The fact that after so many years you still bother to answer questions from comments under old videos and even remake a 4 year old video... You're an amazing teacher, nothing but respect for you!

I have never seen the visualisation of vectors you show at the very end of the video before. This is basically an epiphany for me. Taking the basis to be rows while taking the components to be columns suddenly makes everything I have heard about vectors and covectors make sense. This video has already been truly invaluable to me.

WHAT THE FUCK HOW DID I NEVER MANAGE TO LEARN ABOUT THAT WAY OF MULTIPLYING MATRICES that alone makes this video a literal godsend tbh thank u so much

This remake is very welcome. One of the best introductory courses I found on this plattform.

I'd like to thank you for your care, even on such early lessons like that. Personally, your lectures helped me in a presentation about Schwarzschild's solution to Einstein's Field Equations

I watched your series on general relativity. It was awesome! You make these subjects so navigable. I hope you never stop.

Heaping on the deserved laudations. Your effort is noble and nearly brings me to tears. Thank you. I've studied row and column multiplication, and I started Arfken's Math for physicists, where one of the opening problems is rotating a 2d basis, and I've heard people say "nevermind that torque is a pseudo bi-vector," but I did not know linear algebra and that Arfken stuff was baby tensor math. Your presentation has got me excited, because it might not be as scary as I thought :D Second smily for emphasis :D

Thank you Chris. You answered the exact question I had yesterday in your tensor calculus series. Didn't understand why the jacobian matrix was not transposed. Basis vectors are covariant. Covariant vectors are covectors. Covectors multiply from the left. Thank you

Excellent presentation. It assumes that the student know very little or nothing. This shows experience in teaching and good common sense on the part of the instructor.

Thank you for the remake!

Excellent explanation. Thank you for taking the time ...

Excellent quality as always

Thanks from an experimental physicist--great work.

Hopefully you updated/continue this series!

kudo man. That mistake has caused me much difficulty for years.

Great video chris!

I was reading quite a few books on Tensors for beginners, and then i came across your series of Tensors for beginners(many years ago when you just started making those initial videos)..... Initially it felt the best source for learning tensors however after the error in this particular video and then the correction video you made earlier it all felt so much confusing that i tried many times on multiple occasions to clear things up and make a coherent sense of all the books i have and your (this particular) video but i couldn't.... Ultimately each time i had to drop this topic and move further only to get stuck in advance as I didn't know basics well...... I really hope this time when I'm going to give a new try with this video, i really can make a sense of all the books and your videos and finally understand the proper basics of it.... Meanwhile all these years i was just wondering you moved ahead continuing your series all the way up to General relativity and it made me wonder even if you could revisit your older videos and make a correction for people like me so I can reach the advance videos with you as well..... In hopes i always had your notifications ON...... And woah ho finally you did that today ..... I am so so much grateful for you, for that ...... I'm a high school physics teacher, but Tensors has always been something i really really want to understand and learn properly....and you are the best source I have found till date, again thankyou for this correction video

Wow I just found this video for the first time and haven't watched it yet but I'm sure glad whatever mistake was in the original one I missed because by the sounds of these comments it was a doozy.

Thanks, EigenChris. Actually, your original video on this subject with your mistake was a good pedagogic device to point the matrix entries of the individual equations. You are one of kind, an excellent teacher.

Thanks a lot. Still hope to see you doing stuff on Dirac spinors, QM and QFT!

to understand all of this you really have this notation and linear algebra manipulations under your belt, same if true for quantumfieldtheory, I'm often lost in notation 😂

Hey, thanks for this series! Just a question - In the very beginning, you say that the matrix representing the "forward transformation" eats vectors in the old, blue basis and outputs vectors in the new, red basis. But this doesn't quite sit right with me. If I feed the vector v with components (1, 0) into this matrix via F*v, the result is (2, 1). So clearly, what seems to be happening is F eats vectors in the red, what you call "new" basis, such as (1, 0), and outputs vectors in the blue, what you call "old" basis...

awesome! I get that they are inverses of each other… is seems like it would be easy to directly derive one from the other… just by reversing the left-right vertical and flipping the sign on the right-left vertical… but not sure how to prove that.

very good

Thanks

Good morning, Could you please suggest a book which explains the subject in a similar way that you do? With exercises as well. Thank you very much, in advance.

