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
+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 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.
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.
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.
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.
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 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.
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
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
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.
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.
"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
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.
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.
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
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.
Our professor at our university in Germany suggested your TH-cam 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.
Oh my GOD it finally just clicked for me what an _indentity_ function really is! They're the _starting points_ for the basis vectors! Oh my GOD. I just gained a brand new intuition for the first time since taking Diff Eq 7 years ago. Thank you Grant!
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)
***** 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.
+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 channel proves the power of being able to convey ideas to others. Professors couldn't do it like this guy, because they are only good at what they teach, but not good at imparting the knowledge. I feel like I can finally enjoy learning for once
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!
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?
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 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
@@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
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
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.
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 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.
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
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!
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.
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.
You are my hero man. I went from hating math to finding it beautiful from your videos. I can't understand the math without a visual representation its like for my brain without a visual the numbers are not real so its been hard for me to understand. You make it not only digestible for me but borderline enjoyable. I am a more educated person because of you. Thank you.
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
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.
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.
Also, it's useful to think of vectors as hop instructions. For example, [1, 2] under the ihat and jhat basis vectors means hop 1 in the ihat direction and hop 2 in the jhat direction. Under different transformed basis vecors, T(ihat) and T(jhat), [1, 2] means hop 1 in the T(ihat) direction and hop 2 in the T(jhat) direction.
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.
4:06 matrix multiplication order 1. rotation 2. shear s(r([x,y])) s*r ✓ r*s X n * n-1 * ... * 2 * 1 * [x,y] ✓ n(n-1(...(2(1([x,y])))...))✓ The order of calculation *from now on* doesn't matter. If we start on the right with 2*1, than 3*(21) and so on, will yield the same result as if we start on the left with n*n-1, than (n n-1)*n-2 and so on.
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.
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 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)
@3Blue1Brown , at 3:18 , how did you decide the value of Shear Matrix after rotating the original coordinate system by the Rotation matrix? Not only, there but further also, would you please explain me more how do you decide the value of another matrix after rotating the basis i-head and j-head with the first matrix (rotation matrix).
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!
at 7:40 the shear is applied by fixing i^ and rotating j^ to the right. But then, at 7:56 , the shear is calculated by fixing j^ and rotating i^. Somehow I didn't get the concept of shear. Can someone help ? (Because if the 2nd shear is calculated like how the 1st one is calculated, the resultant transformation will be the same)
although it is quite late to answer but i would like to point out that the shear concept isn't defined based on i-hat or j-hat - something like fixing the i-hat and rotating j-hat or vice versa. Instead, it is defined by a 2x2 matrix whose the first column is 1, 0 and the second column is 1, 1. if we do the shear, then rotate (whose transformation matrix is 0, 1 and -1, 0 for first and second column respectively), the combination transformation is 0, 1 / -1, 1, which shows the position of i-hat and j-hat (which is the same in the video) if we do the rotation, then do the shear, the columns of combination transformation are 1, 1 (first matrix column) and -1, 0 (second matrix column) which show the position of i-hat and j-hat respectively (same in the video - clearly the order does matter) please write it down and do the calculation yourself to verify
Yup Sanath !! am having the same doubt. Am unable to visualize (as in geometric intuition wise) the successive transformations. How is i hat and j hat affected in the second transformation ? ( considering matrix multiplication as two sequential linear transformations )
No.....shear transformation means rotation of j-hat of the REFERENCE COORDINATE SYSTEM j'.Actually transformation is applied on the reference coordinate system. When rotation is done coordinate system changes and NEW COORDINATE SYSTEM becomes the reference ON WHICH the shear is applied. In the video new coordinate system is not shown whose j-hat is rotating by 45 degree. Here L(i) coincidentally matches with j' (of NEW COORDINATE SYSTEM) and that is why it is seen rotating by 45 degree while actually j'(NEW COORDINATE SYSTEM) is rotating by 45 degree. Analogy: multiplying a number by 12 is same as first multiplying that number by 3 and then multiplying the NEW OBTAINED NUMBER by 4. What you are implying is multiplying a number by 3 and again multiplying THAT number by 4 which makes no sense.
If you do the rotation, i hat is on the y coordinate. Shear transformation makes everything on y coordinate go to (1,1) line. As the same, if you do the rotation, j hat is on the x coordinate. Shear transformation makes everything on x coordinate stuck the same point.
