You are a gift to mankind.

This series is basically perfect, congratulations. It's strange that this subject is usually taught so badly. I attended a world class university and at no point did they properly discuss the intuition and motivation for the formal concepts.

"Unlearn what you've just learned" - finally something i can do

I first watched this series 3 years ago. But each year, I rewatch it as I learn more and more maths. Each time, I realize that when I previously watched it, I missed something important and beautiful. It's amazing how much information there is in such a work, that can be easely be missed by the untrained eye, who can't appreciate the hidden connections of math.

Its a shame that mathematics has become such an infamous subject for the average person, otherwise this channel would be far more popular, which it really deserves.

I wish there was a 3blue1brown of physics also

I have a phD in physics. I was really good at geometry during my study years... and I have NEVER seen this. And not one of my (very good) professors talked about this in this way. I am REALLY shocked, in a good way. You are the best Math Theacher on line that there is. We need one of you for every language on Earth!.

The much simpler real life application, I think, for finding a "dot product": Imagine you and your friend are pulling a big rock, tied by 2 ropes. Your friend's rope is making an angle ø, with your rope, like V shape. Now the *dot product* gives the actual amount of force your friend is adding to yours, given there is ø angle between you and your friend's direction of rope pulling. If your friend stands in same line as you, pulls exactly in the same direction where you want to move the stone towards, (or both are pulling the same rope), angle ø=0, i.e Cos(0)=1, then thats the most efficient way of pulling the stone. Bcz all his force magnitude will absolutely add up to yours. Otherwise, atleast, the lesser the angle ø between the forces, the higher the total magnitude will be. But if your friends rope (force) direction is making some angle, ø with yours, then though he's putting some X amount of absolute Force, only a fraction of X (projection of your friend's on yours) will add up to yours. If your friend makes an angle 90 degree with yours, i.e. he is pulling in perpendicular, then he's not adding up anything to your effort, cos(90)=0. But if your friend is pulling the rope in opposite direction, cos(180)=-1==>Means your friend is working against you. Then the rock moves in the direction where the force is higher, but the amount of displacement would be much lesser, yourForce-friendForce. This is also used in the Newton's second law of motion= F = m*a cos@. So basically the Dot Product tells us, how much a vector is working FOR/AGAINST the other. It's called cosine similarities! This is also used in comparing 2 things (vectors) like how similar are they, do they add up to the entirety or negating each other.. Man.. How I wish I had a board to write down, draw and explain..

"Sometimes you realize that it's easier to understand it (a vector) not as an arrow in space, but as the physical embodiment of a linear transformation - it's as if a vector is a conceptual shorthand for a linear transformation." Definitely one of the most beautiful ideas I have ever learned, thank you for articulating it so well.

This series of videos is going to last. I think presents an almost purely geometrical explanation of linear algebra that can't be found anywhere else at the moment. I believe this could be done with every math subject. As an undergraduate student of mathematics, I often find myself struggling to translate what the books say into a more intuitive mental image of what's going on. Of course, some subjects (e.g. linear algebra) are easier to visualize than others. But then again, mathematicians always try to build mental images of what they're working on, even if it is a very abstract subject. These mental images are crucial, and in my opinion they should be taught along with the (undeniably necessary) formal arguments. These videos show that animations are a very powerful tool to do this. All of this could be said about your Multivariable Calculus Course as well. Now, I realize it is a lot of work, but it would be great if you did the same thing with some other undergraduate subjects. Have you already considered this? If so, what subjects?

After viewed almost 10 times, finally I interpret the essence of this episode .

I'm saving this playlist for my children.

at this point, i've been pavlov conditioned to associate that piano sound with "you gon learn some shit"

7:02 i feel stupid for just realizing that theres always "three blue and one brown" π in those talking animations