Is there a deep reason this video series multiplies vectors from the left like xA instead of from the right? Or is it just the convention you felt was pedagogically superior?

Sir can you tell the refrence book for tensor

Why do we notate basis vectors as row vectors? Is it because basis vectors transform like covectors? Would you then notate basis covectors as column vectors?

thanks

is there going to be a playlist for this?

How you changed the summation signs in 8:47 Is there any property? IDK

Great video! Two questions/comments from a real beginner: in the linear algebra video series by 3b1b, the vector was to the right and the transformation to the left, echoing f(x). In matrix multiplication, it means that we frist apply the transformaiton to the right and then the transformation to the left. Why is the order different here? The second comment is that although the building of the new basis vectors out of the unit vectors looks visually pretty intuitive, the opposite is not true. In fact it is not clear at all whether you're applying the same "visual" procedure and why you get for example a -1

Is the fact that tensors are coordinate system invariant the reason they are used in relativity, since then it would not matter what frame of reference we are operating in?

shouldn't the backward transformation be the inverse of the forward one?

If I understand the backward Matrix is equal to the inverse of the Forward Matrix? (I do not speak English so much, and if I had a mistake I'm sorry 😅)

is [e1 e2] a covector? or is it informal to write it this way? thx

Is there a specific reason why we write the vectors as row vectors and multiply from the left instead of column vector multiplied on the right?

Please clear my one confusion, here inverse means that when we go from old basis to new basis we had ro take inverse of the new basis. M i correct ?

At minutes 6:50 you say how to determine the coefficients for a summation I do not understand your choice of linear term coefficients to express E1 tilde and E2 tilde. You have a myriad of terms to select the components from but you pick F12 and Fn1. I do not understand why those two. Thanks

As far as vector spaces by the definition it seems like all vectors in the entire universe are in the same vector space since all vectors have the same 3 V,S,+,; collection. Is that correct. Based on that it seems vector space never have a vector outside of the one and only vector space.

Is there something fundamental about the blue old basis? It seems like they are unit basis. For the sake of argument, what if the old basis was the red one and then you wanted to transform to the new blue basis… then yould need to use the forward transformation still. It almost seems like you can always just use the forward transformation as long as you switch, which one is old and which one is new

Sir,how do we know that we need 1/4 of e1 tilde to construct e1

Writing vectors as row vectors means that those are not vectors but 1-forms?

But e_i = \sum_i e_i ? then the the summation here only represents the value of i taken into consideration?

Bro @8.48 how did you changed the position of sigmas without checking the limits for the variables

What was the mistake?

3:10 Using the forward matrix: [1, 0][2, -1/2; 1, 1/4] = [2 1] Since the forward matrix is supposed to transform the basis vectors from the one without a ~ to the one with a ~, isn't this actually the backwards matrix since it transforms the basis vectors from the one with a ~ to the one without a ~?

If you made a text book I would buy 10

👍

Fkj? Do you mean Fki ?

well that's unexpected to see

Leopold Kronecker, German.

The audio sounds worse. People usually get better audio, not worse, so I'm probably doing something wrong.

Tensors are just objects that behave like tensors

Sir, why u r not regular in uploading the vedioes

Fji

Sorry Fkj is right

it's more like brugging not teaching

What do you mean by "summarizes ... nice and simply" when that just looks like an ordinary matrix multiplication? It doesn't seem to be saying anything that isn't simpler than ẽᵢ = eⱼF. (The fact that the shorter version is completely representable in a KZfaq comment just hurts the more verbose version more.) Similarly, δ is a lot harder to type than I, is introduced much later, but seems to mean the exact same thing. Iv = v is one of the first things that you learn about matrices, alongside MM¯¹ = M¯¹M = I. I'm just failing to see the reasoning for using δ instead of just the identity matrix, or using explicit summation over the matrix (and probably later tensor) product. (Actually, I'm pretty sure the tensor product isn't a direct extension of the matrix product like how the matrix product is an extension of the dot product, but that just begs the question of why there isn't such an extension to begin with.)