9:06 got a question there on (AB)C vs A(BC), Isn't (AB)C is apply B then A then C; and A(BC) is apply C then B then A? don't we always start calculate inside the brackets first?
@Ahmad Alkadri wrote a nice short comment explaining this in more detail... like you said for A(BC), you apply A on (BC) which means you start with C then apply B then A. In (AB)C, you are applying the overall effect of (AB) on C, so similar to the earlier case you start with C. Since (AB) is just applying A on B, you get a sequence where you apply C then B then A -- the same order seen in both A(BC) and (AB)C.
Nice question, I got confused by the same thing. I tell myself that you *have* to evaluate right to left, even if there is a pair of parentheses on the left, but I don't really get why.
@@w0ttheh3ll Here try thinking of it this way, whichever of the 3 transformations, represented by A,B and C, you are performing on the other will be the one on the right. So: (AB)C: We start in the parentheses which means we start with A and B, and then since B is on the right hand side this means we are applying B to A (which is written AB), then we go outside the parentheses and since C is on the right we apply C to B applied to A which is the same as writing ABC A(BC): Again we start inside the parentheses, so we have B and C with C on the right so we have C applied to B (which is written BC) now we go outside the parentheses and we have A and BC and since BC is on the right hand side we have C applied to B (BC) applied to A which can be written ABC Thus no matter where the parentheses are if we go through the equality on both sides we end up with ABC so they are equal to one another
@@w0ttheh3ll I understand it as in each matrix are instructions that tell you how to skew your field (by skew I mean transform) And you can combine 2 matrixes to get a new set of instructions (as in multiplying them) So in A(BC) you skew your field first with C then B then A (pretty straight forward) In (AB)C you first make a new set of instructions from AB let's call it D and it is the same as following AB Now you have DC so skew your field using C then using D (which is the same as AB) so in the end you are basically skewing it with ABC again
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.
When I learnt this at university I really struggled to understand what the lecturer was trying to say. Having the visual representation of vectors being transformed felt so much easier to learn. Instead of a decent chunk of a two hour lecture, this is just two 10 minute videos. Crazy to me, well done.
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.
Hi man I'm an engineering student from Venezuela and I would like to translate your videos to Spanish so that other stundents from my university can use them as well. I would also like to make some videos of my own but I don't really know how to start. How did you learn to make your videos? What program do you use? could we exange contact information? PS: I relly love your videos. The deep yet friendly approach you use to teach math its refreshing. Most teachers i've had classes with focus to heavily on the operational part of maths without ever explaining the true meaning of the tools they teach.
Like saleem khatib pointed out, he uses Python to generate his graphics and animate them. Search "manim github" in google, or go to github.com/3b1b/manim He has not documented how to use his code, and it has no interface. It would require prior knowledge of Python to work with.
I apologize for my ignorance but what do you mean with "He has not documented how to use his code"? Isn't it used the same way by everyone? Anyway thanks for the info, I'll get to it.
For those of you who are programmers, think about it as traversing a 2D array with a nested for loop. You do Row A[i] * Column B[j]. Increment j in the inner for loop and increment i in the outer for loop. Thanks for the video, 3B1B!
in 7:57, I didn’t get the shear part, don’t shear keep the i-hat and move the j-hat based on the last video? Why it’s moving I-hat hear? Am I missing something?
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 😁😁
Something seemed wrong with how we were expected to intuitively figure out that matrix multiplication was associative. If we are to think of applying them as functions. Then It's *not* intuitively obvious that A(BC) and (AB)C are equivalent. In the first case. You "pass" C to B, and get what I'll call D1. Then you "pass" D1 to A and presumably get E. In the second case. You "Pass" B to A and get what I'll call D2. Then you "Pass" C to D2 and presumably get E. But It doesn't seem at all intuitive to me at all that D1 applied to A should result in the same value as C applied to D2. D1 and D2 look very different to my imagination. You're right to say that sure seemed like cheating!
Don't even think matrices, think about A, B, and C representing transformations. If you have A(BC) then you are applying the transformation C, then B, then A. If you have (AB)C, then you are applying the transformation C, then B, then A. Those are exactly the same thing. You just need to recognize that matrices are a way of representing transformations.
Applying matrix transformations is cumulative, like normal addition and multiplication. Regardless of the order they're applied (as long as the left one is applied to right one), you'll end up with the same vector. A matrix can't _take away_ from a transformation. A reversal of a transform is like adding a negative number or multiplying by a reciprocal, rather than performing subtraction or division (which _are_ non-associative).