For anyone who's lost like I was hours ago, I think I'm beginning to grasp it. See the number line as a 1D vector space with basis vector u-hat, where u-hat has coordinates ux and uy. The basis vectors i-hat and j-hat define the 2D coordinate system. The coordinates ux and uy are projections of u-hat onto the x-axis and y-axis respectively. Say you want to transform this 2D space into the 1D number line, such that L(i-hat) and L(j-hat) are projections of i-hat and j-hat onto the number line. You'll need a 1x2 matrix that will look like [L(i-hat) L(j-hat)]. As I wrote earlier, ux and uy are projections of u-hat onto the x-axis (i-hat) and y-axis (j-hat). Because of symmetry, the projections of i-hat and j-hat onto u-hat (the number line) will also be ux and uy. So, L(i-hat) = ux, L(j-hat) = uy! The matrix will be [ux uy]. Note that ANY vector, v, in the original 2D space is a linear combination of the basis vectors i-hat and j-hat. After the transformation, this still holds true: v = c1 i-hat + c2 j-hat and L(v) = c1 L(i-hat) + c2 L(j-hat). So, because L(i-hat) and L(j-hat) are projections of i-hat and j-hat, L(v) is a projection of v onto the number line! Now, the vector equivalent of the transformation matrix [ux uy] is just u-hat. If you take the dot product between a vector v and u-hat, you do exactly the same as you do with the transformation; you project v onto u-hat! So, to generalize: Taking the dot product of vectors v and w is equivalent to transforming the vector v by the matrix [wx wy]. But also equivalent to transforming the vector w by [vx vy].

To those who don't quite get the bit about the duality and how dot product could occur from that, after hours of thinking, I think I might have figured out a more direct explanation that could help you. 7:48 || 3b1b takes a copy of the number line and pastes it on the two-dimensional matrix grid, such that it is slanted and the "0" is at the origin. It is important to note that U-hat (the unit vector of the new number line) has the same length as I-hat and J-hat. We will see why in a moment. 7:50 || 3b1b explains why the new number line is a legal move, and that a function that converts 2-D vectors into a number on the new number line exists. 8:53 || 3b1b tells us that it is important to know where I-hat and J-hat land in the new number line. He explained why earlier, from 5:21 to 6:18. To add on to what 3b1b said, knowing where I-hat and J-hat lands in the new number line allows us to define ALL vectors that exist in the 2-D space just as they are in the form a * I-hat + b * J-hat. Suppose L(V) is the function to transform a given vector V onto the new number line, where V is a 2-D vector of the form a * I-hat + J-hat. Let L(I-hat) and L(J-hat) be where the I-hat and J-hat lands on the new number line respectively. Thus, L(V) = a * L(I-hat) + b * L(J-hat). a and b are given from vector V. If we can find what L(I-hat) and L(J-hat) are, then we know L(V) is possible. 8:59 || 3b1b explains to us that an arbitrary U-hat (with length equal to the lengths of I-hat and J-hat) with an x-component of Ux and y-component of Uy, has L(I-hat) and L(J-hat) as its x-component and y-component respectively. This proof can only work if U-hat has the same length as the lengths of I-hat and J-hat. Suppose otherwise, then the line of symmetry would no longer exist and the proof no longer exists. 10:07 || 3b1b shows us where the dot product comes in. At this point, 3b1b has shown us the dot product between U-hat and another vector, say, V, is really just V projected onto the number line that U-hat sits on. We are meant to prove for two vectors, say V and W, and not one of them with an arbitrary unit vector. At 10:33, 3b1b tackles this by considering projections onto non-unit vectors, i.e, projecting V on W or vice versa. He said that we should scale up the U-hat by a factor. In his example, he used 3. This took me a while to understand, but I think I finally got it. I don't think he explicitly stated this, but the new number line with U-hat, should lie on one of the vector you are trying to project on. Let's say that we want to project vector W on vector V, then the new number line should be on vector V. We need to use a unit vector, in this case U-hat, for this to work. This was explained in 8:59, I have written about it above as well. Now, we have a projection of vector W on unit vector U-hat. However, we want the projection of vector W on vector V, and not its unit vector. This is where the scaling is necessary. Since U-hat is a unit vector of the number line that vector V sits on, that would mean that vector V is a multiple of U-hat. We know this is true because they are perfectly coincident, i.e, they share the same gradient and points. Now that we know that vector V is a multiple of U-hat, how much do we need to scale U-hat by to get vector V? The answer is just the length of vector V divided by the length of U-hat. It is like when a * b = c, and you know a and c, and b is the scaling factor, then you can divide a from both sides to get b = c / a. Recall that the length of U-hat is the same as the length of I-hat and J-hat. I-hat and J-hat both have a length of one, therefore, U-hat also has a length of one. Thus, we need to scale U-hat by the length of vector V divided by one, i.e length of vector V. For the dot product, I am going to be using the asterisk (*). So A * B is the dot product between A and B. Currently, we have (U-hat) * (vector W), or [Ux Uy] [A B] (A and B are meant to be on top of each other, A and B are arbitrary letters that I have picked to represent the x and y components of vector W respectively). Scaling up U-hat by length of vector V (denoted by |V|), |V| x (U-hat) * vector W. This becomes |V|(Ux x A + Uy x B) = |V| x Ux x A + |V| x Uy x B. Let C and D be the x and y components of vector V respectively. C = |V| x Ux and D = |V| x Uy. Finally, we have AC + BD, which is the outcome of vector V * vector W. The new number line can be placed on vector V instead, and by the same arguments, we should have the same results. To summarise, (1) get U-hat and place it on one of the vectors, say V. (2) Project other vector onto U-hat, so that it is projected onto the new number line. (3) Scale U-hat up to get the original vector V. If it helps anyone, I could have this down on paper and link it here.

