had second thought why "Am I doing this, for what". after several years you showed me the true value of math . Thank god the Internet exists and I learned English
26 dislikes?? Really? I bet they are just spams. Excellent series and intution. Probably the best explanation I have ever had.
haters, poeple who don't believe in math (i assume there are some; i know it's nonsensical though), frustrated content creators and/or jealous people.
@@zokalyx bruh what? nowadays there are some people who don't believe in maths? ehhh the world's getting crazier
Never before looked at the matrices like this. This is mind blowing. You are the best!
Omg, it is so simple and intuitive! But same time beautiful! Thank you 3b1b!
I was a curious student back then, the way they taught me made me sleep in the class. I always felt why the hell I need to just memorize these shit. If this was the way they taught us. Everything would have been different now. 😑, Thanks a lot ❤
Confession: I have a notification set up for when videos in this series are released
I'm glad to fill this hole in my school traditional learning. Thanks Mr. 3Blue1Brown!
the best math videos i have ever seen... thank you sir for making these concepts soo clear, these are very helpful
The only thing I’ve learned for certain after all this is that pretty much nobody with a PhD teaching linear algebra (and at exhorbitant tuition rates) actually knows how to teach it. Halfway through my first linear algebra course and it has only disappointed thus far. Too stressful and confusing, despite it not being terribly difficult. But these videos give it so much more nuance.
This lecture serie makes matrix manipulation more intuitive and linear algebra as a whole more understanding, by showing the 'why?'.
Thanks for the time and effort you've put into this.
I have one question now, at 03:19, according to the example, shouldn't the sheer matrix be
| 1 -1 |
| 1 0 |
Instead it's written as
| 1 1 |
| 0 1 |
Thanks for the clarification
No. The first matrix you gave is the output of the composition, ie the positions where the original basis vectors end up after you apply the rotation and the shear. The shear matrix corresponds to the transformation of the shear itself (or, the endpoints of the basis vectors if the shear is performed without doing the rotation first). Sorry if I didn’t explain this very well but I hope I did
Now I love linear Algebra, these video are the best explanation I've ever had. You managed Algebra in a less abstract way, and it is the best way to learn it. I'm going to take my first Univeristy Algebra exam in two weeks, and if I pass it will be thanks to you too!!!
These are so simple, yet so deep. I am literially watching with tears because I am feeling I actually start learning linear algebra from this moment.
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"Grant's New Etude" by Vincent Rubinetti
Wow. Really helpful. You must be feeling so happy educating others. Keep doing that
I’ve been struggling with Linear Algebra for a long time because I always feel that matrix is abstract and very hard to imagine what a matrix really looks like. Thanks for your series especially all these vivid animations and they really make the knowledge points make much more sense.
Accidentally stumbled on this video. Awesome explanation. Thanks a lot for creating and sharing this.
No one:
Grant: *Smoosh* those vectors.
"Good news for the Hebrew readers and bad news for the rest of us" had me on the floor😂
Grant, you've done great work on linear algebra, it's became much more easier to understand it, & enjoy the visuals. Thank u 👏🎉
Gracious man ! I've been looking around for this video for the past 22 years.
Thank you for this! One question: Is it the case that the first matrix (the one on the right) transforms i-hat and j-hat, and then the second matrix transforms i-hat and j-hat again? What is the best way to conceptualize that? I'm just struggling with the intuition of moving from the first to the second transformation. I can't picture that in my head.
At 7:30, why is J the one sheared in the first example, but then î is the one sheared in the second? Wouldn't J also be sheared in the second too, and not I? That combination would be commutative if so
i too have this exact same question. previously he sheared j keeping i fixed and second time he sheared i keeping j fixed.
In the second example, the rotated i-hat is in the original j-hat position, so the shear now operates on whatever is in the original j-hat position (i.e. shear operates on the transformed i-hat).
The author chose this example to show that, in general, matrix multiplication is not commutative. You could work up another example to show commutativity in a specific case; e.g. scaling by 3 with rotation by +π/2, followed by scaling by 1/3 with rotation by -π/2.
vk2ig By the example you gave, commutative property does work in this case, right?
I'm tripping on this too, I had to transform it myself to really understand what's going on. To anyone confused about that part, remember that the second transformation basically transforms the TRANSFORMED î and ĵ, so not the original î and ĵ, let's refer to the transformed vectors as î' and ĵ'.
When you do shear first on the original basis vectors, only the ĵ part moved while î' stays the same, so in this particular case only vectors that have value on the y axis (I know it's not really interchangeable but because the basis vectors lie on the x and y axis I'll just use them for simplicity) is sheared, so ĵ is sheared into ĵ'.
But when you do rotate first, ĵ' lies flat on the x axis, while î' now have a non zero scale on the y axis. Because the shear transformation in this case only affects those with y value, î' is sheared while the ĵ' stays the same because it has 0 y value.
I tripped over this one, too. The thing with transformation matrices is, that they are only an array of components not including a vector basis. So the transformation matrices in this example always transform with respect to the "x- and y-axis", because we chose to represent our vector space in this representation. If you would use a transformation tensor, you would have a transformation with respect to a basis. In our case î and ĵ. So it would be clear, that M1 and(!) M2 are transformations with respect to basis î and ĵ. This implies that M2 (the shear) would always "shear" the components of vectors in ĵ-direction no matter how many transformations were executed before.
I'm so glad I found this channel and so grateful for these videos.
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Multiplying two matrices has the same geometric meaning as applying one transformation then another. Right to left.
"Reading right to left is strange"
Manga readers: ..............
you mean right to left
Well... Manga readers are a bunch of weirdos, so...
Any person who knows a semitic language: ........
@@smolkafilip Omae wa mou shindeiru
just kidding
This series is absolutely incredible. I can't believe I learned all these computations in school and never understood what they really represented.
amazing. Thank you for that beautiful explanation. After that horrible class with a teacher who doesn't even know how to use zoom and just reads the book, this entrance to linear algebra really made me understand and love it.