But when you take A and B in brackets it means you multiply those matrices first, then apply transformation C, and the the product AB. But what is the product intuitively? You cant imagine the product of matrices without applying in to a vector, so it's not intuitive.
Taras Pokalchuk Numerically, you're only *multiplying* the matrices' entries and then *adding* them in a certain way, and since both operations are associative, so will the entire operation done on the matrix. You can think of a matrix as a collection of vectors (its columns), or you can think of a vector as a 1×n matrix. The reason that they're not commutative (*_AB_*_ ≠ _*_BA_*) is that vectors and matrices are ordered (_[x y] ≠ [y x]_), but even with a vector, *v*, you can still say that (*AB*)*v* = *A*(*Bv*).
Mark Cidade I think what's key to understand here, is we're talking about an intuitive proof of associativity. Not that Matrix multiplication is associative. Everyone here accepts that. Me and few others are saying that intuitively associative proof didn't seem that intuitive, because it cheated. and it cheated because it relied on the properties of associativity to demonstrate that matrix multiplication was associative. It assumed that (AB)C was equivalent to A(BC) because in both operations, C would effectively be applied to B as a first step. That assumes associativity, To evaluate (AB)C you can't just say: "OK, apply C to B, then...", that's not the order of operations that is described here. You must apply B to A, and get a transformation. D. What''s D look like? I dunno. Don't misunderstand the topic. we don't get the proof. We get what the proof proved.
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!!!
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
I am 20 now,have learned matrices since I was 12,but this knowledge is blowing my mind,how inadequate our syllabus is and how cool it is to learn this after all these years of partial knowledge
i like how much thought Grant puts in his videos. for example, 3:50 "composition" is colored from teal to pink indicating application of two transformations one after the other. hats off!
I am a little confused why m1m2 =/ m2m1 if we are also saying (AB)C = A(BC). Are we not performing a sequence of matrix multiplications in both but in the example using m1m2 the order matters but then how does the order then not matter for the ABC example of matrix multiplication?
The order is the same for (AB)C = A(BC), you are applying first transformation B, then transformation A. It's just that in (AB)C you are computing first the composite linear transformation AB and then applying it directly to C, and in A(BC) you are applying first transformation B and then A to the resulting space. It's the same order of transformations, just different order of doing the operations.
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 ❤
I want to cry right now, I have a midterm tomorrow and I’ve conceptually learned more in the past 40 minutes watching this than the past month I’ve been working on my linear algebra course
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.
God level explanation , a mystery that was bothering me since I started studying matrices finally got resolved for me. I always refer your videos for past 5 years whenever I got stuck to a mathematical explanation , but somehow missed out this gem linear algebra series. I’m thankful to you 😀.
In case we had used [[1, 0], [-1, 1]] we would have got [ [0 , -1] , [1 , 1] ] , which is not what we're doing here. Whereas , when we use [ [1 , 1] , [0 , 1] ] we get the desired composition. Additionally, the shear is taken with reference to the original basis and not after rotation.
There are some scripts like the Traditional Mongolian script where you read from top to bottom, so would that make them better at reading vectors and matrices?
In the "shear" + "rotation" example. If you do "rotation" then "shear", in the video is seems the "shear" transformation is rotating the L(i) instead of L(j)? Should the "shear" transformation be rotating j-hat 45 degree clock-wise?
No.....shear transformation means rotation of j-hat of the REFERENCE COORDINATE SYSTEM j'.Actually transformation is applied on the reference coordinate system. When rotation is done coordinate system changes and NEW COORDINATE SYSTEM becomes the reference ON WHICH the shear is applied. In the video new coordinate system is not shown whose j-hat is rotating by 45 degree. Here L(i) coincidentally matches with j' (of NEW COORDINATE SYSTEM) and that is why it is seen rotating by 45 degree while actually j'(NEW COORDINATE SYSTEM) is rotating by 45 degree. Analogy: multiplying a number by 12 is same as first multiplying that number by 3 and then multiplying the NEW OBTAINED NUMBER by 4. What you are implying is multiplying a number by 3 and again multiplying THAT number by 4 which makes no sense.
@@MrEjok New coordinate system is just the coordinate system obtained by applying rotation(anticlockwise) to the orginal coordinate system. Thus i of original system coincides with j of the new system. and j of the original system is coincide with - i of the new system.. The new coordinate system will look exactly like normal system(x and y). It is original system(axes) which will look rotated by 90 in this system.