Man, even with your fancy well-animated and explained examples, I was having a very hard time to understand this concept and took some days when I finally got it. No, it's not your fault. I think it's simply because english is not my main language so my learning curve rely more on visual examples than anything (my english is not poor, but it's just not fluent enough to understand everything at the same pace of natural english speakers). However, I took this difficulty as good thing, as I REALLY learned all the subject of your linear algebra series by wathing every video multiple times. I just want to thank you for your effort to bring us this different method of teaching math which is actually much more stimulant than the old fashion way most of us are used to (copy and paste formulas). And your passion about how cool is this subject gives me even more inspiration to learn. Thanks from a brazilian guy.

Holy moly, this was deep. I feel as if I don't have to memorize anything for my upcoming linear algebra test -- it's all intuition! Thanks so much!

It took me a while to understand the point he was making(with regards to how projection occurs), but it makes sense in the context of previous chapters: _(3-D Example below, but it generalizes pretty intuitively imo)_ 1) *Linear Transformations and Matrix Multiplication→* Multiplying two matrices can be conceptualized as applying one linear transformation after another. I will make sense of the dot product in this context. 2) *Column Space→* The non-square vector on the right side of the dot product has column space one(rank = 1); thus, applying it as a linear transformation will shrink your original vector down into 1-dimensional space. → This is what Grant was getting at using û as the basis vector for 1-dimensional space. 3) *Basis→* The non-square matrix on the left can be conceptualized as the linearly-transformed î-intermediate, ĵ-intermediate, and k̂-intermediate coordinates(where the basis vectors land during the second linear transformation). The non-square matrix on the right can be conceptualized as just 1 basis vector(û) with 3 coordinates of landing. 4) *Dot Product→* Conceptualized as the projection of a 3-d vector onto 1-d space(the right matrix) where the product of transformed basis vectors(left matrix * right matrix) determines the extent to which each dimension contributes. *What this means→ * 1) The dot product represents the extent to which 2-vectors combine in each dimension, represented by a singular value. 2) Column vectors can be conceptualized as a linear transformation from n-dimensional space onto 1-dimensional space.

I think I am starting to understand it now. It really helps me to see it this way. I finally got more than halfway through your series and my brain is finally starting to put pieces together. I will keep watching and re-watching until it sticks. I'm gonna try to solve some problems now to see if I can solve them and also understand them better than I have before. Thank you for these!

Sincerely, this series of videos has taught me more than the whole Linear Algebra subject I took many years ago when I was studying Computer Science. It addresses precisely the most important problem I found back them and that is the lack of geometrical interpretation. Now I'm a Postdoc in Computer Graphics and still find myself finally understanding some details here and there. Thank you very much. I would gladly pay tons for something like this applied to more advanced concepts.

I love the small pauses that give you a chance to reflect on maths' significance and beauty.

