Least Squares Formula PROOF

[@martinzapata7289]

This proof uses the minimising vector theorem, but I guess it’s fine not to explain it since the presentation is very visual. Very clear, I like it!

[@JUXIX101]

Best video I've seen on this topic thus far

[@virtually_passed]

Thanks 😊

[@kevinnania3652]

Absolutely beautiful.

[@jaafars.mahdawi6911]

Indeed we liked it. Very. Well done.

[@HimuraK1]

Excellent video! Thank you!

[@LQC783]

Perfect explanation. Thank you!

[@virtually_passed]

Glad you enjoyed it!

[@SamKwak-q4m]

Fantastic. Keep it up!

[@chadx8269]

Very nice, now its stuck in my brain.

[@posi0504]

Thank you!!!!

[@ShahbazKhalilli]

this was insaneeee

[@noonesperfect]

Great explanation.

[@virtually_passed]

Thanks :)

[@chandraprakash934]

Amazing ! keep up the good work :)

[@virtually_passed]

Thanks!

[@aniruddhvasishta8334]

Awesome! When I saw that everything was on a plane I assumed you would find the orthogonal projection of b with the projection matrix that I vaguely recall from my intro linalg class. This solution seems more elegant tho.

[@virtually_passed]

Thanks for the kind words. Using the projection vector formula is indeed another method that can work :)

[@ShalevWen]

Since A has only 2 columns, its rank is at most 2 and so is the rank of A^t*A, which means that if you have more than 2 data points there is no inverse

[@MrRyanroberson1]

Notice (AB)^-1 = B^-+ A^-1 and matrices are associative, therefore the final result is A^-1 b after simplification, so really this can be taken as a fancy way of inverting irregular shaped matrices

[@chadx8269]

Assuming (transpose A) has a left inverse. Then x= (inverse A)*b. But what if (Inverse (transpose A))* (transpose A) not equal to Identity.

[@SuperSeaguard]

I wish I had visualizations like this while learning calculus / differential equations. We humans are visual learners. Math without visualization is very difficult to learn.

[@martinsanchez-hw4fi]

Wonderful

[@virtually_passed]

Thanks!

[@zeroheisenburg3480]

Cool Video. May I asked what did you use to draw animation for 3d planes and coordinates system?

[@virtually_passed]

Hey I used ManimCE. It's a python library made by 3b1b

[@DavidKenny64]

How much more complicated would it be if the pins were not in the shape of a circle, but maybe the shape of a horse? A picture in a silhouette.

[@virtually_passed]

I think you meant to comment on my string art video? :} for the method shown in the video it's no additional hassle at all! It's the exact same method. The A matrix will just change. Fun fact: depending on which way you wrap your string around the nails, you can get a slightly different ending angle and starting angle, so you can effectively get 4x nails. This irregular spacing of nails can be stored in the A matrix

[@DavidKenny64]

@@virtually_passed You are correct. I had followed the link in your string art video to this one and then commented here. Do you plan to sell or release the calculations for people who just want to create computer art?

[@tsunningwah3471]

what does angle mean in higher dimension say a dot b = |a||b|cos theta what does theta mean if a b are like 5 dimension vectors? thanks

[@virtually_passed]

Intuitively I can only tell you what it means up to 3 dimensions. Imagine drawing two vectors in 3 dimensions. Then draw a plane that is spanned by both of those vectors. The angle on that 2 dimensional plane between these vectors will be theta. After 3 dimensions all intuition falls apart. I can only loosely tell you that theta is a measure of how 'similar' the two vectors are.

[@tsunningwah3471]

@ thanks

[@dothuong-f1e]

Help me explain sensitivity of variable A

[@nehalkalita]

Which software do you use for the animations?

[@virtually_passed]

I used Manim. A free python library

[@nehalkalita]

Okay. Since the background colour is not black and the font is different unlike many other videos of Manim, I could not figure out it was an output from the same tool.

[@AlokKumar-kk8pj]

Sir please don't put background music. Great content

[@virtually_passed]

Thanks for the feedback!

[@halmirofigliolo1637]

@@virtually_passed I like it

[@gigastein3151]

Sir can you make a whole playlist of mechanics, I really understand you really well. Please sir

[@virtually_passed]

Thanks for the comment. It's in the pipeline!

[@gigastein3151]

@@virtually_passed Thanks sir, idk you are one of the best for mechanics, I am struggling with mechanics and your summary helped very much. I hope you will upload all of the playlist of the mechanics. Thanks for you efforts.

[@gigastein3151]

@@virtually_passed sir do you use any social media, as if I have any conceptual doubt. I will ask you.

[@virtually_passed]

@@gigastein3151 I have a Facebook page. But you can email me at [e-mail]

[@gigastein3151]

@@virtually_passed oh do you use applications like discord

[@juanramonvazquez3212]

bro, change the title while not many people has seen it yet lol (typo)

[@virtually_passed]

Thanks for the warning! I've changed it. For those not knowing what this is about; I stupidly misspelled the word "squares" :x

[@dariusgoh5314]

Doesn’t this proof only make sense for when A has 2 columns . Then the column space of A is a plane

[@一個平民]

This video used a 3d to 2d projection for better visualization But i believe those concept used (projection, orthogonality, dot product etc) are still viable in higher dimensional spaces so the prove should be similar

[@virtually_passed]

Yes you're right. But this idea can be generalized to N dimensions. Usually the projection is used and not a manual dot product as I showed. I did that because that seemed more intuitive and simple to me