@@MrEjok @3:40 the new coordinate system axes are in same direction as conventional direction. Also in that figure original cordinate system axes are shown with bold arrows.
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.
Hi, I've got a question... This may sound really silly but do matrices always have to do with geometry? As in, could I use matrices to express a set of values of a table of values or something? If so, then would this render the dot product of two of these matrices kind of "meaningless", or would it still indicate something relevant about the two matrices? I'm not sure if I'm expressing myself very well...
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.
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
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
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...
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 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.
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
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 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
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
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 TH-cam 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 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
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
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
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.
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.
Our professor at our university in Germany suggested your TH-cam 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.
That's what a great teacher would do. This is the future of education.
Oh my GOD it finally just clicked for me what an _indentity_ function really is! They're the _starting points_ for the basis vectors! Oh my GOD. I just gained a brand new intuition for the first time since taking Diff Eq 7 years ago. Thank you Grant!
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
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
This channel proves the power of being able to convey ideas to others. Professors couldn't do it like this guy, because they are only good at what they teach, but not good at imparting the knowledge. I feel like I can finally enjoy learning for once
the amount of effort he puts into these videos is incredible , I really appreciate his work!
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!
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
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
Thanks
Worth every penny. Thanks for such comprehensive and well developed courses and information. 😀
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
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
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.
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!
Accidentally stumbled on this video. Awesome explanation. Thanks a lot for creating and sharing this.
My ears feeling educated
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.
4:26
As a Hebrew reader, I find this good news indeed
We are obliged, man. Just Thanks❤, just ❤
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
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!
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.
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.
You are my hero man. I went from hating math to finding it beautiful from your videos. I can't understand the math without a visual representation its like for my brain without a visual the numbers are not real so its been hard for me to understand. You make it not only digestible for me but borderline enjoyable. I am a more educated person because of you. Thank you.
“It’s horrible ,just horrible” 8:52
I feel you man
26 dislikes?? Really? I bet they are just spams. Excellent series and intution. Probably the best explanation I have ever had.
@@ゾカリクゾ bruh what? nowadays there are some people who don't believe in maths? ehhh the world's getting crazier
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.
Also, it's useful to think of vectors as hop instructions. For example, [1, 2] under the ihat and jhat basis vectors means hop 1 in the ihat direction and hop 2 in the jhat direction. Under different transformed basis vecors, T(ihat) and T(jhat), [1, 2] means hop 1 in the T(ihat) direction and hop 2 in the T(jhat) direction.
I am actually crying of how beautiful this playlist is. Sir, you truly are exceptionally inspirational.
I'm so happy I found this channel!!!
That's the way it should have been taught.
Just subscribed to your Patreon and bought the 3b1b hoodie merchandise. This series is... exceptional.
Thanks so much!
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.
4:06 matrix multiplication order
1. rotation
2. shear
s(r([x,y]))
s*r ✓
r*s X
n * n-1 * ... * 2 * 1 * [x,y] ✓
n(n-1(...(2(1([x,y])))...))✓
The order of calculation *from now on* doesn't matter.
If we start on the right with 2*1, than 3*(21) and so on, will yield the same result as
if we start on the left with n*n-1, than (n n-1)*n-2 and so on.
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.
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💟
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!
This is truly eye-opening. Thank you very much!
Thanks!
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
@3Blue1Brown , at 3:18 , how did you decide the value of Shear Matrix after rotating the original coordinate system by the Rotation matrix? Not only, there but further also, would you please explain me more how do you decide the value of another matrix after rotating the basis i-head and j-head with the first matrix (rotation matrix).
Sir,Did you get it?
Brother,Did you understand it? I am unable to understand it?
Yes TH-cam serve me lot of ads , afterall it's for the great tutor.