If anyone's having trouble understanding, this is how I've come to think of it after watching the video a few times: 1. To turn 1 unit (u) of the number line into a vector, you'll have to project it from the number line onto the xy-plane. That projection will be marked by u's coordinates (ux, uy), as shown at 9:30 2. To go back to the number line from the xy-plane, move ux units along the x-axis, and then move uy units parallel to the y-axis, and you'll get back to u on the number line. 3. To turn any vector on the xy-plane into a number on the number line, you'll also have to mark its location in the xy-plane by taking note of its coordinates, (x,y). Then if you scale those coordinates by u's coordinates, you'll be able to get to the number line by walking x*ux units along the x-axis, then y*uy units parallel to the y-axis. And x*ux + y*uy is basically just the dot product of [x, y] and u.

7:22 "unlearn what you have learned" makes me want to smash the like button more times than KZfaq will allow, Master Yoda.

It took me this much to realize why your function always makes 2 7 To [1.8]. It's e 2.71828...

In computer graphics, I think there is a _far_ more intuitive way of understanding the dot product - it is a measure of similarity between two unit length vectors, 1 when identical, 0 when orthogonal, and -1 when perfectly opposite. (This can be intuited by the fact that a . b = |a||b|cos(theta)). This comes up _constantly_ when writing lighting shaders, since the amount of diffuse light that bounces off a surface is generally L . N, where L is the direction from the surface to the light source and N is the normal of the surface. (This is fantastic because it has a direct visual interpretation.) Also, interestingly enough, I believe Google employs a concept of "cosine similarity" on vectors of words when ranking websites for search.

I appreciate that you use a black background. Its easy on my eyes because i get headaches very easily

Thank you. Thank you so much! Dot products finally make sense on what they physically represent. However, I will say that this video is presented in a confusing order: it mentions that dot products relate to projections before proving why much later. I had to watch the second half of the video first in order to understand the first part's discussion of why order does not matter.

You've done a truly excellent job with these videos. The geometric insight into linear algebra is wonderful, and I say that as someone who used linear algebra almost daily during my career at NASA.

I've been a graphics programmer for over 8 years, I've also written countless collision systems, animation engines, physics engines, and your videos are still blowing my mind. How is this not the standard way of teaching linear algebra in school/uni?

Currently learning about this in much less detail in my trig class, but I really appreciate the number line translation you're explaining here. I typically need to hear/see a concept a few different ways to really get it hard wired into the brain and this number line translation helped a lot, thank you!

Had to watch this multiple times to understand. Congrats! You blew my mind.

The last four minutes of this lecture are absolutely beautiful!!! Thank 3B1B for visualizing these and making such beautiful connections between different concepts!!!

Funny, how the things we have been taught for years finally makes sense. Beautiful work.

I went through it for more than 1 hour pausing at each point made. I have never attended college and learned through textbooks recommended by first class colleges and the Khan Academy teachings (in particular) to get my engineering credentials for a living. I jammed into my head all these math proofs like an Ape and try to relate them to the real world of tangible things. Thank God that I found You to light up the bulbs and cast away the shadows (vague conjectures of realities). Example on dot product. You sum it beautifully as a linear transformation from 2D space not defined as numerical vectors but projecting space onto a diagonal copy of a number line. I take this and others as wings to fly now. Yes. I signed onto Patreon as a small token of appreciation.

Gonna have to rewatch this one a few times before i understand what is going on.

Your channel should trend monthly. I want to believe there is a reason why most people find math repulsive. A reason somehow more meaningful than the way it is taught in school. I think that knowing these types of things gives you a satisfying and powerful understanding of almost everything around you. Maybe through some divine phenomenon, that power is kept in the hands of few people, because if everyone found this stuff easy, well, they might not use it for good. But then again what am i saying think of how advanced we would be as a civilization there would be no room for evil. I dont kno

This one took some time to wrap my mind around but it is truly amazing! Thank you for this series, it's a great help to develop an intuition about linear algebra and all its applications.

Every time I re-watch this video, I feel happier and more realized, I feel hope again, and I am not exaggerating. Thanks for your work, sir.

The great math class I never had in college, so insightful, amazed by the way you described those math concepts in a tangible way. Great work!

Woow just wow I have seen this video so many times already and it just keeps blowing my mind This is so beautiful I hope that second series is coming soon for all the Jordan, IP and spectral stuff Thank you very much!

