Visualising matrix elements as dials is genius! Glad to see a new video and excited for what comes next. :)

Best tutorial. It doesn't give you a formula to summarize, it gives INTUITION. Greate job

I kinda knew this but seeing it really helps. Especially that third row.

Great visualization with the "dials" in the matrix and very nice way of connecting 2D transformations with the side of a 3D cube. I never saw it so neatly presented.

I really enjoyed this explanation - one of the best I have seen. I particularly liked your dial visualisation! I always knew it did that but seeing each dial and how the vertices moved accordingly really made it click - especially in the 3D mapped to 2D case.

your detailed explanation in algorithm archive really amazed me!!

Very cool! The perspective of holding all matrix values except one constant and shifting it while seeing the effect on the square is new to me. Great stuff, keep it up, subscribed!

No budy in the world had ever taught math like this!! Thanks a lot ❤

So mesmerizing.. A beautiful intuitive explanation I've ever seen! Thank you bro!

This was an interesting and helpful visualization of affine transformations. I liked linear algebra, programming, and their applications in games, and I always wondered how to visualize the points of the 2d affine transformation in 3d space. I also never noticed that the rotation matrix could also be a composition of shearing and scaling (but also I learned that it is also a composition of two reflections).

That is so cool can't wait to see what's next.

Wow, never thought of visualizing the elements as controling shearing and scaling. And consolidating 2D affine into 3D linear is just wonderful

This visuals are great. Really enjoyed this video.

An excellent piece of video. Teaches you 2 days of studying the topic on the article. I can't imagine the effort being put onto these visualization tho. Thanks!

That was a great video to get a intuition. Hope to see your next video soon.

Really useful - quick and comprehensible. Thank you!

That was a great explanation and clearly showed the connection and translation (pun intended) from numbers to graphics. 👍

This is the best explanation of affine transformation out there

Please make more videos this is great

This is a great video about tensors ! You should totally do a video about fluid mechanics and Reynolds stresses ^^

Great video!! Jusr finished linear algebra course and watching those kind of explanations is awesome and inspiring!

Absolutely amazing explanation, thank you so much! Way more intuitive than my course material.

Great Visualization

your visualization works are awesome!

You are a god at communicating this stuff dude, thank you!

This is a very clear intuitive explanation man, thanks!

You are talented. Please upload more of your videos.

with visualizations, the affine transformations are illustrated in a very simple and easy to understand manner. very thanks

Awesome video. Super easy to follow

actually great, and intuitively easy to understand, explanation

Thanks brother! Great video

That was an amazing perspective to look at it..

Beautiful :). Great animations; very intutive.

Thats insane , a nother video about it please , the couple passed days ive just start digging about it , and yeah here u ar . More detailed video would be pleasuer. Great work man !

Best explanation for affine transformations

My God, if only this sort of explanation was available back in the 70's when I was studying calculus and advanced math. Eventually, I even had a set of microfilms of Oliver Heaviside's notebooks and was working through and following his work. Eventually I moved on to software work and donated it all to the Brown Univ. Electrical Engineering Dept. where, I'm sure, it was long since lost and forgotten.

yay, new leios~

Sometimes i really feel tht this is very underrated math channel ❤️🔥

The concept is super awesome!! Hope every high school teacher introduces matrix in such intuitive ways!!

Great idea with the dials!

Fantastically explained! 👍🙂

You are amazing brother ! Halelujah

great explanation

absolutely amazing

so cool explanation!

Thanks a lot，it's clear and intuitive.

What a great and clear video!

awesome explanation

woahh, was just learning opengl and wondering why the need of vec4/mat4 instead of vec3/mat3x3. Really well made video!

I love the use of dialogy to solve the matrix math problem of representation. In a quantum system of adjustment, a "::" has toggles and sliders to shift coupled AI perception. Since a "::", ":;", ";:", etc. is bijective and modular between containers.

The science boss is back! I must say Jimmy that you are looking affine! Great video, but what I'm really wondering is..... how does this transformation relate to the tesseract?

In my book, matrices were explained with some abstract definition (great for math student's first course...), and as a system of linear equations From 3blue1brown's videos I've learned to see the columns as where the basis vectors land, which helped a great deal in the visualisations Now this seeing the numbers as dials really makes me feel like I understand how to look at the numbers of a matrix and see it's effect as a transformation, especially combined with 3b1b's basis column visuals

I was a little put off by the exaggeratedly compassionate soft warm voice but the visualization gave me a really nice mind blown moment. Thanks!

Those are really cool ideas! I certainly like more the one that translation by using augmented matrix is basically a shear using 4th dimension. But the one about rotation = shear + scale is certainly adds to understanding to how a shear works for me.

My intuition is that the columns in the matrix tell you where the (tips of) the unit vectors end up. In my mind, I also picture it as viewing from the top at z=1 so that the last column works properly. That way I can quickly create a grid from those vectors and draw the shape in that transformed grid

I've just started watching , it's so good, have you been teaching?