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!
at 7:40 the shear is applied by fixing i^ and rotating j^ to the right. But then, at 7:56 , the shear is calculated by fixing j^ and rotating i^. Somehow I didn't get the concept of shear. Can someone help ? (Because if the 2nd shear is calculated like how the 1st one is calculated, the resultant transformation will be the same)
although it is quite late to answer but i would like to point out that the shear concept isn't defined based on i-hat or j-hat - something like fixing the i-hat and rotating j-hat or vice versa. Instead, it is defined by a 2x2 matrix whose the first column is 1, 0 and the second column is 1, 1.
if we do the shear, then rotate (whose transformation matrix is 0, 1 and -1, 0 for first and second column respectively), the combination transformation is 0, 1 / -1, 1, which shows the position of i-hat and j-hat (which is the same in the video)
if we do the rotation, then do the shear, the columns of combination transformation are 1, 1 (first matrix column) and -1, 0 (second matrix column) which show the position of i-hat and j-hat respectively (same in the video - clearly the order does matter)
please write it down and do the calculation yourself to verify
Yup Sanath !! am having the same doubt. Am unable to visualize (as in geometric intuition wise) the successive transformations. How is i hat and j hat affected in the second transformation ? ( considering matrix multiplication as two sequential linear transformations )
No.....shear transformation means rotation of j-hat of the REFERENCE COORDINATE SYSTEM j'.Actually transformation is applied on the reference coordinate system. When rotation is done coordinate system changes and NEW COORDINATE SYSTEM becomes the reference ON WHICH the shear is applied. In the video new coordinate system is not shown whose j-hat is rotating by 45 degree. Here L(i) coincidentally matches with j' (of NEW COORDINATE SYSTEM) and that is why it is seen rotating by 45 degree while actually j'(NEW COORDINATE SYSTEM) is rotating by 45 degree.
Analogy:
multiplying a number by 12 is same as first multiplying that number by 3 and then multiplying the NEW OBTAINED NUMBER by 4. What you are implying is multiplying a number by 3 and again multiplying THAT number by 4 which makes no sense.
I was wondering the same! Thanks for commenting this!
If you do the rotation, i hat is on the y coordinate. Shear transformation makes everything on y coordinate go to (1,1) line. As the same, if you do the rotation, j hat is on the x coordinate. Shear transformation makes everything on x coordinate stuck the same point.
what the hell! i have been all learned the matrices without knowing what it is practical....thanks a lot man!
I dont mind how many ads i get when i watch these videos!! Its for a Great Cause !! Mind Blowing Series!!!
9:06
got a question there on (AB)C vs A(BC),
Isn't (AB)C is apply B then A then C;
and A(BC) is apply C then B then A?
don't we always start calculate inside the brackets first?
@Ahmad Alkadri
wrote a nice short comment explaining this in more detail... like you said for A(BC), you apply A on (BC) which means you start with C then apply B then A. In (AB)C, you are applying the overall effect of (AB) on C, so similar to the earlier case you start with C. Since (AB) is just applying A on B, you get a sequence where you apply C then B then A -- the same order seen in both A(BC) and (AB)C.
Nice question, I got confused by the same thing. I tell myself that you *have* to evaluate right to left, even if there is a pair of parentheses on the left, but I don't really get why.
@@w0ttheh3ll Here try thinking of it this way, whichever of the 3 transformations, represented by A,B and C, you are performing on the other will be the one on the right. So:
(AB)C: We start in the parentheses which means we start with A and B, and then since B is on the right hand side this means we are applying B to A (which is written AB), then we go outside the parentheses and since C is on the right we apply C to B applied to A which is the same as writing ABC
A(BC): Again we start inside the parentheses, so we have B and C with C on the right so we have C applied to B (which is written BC) now we go outside the parentheses and we have A and BC and since BC is on the right hand side we have C applied to B (BC) applied to A which can be written ABC
Thus no matter where the parentheses are if we go through the equality on both sides we end up with ABC so they are equal to one another
@@w0ttheh3ll I understand it as in each matrix are instructions that tell you how to skew your field (by skew I mean transform)
And you can combine 2 matrixes to get a new set of instructions (as in multiplying them)
So in A(BC) you skew your field first with C then B then A (pretty straight forward)
In (AB)C you first make a new set of instructions from AB let's call it D and it is the same as following AB
Now you have DC so skew your field using C then using D (which is the same as AB) so in the end you are basically skewing it with ABC again
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.
Binge watching this series before my linear final tomorrow. I gotta say, your channel is amazing! You clearly have passion for this subject.
When I learnt this at university I really struggled to understand what the lecturer was trying to say. Having the visual representation of vectors being transformed felt so much easier to learn. Instead of a decent chunk of a two hour lecture, this is just two 10 minute videos. Crazy to me, well done.
I love how fancy and difficult these things look, then you just explain it a bit by bit and it all makes sense.
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.