Ok, I love the series so far. Chapter 7, however, took me by surprise. I spent three days and scribbled out 17 pages of notes trying to assimilate Chapter 7. I still catch myself daydreaming, trying to visualize the contortions those poor unit vectors must go through to exit 2-D space and find their new home on the number line. I admit I'm a little fearful of tackling cross products in Chapter 8.

I am simply overwhelmed by the clarity that I get after watching your video. Inspiring. Keep making these videos. Please.

This is just . . . Beautiful! I just feel so lucky to have discovered your channel, because, no teacher, no professor, no matter how we advance in our fields that are related to maths, explains like this and shows the deep essence of the science (except if your main specialization is mathematics).

I can’t know how to thank you more. Those videos are brilliant! As someone with a computer science PhD, linear algebra never looked that meaningful to me before. You cannot guess how many “aha” moments I got during your videos. Before anything else, these videos should be shown to students in every LA course all over the world. Thank you!

For anyone who is still struggling to grasp, what finally made this concept click for me was to think of the "dual" of the vector as the UNIQUE LINE in space that now represents your 1D number line. A number line can live anywhere in a 2D space (a 1,0 number line would be the straight line along the X-axis, a 0,1 number line would be the straight line along the Y-axis, or the U-hat number line that was diagonal through the origin looked like coordinates of ~(0.5,0.5)) So, 2D space in a sense remains absolute. When we project onto a 1D number line, we're essentially saying "this is your PARTICULAR 1D "playing field" (number line) right now and it lives in a unique place within 2D space" but there are an truly infinite number of 1D number lines capable of being generated from the 2D space. The three examples I gave are just a subset of all the possibilities. Thus, because each number line is unique (the 1,0 number line doesn't LOOK like the 0.5,0.5 number line in 2D space), each projection onto 1D space is unique, as represented by the unique 2D vector that codifies where the number line "originally lived" in 2D space.

I've had rough 4 years of undergraduate studies and got two more semesters extended. In my last semester before graduation, watching your videos on vectors gives me strength, 'cause I'm taking Vector Calculus. I would get nothing out of my course, if not for your beautiful explanation and eye-opening visuals. Now I am sure that if I can understand these videos, then practically anyone can enjoy, let alone learn, mathematics. Thank you so much! P.S. I keep rewatching your course playlists to root the concepts and entertain, as well.

This is the BEST explanation I’ve ever seen. This is exactly what’s happening behind primal-dual optimization. Thank you!

Great video! I'm now taking functional analysis where duality is a crucial topic. This video is incredibly helpful for building my understanding. Returning to intuitively simpler vector spaces and understanding duality here helps a lot.

I am surprised how these 15-20 minutes of videos are teaching me more than my high school courses that lasted for more than 9 months.

you nailed this one! i also learned this concept almost at the end of my lineair algebra course, it just makes way more sense that way, and you can really dive into the concepts of inner product spaces

These videos are amazing! My only criticism is that you started using "projection" and "unit vector" without fulling explaining what they were. As someone with no background in linear algebra, I had to look these up.

Sal introduced me to math, 3Blue1Brown made me appreciate and fall in love with it

If this feels a bit overwhelming, ask drunken Grant to explain this to you. You reach him by setting the video speed at 0.5x. He's just as smart as regular Grant but has a more laid back style.

Well done guys, this just blew me away! I have struggled for a time to really intuitively understand the notion of a vector space and its dual. The way tensors and forms are framed in this language is something I've used for a Long time but never truly understood intuitively. This video (and the one on determinants) was an epiphany for me and everything I've struggled with (tensors, forms, differential geometry etc) just fell into place. I feel humbled that something I thought I knew so well, seen in a different way could illuminate so much more. I cannot recommend these videos enough.

Wow! This one was quite tough but really awesome once I stopped to think about it. The toughest bit was understanding why exactly duality holds for non-unit vectors, but I think I've finally got it. Thanks for this series, it's great and really approachable, like most of the stuff on your channel.

