This solved about 5 years of my confusion in 10 minutes. Thank you so much.

How the hell you only have 13 subs.... Is one of the best explanation I ve found!!!!

The best explanation of transpose matrix on the internet. Very clear voice and simple visualizations!

I love how geometric and intuitive this explanation is. It seems as though a lot of math lectures and explanations often end up deep into the symbols that the higher-level intuition and meaning gets lost. Not that symbolic explanations are bad per se, but it definitely helps to be able to understand and visualize things at a high level like this. Thanks for the excellent explanation.

Super helpful, this is not usually explained in textbooks. Authors just assume you know what is happing. Couldn't find better explanation elsewhere. Thanks a lot.

Thank you sir, You really provided a good visualization for how matrices distort/stretch data. A_transpose and A_dodge rotate the date in same manner but magnify differently! Visually , since they have the same rotational effects, the orientations of the basis vectors does not change, so their subspaces, particularly the null space and column(range) spaces remain intact. But still A_transpose and A_dodge distort data shapes differently!

Amazing job Ben. Explained with such conciseness and clarity

Thank You Ben! It was a lovely experience watching your video.

This is really good, thank you so much. Was confused by the fundamentals of subspaces and what do they mean in terms of visualization.

What a incredible video you upleaded!

Nicely done. Thank you for this.

Loved IT. Best video on transpose on youtube.

Incredibly good video. Thanks!

Wow! You are such a great teacher!!!!

A homomorphic function maps K^n to K^m sending each element of K^n to a subspace of K^m. A^T the transpose maps K*^m to K*^n sending each element of K*^m to a subspace of K*^n. where K* is the dual space of K and it must satisfy A^T(f)= f(A), f is a linear transformation in A^T. the 4 spaces are the kernel and the image of A and A^T. now it happens that K is always isomorphic to K* if both n and m are finite natural numbers. so you can kind of think of A^T as a function from K^m to K^n. in the example: A maps R^2 to a line in R^3, ima(A) and ker(A) are both lines. A^T maps R*^3 to a line in R*^2, ima(A^T) is a line and ker(A^T) is a plane.

Nice geometric intuition. Note that the SVD actually rotates the image and coimage from/to the x axis (or xy-plane, etc) so the diagonal matrix can do its job. In general, 'rotation into the same direction as the image' is not well-defined. You could extend your intuition to motivate A+ = (A^T A)^-1 A^T, since A^T A maps the image back to the image, swished around.

3:10 “Applying A squishes R^2 like this” doesn’t click in my head. The matrix A is a mapping from R^2 to R^3, so I’m not sure why the output vector is discussed in R^2 instead of R^3…?

@miikavuorio9190

2 жыл бұрын

The way I understand it is simply that he's breaking down the transformation of A into two steps, first one which squishes R^2 and then another which maps that squished R^2 into R^3. It was simply a way of explaining it in order to make the transformation easier to intuit where the null space of A is

@pauselab5569

4 ай бұрын

this is a result from the first isomorphism theorem and homomorphism decomposition. A is a homomorphism from R^2 to R^3. ima(A) is given in this example to be a line. R^2/ker(A) is isomorphic to ima(A) which shows that ima(A) is also a line. now do a homomorphism decomposition. Function 1 is a homomorphism from R^2 to R^2/ker(A) and function 2 is an isomorphism from R^2/ker(A) to ima(A). the composition of function 1 then 2 is the same as the homomorphism A. you just break down the function which makes it easier to understand.

Thanks a lot!!!🎉😊

This was helpful, thanks!

Wow.... Superb VIdeo.... Best video ever seen.. : 11:06

Great video! Got it now!

Too overlooked channel. Instant sunscribe!

I understand up to the point where you're breaking down how to transform the R(A+) subspace to the R(A) subspace. Why does the 1st rotation V^T have to be orthogonal?

Can you also explain the difference between transpose(A) and inverse(A)?

Great content!

nice, i somehow got a feel of what transpose means

There are unknown way to visualize subspace, or vector spaces. You can stretching the width of the x axis, for example, in the right line of a 3d stereo image, and also get depth, as shown below. L R |____| |______| This because the z axis uses x to get depth. Which means that you can get double depth to the image.... 4d depth??? :O p.s You're good teacher!

Damn this is so good !

You are the only one who is clearing this transpose doubt....i understand few things here But there is one doubt While you explaining the both range on same space through the example I didnt get the point why the vector length on R(A+) is different from R(A) Could you clear this doubt ? I know it will be not easy in text to explain but if you can with same video you can include example in more defined way Really thanks for this video i learned a bit more today

@cocoabutter5888

Жыл бұрын

A might be scaling vectors as well when moving from R2 to R3. So the length of a vector in R(A) may not be the same as the length before the mapping to R3 (i.e, a vector in R(A+)) is my guess.

Can someone please explain, if you can, how can a length which is 1D “a line” (Length of cross product vector) be equal to the area which is 2D, how can something that is 1 dimensional be equal in length to the Area which is 2 dimensional (and has width plus length as information)?

Realy cool!

What did you use to record this?

8:10 "V is an ortogonal/orthonormal matrix" why V has to be ortogonal? Every shear on the plane can be represented with an ortogonal matrix?

And if you can maybe to make a video in this style which more focus about the transpose matrix i dont think i really understand it well and in a short exploration in the internet it’s look like i am not the only one

Great video! but I that A should be [[-1,1,1], [-1,1,1]] shouldn't it?

@Andrew-fg3jj

Жыл бұрын

I feel the same

Why don't you make more videos buddy ❤️

Just so I'm certain, Range as used in this video is functionally the same as column space correct?

@It_is_Jake

2 жыл бұрын

Range could be of column space or of null space. Range is the number of dimensions that column or null space has.

@It_is_Jake

2 жыл бұрын

So, I feel you are not right. Range is a characteristic of column space, row space, null space, or left null space.

❤❤

at 2:13...why isn't it R^3? The line is in R ^3, not R^2.

3:10 Why ? Why all the vectors in r2 goes like that it’s not clear to me ... All besides that an amazing explanation

@omridrori3286

3 жыл бұрын

I think it’s not true because for example (1,2) Is going to (3,3,3) it is not going to (1,1) before

@miikavuorio9190

2 жыл бұрын

Yea, it's not a mathematically rigorous explanation of A. However, I believe it is worded like that to make it easier to intuit where the null space of A lands. He breaks down A into two steps, firstly the squishing of R^2 to the range of R transpose and then the mapping of that to the range of A

YOU ARE FUCKING AMAZING

So if my sigmas are 1 that means my transpose is my ingular value decomposition

I don't understand why we don't have two non zero elements in the scaling matrix sigma.

great video, appreciate your work, only 1 comment, could you speak more clearly. I am not a native speaker, I had a hard time follow your lecture as your words stick together

wish there was a way to make the voice in videos less... wet... makes it hard to listen to for some people