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Keep the good work up! Amazing video! Thanks for sharing! My only suggestion would be lowering just a tiny bit the music in the background. The rest is amazing!

Beautiful!!!

Thankyou for making this great masterpiece! Absolutely mind blowing to see how delightful linear algebra is :D

This is a really clear and good explanation. The introduction of the transpose in basic courses is deceptively easy. Even a child could understand the definition of switching rows and columns. This hides the fact that the transpose is actually one of the deepest matrix operations, mathematically, you encounter in the first courses. You just don't typically realize it.

You made that transpose video for fun ? Legendary.

I am going to watch this again ❤

Is there a real-life situation when we want to find the M transformation that preserve the dot product? ... Super neat exploration and I'm trying to understand better. Thank you!

Simply amazing, it is by far the best video I've found about this topic.

The easiest looking matrix operation has the most difficult geometric intuition. A very good explanation indeed! Thank you

from my heart , i would like to say thank you , with high appreciate for all of your videos, i believe your video are helping alot of student are struggling with LA.😊😊😊

5:02 vTw is a 1 by 1 matrix and the dot product is a number so these don't seem to be the same thing

This is the most underrated video ever! It should have the same amount of views as 3Blue1Brown has

Truly a superb explanation!!

Wow, finally someone that explained it well! magnificant!

This is amazing! When you discussed how the dot product of x and v retains by applying different (but almost similar) matrix transformations in 15:11, a nice way to look at it also is through the formula of the dot product (x*v) = ||x|| ||v|| cos\theta. Since x and v are essentially applied with the same orthogonal matrices from the decomposition, theta will be unchanged. But scaling x using Sigma and scaling v using the Sigma inverse will tell that ||x|| is scaled "up" as much as ||v|| is scaled "down" and so the dot product is retained. regardless, great visual intuition! I'm loving the current YT community that teaches mathematics.

Perfect explanition doesn't exsi...

I wish you had shown what A transpose does to the output of A. I get that it isn't clean, but its also what we came here for.

Awesome -- thanks! Do you share your Manim (presumably) code on Github for other presenters to learn from?

+1 sub

didn't quite get it, i'll be referring back to this video a bunch of times in the future throughout my engineering degree. I don't even have to know this yet, but I find it extremely helpful to understand concepts instead of just memorizing the exercises that one must solve in order to pass the test. Thank you for putting this knowledge online <3

Thank you so much for this!

鼠疫

累人

So if I understand correctly, is a transpose sort of like an "inversion" for the reflection of Ax over the line y=x, which brings you back to x?

Man you f****ing Rock!!! 😮 the best video on the subject! And i have watched many…. Top notch explanation and video

11:48 Thank you.

Great explanation!

I came here from MIT ocw and this video is too good!!! Thank you.

Thank you.

Dunno what your occupation is but you clearly have an educator/ideas sharing talent.

Bro casually explained SVD in 20 seconds better than most can do while covering a different topic.

was doing some graphics programming and this clears things up. thank you!

hi there, fellow math educator :) this is my first time watching your channel. great visuals and explanations! my only criticism is that the music was a bit loud and distracting. at the very least i'd say reduce the music volume (or do away with it), and if you keep music, i'd choose music with much lower tempo. the choice for this video felt a bit too 'fast' personally. but otherwise, great content! i'm looking forward to future videos. best of luck :)

This video is so beautiful !

Such an amazing video. I'm shocked that you only have 924 subscribers. You explained this so well, and so elegantly, it really is truly amazing. Linear algebra is so beautiful. Thank you.

Thanks man i discovered retrosynthwave!! Have an icecream from my side!

Excellent exposition!!!

Can yu explain what is cofactor geometrically?..❤

Bro you're 2nd 3B1B! Keep going✨🔥

This was very well done!

As a graduate student in physics, this was very helpful in grounding the definition of unitary transformations, thanks so much and beautiful video!

underrated af

I remember asking my linear algebra teacher this exact thing and he just looked at me weird and said, "you just change rows and column". I stopped asking him stuff after that.

You are going to blow up in millions very quickly...mark my words!! Commenting here to get atleast thousands of likes from million views😉😁

I don't really get the premise of the explanation. Wouldn't there be infinitely many new vectors v bar that satisfy x dot v = x bar dot v bar with x bar = A x and thus infinetly many matrices to get us there? What is so special about the transpose of the inverse of A then? It does of course satisfy the equation, but so could infinetly many other matrices, no?

Wowwwwwwwww22❤❤❤❤❤❤

Beautiful 😻

Is the choice of x and v as names for the vectors instead of u and v a deliberate choise so that students don't mess up because u and v are very similarly written ? If yes, this is another proof of the care you put in the video who is very good

I like your video as i often get confused while watching other linearly algebra videos. The recap part in the beginning is very neat.

Thank you for the video. Very clear explanation of transpose and SVD. I am motivated by applications… The SVD is so insanely powerful! Could you make a video that illustrates how the unitary rotation/scale/rotation of the SVD solve a problem? That would be so helpful! Thank you for sharing.