This is the most enlightening video I’ve ever watched on YT and probably in my entire life... I still can’t stop smiling :’) thanks a ton, Grant! 🤲🏼

Man, this is the best explanation I have seen for Linear Algebra, excellent series, I can't wait for the next video. It felt great to find the series just at the moment I'm starting to learn Linear Algebra at the University, it has helped me to understand the true meaning of what I have been learning. Keep it up!

4+ years later...still my favorite video on KZfaq.

Thank you so much for this video!! I also always had this problem with the fact, that the projection isn't symmetrical but I never thought about it that way!! I now understand it which is absolutely amazing!!

Suddenly, the way the dot product was first introduced in my maths textbook - the length of the first vector times the length of the second vector times the cosine of the angle between them - makes perfect sense!

aaaaand this is where you lost me. lol. back to review i go.

Your teaching is unparalleled. The best explanations, with the animations to build the best intuition. When I think mathematically, I think in terms of 3B1Bs animations.

You, man...You, just make incredible videos on mathematics...I never thought all of this could ever have geometric interpretation. Hats off to you.

I'm no mathematician and I'm just a dilettante when it comes to the subject matter. But boy is it interesting. I always had respect for math and my math professors. Got me all the way to calc III. Love this series of videos. Also completely approve of the Calvin and Hobbes dialogue at the beginning!

I wish I watched your videos back when I was taking my linear algebra course. Life would have been really simple! You are an Ubermensch by all means, Cheers!

I've come back to this video countless times, and each time it helps just as much

Absolutely bloody amazing, it took me almost an hour to pause and rewind multiple times throughout the video, but now that I GET IT, i feel enlightened!

The definition you gave of dot product is only valid under the right hypotheses, which are not always validated in linear algebra since you could take any base of vectors for the set you're representing. Instead, the more general geometric definition gives a clearer view of the geometric properties of the dot product, and makes the formula you give trivial to find under the right circumstances. Great series, thanks a lot for your work.

wow, just wow. this was amazing. I never understood why scalar product actually works...

Bravo sir. This was absolutely brilliant! Having a degree in math from one of the most trending universities, but seeing the first time in years this actually not-so-simple concept the way it was intended to be... Here is my deepest bow.

Only looked this up in preparation for my linear algebra intro course, now I am fully subsumed by the beauty in mathematical manipulations themselves..because of you Grant!!

Something eluded me the first time a saw this. However, after learning about "change of basis" in a rigorous manner, I think the explanations that used "symmetry" and "projections" finally clicked. The diagonal blue line that is spanned by u-hat is a vector (sub)space U; the projections are linear transformations. Since the length of u-hat, i-hat, and j-hat are all the same, then the length of the projection of u-hat to i-hat is equal to i-hat's projection to u-hat, i.e. u_x. Note that the projection of i-hat to u-hat is a projection of i-hat to U. Thus, in a "change of basis", i-hat has a "coordinate" of u_x in U with respect to U's basis which contains one vector, that is {u-hat}. Similarly, the coordinate of j-hat in U with respect to {u-hat} is u_y.

U demonstrate one thing: There are no bad students. Only bad teachers. All the people who thought they hate math, will definitely change their mind after watching these videos. Respect, for all the time and efforts u put into making this available 🙏

The duality of convolution has always amazed me. The method of fractional resampling for signal timing correction is the best example. In the end rather than adjusting the time index of the sample you adjust the filter to its location at the time that index of signal would have occured because of the duality... i used to explain it as like the farnsworth engine in futurama... it doesnt move the ship forward, it pulls space backward.

I LOVE LOVE LOVE your "alternative pacings" to things because it helps set such a CEMENT understanding of things to know "mathematically yeah sure, but *why*?" You do such a good job helping conceptualize such nebulous topics.

One easy way to understand dot product is to relate it with the formula of _cos(a-b)_ , i.e. _Cos(a-b)= Cos(a)Cos(b)+Sin(a)Sin(b)_ if you replace *Sin(a)Sin(b)* with *Cos(90-a)Cos(90-b)* which are essentially the angles the vectors make with y axes if a,b are angles with X-axis. This is exactly the same as dot product of the vector matrices of two unit vectors, as in both cases we are multiplying X and Y component of unit vectors and adding them. You can look up the geometric proof _Cos(a-b)= Cos(a)Cos(b)+Sin(a)Sin(b)_ and think of the line segments as unit vectors and you will understand that we are dong exactly the same thing in both the processes but under different names, that is line segments and vectors. You can do the same for higher dimensions.