Hi man I'm an engineering student from Venezuela and I would like to translate your videos to Spanish so that other stundents from my university can use them as well. I would also like to make some videos of my own but I don't really know how to start. How did you learn to make your videos? What program do you use? could we exange contact information?
PS: I relly love your videos. The deep yet friendly approach you use to teach math its refreshing. Most teachers i've had classes with focus to heavily on the operational part of maths without ever explaining the true meaning of the tools they teach.
I think someone said he uses python programing language
Like saleem khatib pointed out, he uses Python to generate his graphics and animate them. Search "manim github" in google, or go to github.com/3b1b/manim He has not documented how to use his code, and it has no interface. It would require prior knowledge of Python to work with.
How are you still alive? :(
I could help you, with this series from start
I apologize for my ignorance but what do you mean with "He has not documented how to use his code"? Isn't it used the same way by everyone?
Anyway thanks for the info, I'll get to it.
For those of you who are programmers, think about it as traversing a 2D array with a nested for loop. You do Row A[i] * Column B[j].
Increment j in the inner for loop and increment i in the outer for loop.
Thanks for the video, 3B1B!
As soon as the end music hits, I remind myself to like the video because it was so so much helpful
8 years later, a master in physics engineering later, these videos are still an enjoyable and worthwhile watch for me!
in 7:57, I didn’t get the shear part, don’t shear keep the i-hat and move the j-hat based on the last video? Why it’s moving I-hat hear? Am I missing something?
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 😁😁
Something seemed wrong with how we were expected to intuitively figure out that matrix multiplication was associative.
If we are to think of applying them as functions. Then It's *not* intuitively obvious that A(BC) and (AB)C are equivalent.
In the first case. You "pass" C to B, and get what I'll call D1. Then you "pass" D1 to A and presumably get E.
In the second case. You "Pass" B to A and get what I'll call D2. Then you "Pass" C to D2 and presumably get E.
But It doesn't seem at all intuitive to me at all that D1 applied to A should result in the same value as C applied to D2.
D1 and D2 look very different to my imagination. You're right to say that sure seemed like cheating!
Don't even think matrices, think about A, B, and C representing transformations. If you have A(BC) then you are applying the transformation C, then B, then A. If you have (AB)C, then you are applying the transformation C, then B, then A. Those are exactly the same thing. You just need to recognize that matrices are a way of representing transformations.
Applying matrix transformations is cumulative, like normal addition and multiplication. Regardless of the order they're applied (as long as the left one is applied to right one), you'll end up with the same vector. A matrix can't _take away_ from a transformation. A reversal of a transform is like adding a negative number or multiplying by a reciprocal, rather than performing subtraction or division (which _are_ non-associative).
But when you take A and B in brackets it means you multiply those matrices first, then apply transformation C, and the the product AB. But what is the product intuitively? You cant imagine the product of matrices without applying in to a vector, so it's not intuitive.
Taras Pokalchuk Numerically, you're only *multiplying* the matrices' entries and then *adding* them in a certain way, and since both operations are associative, so will the entire operation done on the matrix. You can think of a matrix as a collection of vectors (its columns), or you can think of a vector as a 1×n matrix. The reason that they're not commutative (*_AB_*_ ≠ _*_BA_*) is that vectors and matrices are ordered (_[x y] ≠ [y x]_), but even with a vector, *v*, you can still say that (*AB*)*v* = *A*(*Bv*).
Mark Cidade I think what's key to understand here, is we're talking about an intuitive proof of associativity. Not that Matrix multiplication is associative. Everyone here accepts that.
Me and few others are saying that intuitively associative proof didn't seem that intuitive, because it cheated. and it cheated because it relied on the properties of associativity to demonstrate that matrix multiplication was associative.
It assumed that (AB)C was equivalent to A(BC) because in both operations, C would effectively be applied to B as a first step. That assumes associativity,
To evaluate (AB)C you can't just say: "OK, apply C to B, then...", that's not the order of operations that is described here. You must apply B to A, and get a transformation. D. What''s D look like? I dunno.
Don't misunderstand the topic. we don't get the proof. We get what the proof proved.
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!!!
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
I am 20 now,have learned matrices since I was 12,but this knowledge is blowing my mind,how inadequate our syllabus is and how cool it is to learn this after all these years of partial knowledge
Please like the video and support him. There is no reason to dislike.