The way you explained how translation works with the homogenous coordinate was awesome, I finally understand it now :) If we're only concerned about 2D, what would happen if that third coordinate is not 1?

Great mental model with the horizontal/vertical/diagonal dials. Love it! Any teaser on what topics we get to look forward to with your videos in 2021?

Great idea with matrix of dials!

so great !!

Excellent!

interesting perspective!

Great video! Homogenous coordinates are an awesome topic and (in my opinion) not hacky at all! I'd love to see more on them

that was soo cool 😊

visualisations just make this thing a whole lot more sensible!

I really liked the graphic work you did in the video. Can you please tell me how you did it?

wow, super video

thanks for making this video... makes math so much easier!

Nice. I like the visualizations. You can also do this in an extended complex form with √k instead of i. If I remember correctly, k is cotangent to the angle of a line through the origin from the lower left to the upper right corners. The sign of k defines whether it is stretching or skewing and whether rotations are elliptic or hyperbolic. At 45⁰, it's circular rotation if negative. If positive, the angle is of the asymptote of the rotational hyperbola. It can be more computationally and space efficient than matrices in some cases. You might be able to do the same hacky combination of the affine using a similarly extended quaternion but I haven't tried that yet.

great thank you

hi. james was it? anyways, im one of your mom's students

Great vid! Ive been learning tensors recently so - are there any plans for a relative video?

Its staying in the head for longer after watching this video!

Like before I even watch

Cooler!

I understand matrices in completely different way that allows me to construct them directly. Multiply matrix times vectors by hand writing down complete equations like v.x'=v.x*r1c1+v.y*r1c2+v.z*r1c3 (normally m11... or m00..., i wanted to make clear what is row and column) Now try unit vector v.x=1. You will notice that first column is basically how will transformed unit vector x look like. Second column is for vector y, third for z. If you take any vector you can write it as sum of unit vectors with some scale so it can transform any vector. Do you want rotation by 30 degrees? Ok. Let's start with x. Ok, unit vector x will be tranformed into cos30 in x and sin30 in y -> first column. vector y will be transformed into something pointing left and up, so -sin30 in x and cos30 in y -> second column. Do you want translation for points, but not vectors? Expand matrix to 3x3, assign z=1 to points and z=0 to vectors. Write translation in tx,ty to third column and let r3c3=1. Now if v.z is equal 1, it will be transformed into vector tx,ty and z will remain 1. If original vector had v.z=0 it won't be affected by translation and z will remain 0. You may ignore that z is "borrowed" from 3rd dimension and you may call it w to make it less confusing and compatible with 3D transformations. This way you can even do some stuff like align two 3D objects, e.g to align screw with hole in some mechanical part if you can measure some vectors and points for reference.

HOPE YOU WERE THERE IN MY COLLEGE AS MY PROF.

good video.

After too many long days

Instant like :)

eyyyyy you made a video

wow!

I still have vague idea about matrix and sin/cos. Is there any visualize d material can help me get it easier?

"rotation = scaling + shearing" BAM!!!

nice! i am looking for grid deformation due to straightening a function. say i am traveling along y=0, in our 3d world (z=0), using perspective, i'd see equally deformed "squares" on both my sides. in 2d, the cartesian grid will not change, as the line is straight and parallel to the x-axis. I would like to "see" what happens to the grid if i travel along y=x, and if i make the graph a straight horizontal line. same for y=x^2 and for y=x^3 and for y=sin(x). can you point me in the right direction? thank you!

Could have been awesome if you tried the hypercube that way ^^

May be it was a bit fast but what is the benefit of adding an extra dimension in your last example?

Good

What tool do you use for the animation or simulation.

How do you make animations like in this video? Could you do a “how to”?

may i know how to make these kind of videos ? this looks similar to 3b1b videos , is there any software for such video making ?

hello what is the difference between affine and mathomogeneous matrix

Hey won't you be teaching Julia class this year?

now if only there were a way to easily slice a geometric figure with an n-1 dimensional knife

why do people say an affine transformation means to 'forget the origin' , when you are clearing using the origin.

a green screen could help to look your demo better~?

Affine transforms are just a subset of 3D transforms projected into 2D.. Maybe n to n-1 D too? What does it all mean??? 😖

This is a weird comment. But I feel like your voice doesn't match your face. Like I've never felt this way before in my life and it's really strange. It's not an insult at all. You have both a nice voice and nice face! It's just super weird seeing you sound like that. Idk how to describe it!!! It's crazy. Has anyone said this to you before? Awesome video btw. I'm studying fractal topology and this was super helpful!

@leiosOS hello i have been watching your videos for quite a long while now, and I'd like to ask you if you knew about this programming language called processing? your style of making videos and math would translate (hehe) really well, maybe check it out!

Long time no see. 🤨