Now I know why dot product of two perpendicular vectors is 0. Such an amazing series, loving it.

You know you explain so much in very less words that i have to watch it many times before even thinking of writing it.But when i understand it i feel so awesome.I am even making notes of these and trust me its really hard to note these things and write it in your own words but i think its really a good way of revisiting those thoughts you taught us. Thankyou for your work sir.☺️☺️

this is an incredible supplement (if not replacement) for the precalc class i’m taking, everything makes so much more sense, thank you 3B1B we love you!

Hey Grant I know you're not gonna read this but you're my favorite person after Einstein

Thanks, now I understand. Also, a vector is a n x 1 matrix projecting 1D space into n-D space with Rank one.

This is absolutely great! I think I see for the first time "intuitive" explanation for what det and rank is. I took linear algebra course at the university but we were just using them to solve systems of equations. They just were. I feel almost betrayed that nobody told me this explanation then. (and nobody mentioned anything about eigenvalues !!!) This series is just wonderful! Keep doing the great stuff! And I really like it is strongly focused on essence not on computing: computing I can find anywhere.

I think I have watched the part between 6:30 and 10:00 at least 8 times, but it finally clicked and made geometric sense. Thank you for the cool new way to look at linear algebra. I love how it gives us a more intuitive way to think about vectors.

I really wish they'd emphasised these more in school in the 90's, given how important they are for game programming today.

I'm not going to lie. This one fried my brain a little bit. I need to take a nap then come back to fully process this haha. It seems amazing though.

The most intuitive video I have ever seen in my life.Thank you

9:22 and 10:11 are awesome moments of realizations. This episode is by far the one that most challenges my abstract reasoning ability. Thank you so much for the videos!

2:37 My intuition for "Why the order doesn't matter" was to consider the area of the triangle bounded by the 2 vectors. What all of this means is that you get the area of that triangle by 2 ways.

This is real beauty , my eyes tears sometimes from such elegance !

Had the same question related to the relation between dot product of vectors and projection before starting this video, had a smile on my face when Grant started with this question in middle and tried to explain it. That's the inspiring thing about this channel, diving deep into the why of these relations :)

the animation at 5:12 is just! can't find a word for it Thank you for taking the effort of making this series! I'm currently watching it as a repetition from what I learned (almost only in an abstract manner) in my first semester and it's just perfect to fill in some gaps of knowledge and to get a deeper understanding of what's actually going on. Especially the concept of matrices that convert for example a 2D Vector into a 1D, 2D or 3D vector was totally new to me.

For those who don't know there is a different notation for the dot product. Imagine that you have a vector x and y (of same size), the dot product of those vectors can be noted as x • y but also as x^Ty. The T means transpose which means that you turn the vector *x* on it's side to get the associated matrix as shown on the video.

i found 9:30 super hard to understand visually with what is being said. What i see happening is that to project i-hat onto u you make a 90 degree line from u onto the head of i-hat. To project u onto i-hat you make a 90 degree line from i-hat that crosses the head of u. Super great videos and representations. Its a very nice way of thinking about the cases

My goodness, this single video actually helped me realize what Reisz Representation Theorem is actually saying: this duality of dot products on R^n generalize to abstract vector spaces with abstract inner products. Thank you so very much!

I learned it when i was about 16 and now i'm 27 if someone had taught me the way u taught i would have no need to learn it again... Thanks you very much!!!

This is so unfair. All my life I suffered with linear algebra (I am into data science) and only now the does the author post these videos. :P

Riesz theorem spotted! An amazing video, always teaches me something new

Amazing, These videos really make concepts much easier to understand. Duality is really beautiful.

This video is a gem, this is what is needed in education, I understood each and every part and really say this is never taught it is also hard to teach these topics without actual visualization, there were parts where the graph was not meant to be scaled, I also like the simplicity and focus not giving irrelevant examples, when I will grow I will surely donate to this channel so that it can keep up with these videos for the future generation, I am still 16 and fully support this guy, Hats off huge respect from India sir🎉🎉🎉