I laughed for 5 minutes straight after I realised what he was saying at 9:17
Thank you so much Man. Lots of love from Nepal
i like how much thought Grant puts in his videos. for example, 3:50 "composition" is colored from teal to pink indicating application of two transformations one after the other. hats off!
I am a little confused why m1m2 =/ m2m1 if we are also saying (AB)C = A(BC). Are we not performing a sequence of matrix multiplications in both but in the example using m1m2 the order matters but then how does the order then not matter for the ABC example of matrix multiplication?
The order is the same for (AB)C = A(BC), you are applying first transformation B, then transformation A. It's just that in (AB)C you are computing first the composite linear transformation AB and then applying it directly to C, and in A(BC) you are applying first transformation B and then A to the resulting space. It's the same order of transformations, just different order of doing the operations.
"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
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 ❤
I want to cry right now, I have a midterm tomorrow and I’ve conceptually learned more in the past 40 minutes watching this than the past month I’ve been working on my linear algebra course
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.
God level explanation , a mystery that was bothering me since I started studying matrices finally got resolved for me. I always refer your videos for past 5 years whenever I got stuck to a mathematical explanation , but somehow missed out this gem linear algebra series. I’m thankful to you 😀.
Multiplying two matrices has the same geometric meaning as applying one transformation then another. Right to left.
Awesome !
But I don't understand why the shear at 3:31 is [[1, 1], [0, 1]] and not [[1, 0], [-1, 1]] ?
In case we had used [[1, 0], [-1, 1]] we would have got [ [0 , -1] , [1 , 1] ] , which is not what we're doing here.
Whereas , when we use [ [1 , 1] , [0 , 1] ] we get the desired composition.
Additionally, the shear is taken with reference to the original basis and not after rotation.
@@jayenb4848 the shear transformation should be described as [[1, -1], [1, 0]] if I follow the logic of your last sentence and that's incorrect.
Not just for the Hebrew readers but also the Arabic readers like me!
Hahaha lol
There are some scripts like the Traditional Mongolian script where you read from top to bottom, so would that make them better at reading vectors and matrices?
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.
The best ever explanation of the mathematical concepts. Brilliant
"Good news for the Hebrew readers and bad news for the rest of us" had me on the floor😂
In the "shear" + "rotation" example. If you do "rotation" then "shear", in the video is seems the "shear" transformation is rotating the L(i) instead of L(j)? Should the "shear" transformation be rotating j-hat 45 degree clock-wise?
No.....shear transformation means rotation of j-hat of the REFERENCE COORDINATE SYSTEM j'.Actually transformation is applied on the reference coordinate system. When rotation is done coordinate system changes and NEW COORDINATE SYSTEM becomes the reference ON WHICH the shear is applied. In the video new coordinate system is not shown whose j-hat is rotating by 45 degree. Here L(i) coincidentally matches with j' (of NEW COORDINATE SYSTEM) and that is why it is seen rotating by 45 degree while actually j'(NEW COORDINATE SYSTEM) is rotating by 45 degree.
Analogy:
multiplying a number by 12 is same as first multiplying that number by 3 and then multiplying the NEW OBTAINED NUMBER by 4. What you are implying is multiplying a number by 3 and again multiplying THAT number by 4 which makes no sense.
@@ahmadfaraz9279 I think your explanation just resolved some things in my head, thank you!
@@ahmadfaraz9279 thank you for the explanation. Can you help to understand what a New coordinate system looks like?
@@MrEjok New coordinate system is just the coordinate system obtained by applying rotation(anticlockwise) to the orginal coordinate system. Thus i of original system coincides with j of the new system. and j of the original system is coincide with - i of the new system..
The new coordinate system will look exactly like normal system(x and y). It is original system(axes) which will look rotated by 90 in this system.
@@MrEjok @3:40 the new coordinate system axes are in same direction as conventional direction. Also in that figure original cordinate system axes are shown with bold arrows.
What program are u using to do the animations? They are really nice.
He's programming them himself in python
yes. thanks very much
you can use JavaScript
Reminder to all: programming is a superpower that can be self taught.
When math is intuitive like this, thinking about it becomes a source of pleasure.
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.
Hi, I've got a question... This may sound really silly but do matrices always have to do with geometry? As in, could I use matrices to express a set of values of a table of values or something? If so, then would this render the dot product of two of these matrices kind of "meaningless", or would it still indicate something relevant about the two matrices? I'm not sure if I'm expressing myself very well...