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• 
  @raizan15263 күн бұрын

  beautiful explanation, thank you.

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  @paracyber.x67973 күн бұрын

  mind pothondi raa rei..........su sir

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  @student99bg8 күн бұрын

  When I first read about the theorem in my textbook about polar decomposition my first thoughts were "is this really true? If you can write any matrix A as Q and S (let's use the same letters as in these videos), you can also write S as XDX^T and get A=QXDX^T which is just one rotation matrix multiplied by a diagonal matrix, multiplied by another rotation matrix. That can't be right. We can't express every linear transformation as rotating and reflecting, then scaling and then again rotating and reflecting, right?" So, my intuition was wrong then, this is apparently possible to do with every matrix, if I understood the videos correctly.

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  @MathTheBeautiful7 күн бұрын

  Almost - the exception in the "defective" case

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  @jvmguy9 күн бұрын

  I really like the way you teach this. I thought I knew linear algebra, but this takes things to another level.

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  @MathTheBeautiful8 күн бұрын

  There's always another level!

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  @jonathanr325811 күн бұрын

  idk why the quality on the lemma website is so bad. can't read much that you have written on the board.

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  @kingplunger603311 күн бұрын

  This is not the end yet, is it ?

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  @aayushmehta852311 күн бұрын

  Sir I abousulutly love your teaching may you always be well and teach maths to all of maths lover

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  @DivineScaleOfGod12 күн бұрын

  I have two questions about surfaces. Is there a way to find the inverse of a surface? The other question is we have parametrization by arc lengh, but is there a way to define a parametrization by surface area?

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  @DivineScaleOfGod12 күн бұрын

  Even though differentiation of functions having vectors as arguments is weird to think about you can still make an intuitive approach to it geometrically and see where the errors first occur.

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  @cardinalblues712113 күн бұрын

  Thank you for your amazing videos.Speak of the rotation stuff, could you make a video details how the quarternions works. I have watched so many videos still could not fully understand the concepts .Thanks for your reply in advance

• 
  @escher440116 күн бұрын

  When something is so universal like this I start to wonder if it can be somehow generalized to non-linear transformations like diffeomorphisms

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  @escher440116 күн бұрын

  Can we restrict the form of A using this decomposition if we know that A is nil-potent with index 2?

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  @user-pl7ko4qk6t16 күн бұрын

  At time 43:00 you write the equation for the surface MT (4 values) each of them as a function of the ambience MT (9 values). You make a contraction and now each value of the surface MT is expressed as a function of a PORTION of the ambience MT i.e. only 6 values because beta will never be = 3. Is it correct ?

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  @cardinalblues712117 күн бұрын

  I saw the little messy 😂😂😂😂

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  @MathTheBeautiful16 күн бұрын

  That was my goal

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  @cardinalblues712116 күн бұрын

  @@MathTheBeautiful Sir, Do you plan to do a series on abstract algebra? And the text book on this series of linear algebra? Thank you for your amazing videos !

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  @holyshit92218 күн бұрын

  At first look I wanted to calculate coefficients of Legendre polynomials but for numerical methods we don't need them We need nodes (roots) and weights which can be calculated from eigen problem with symmetric tridiagonal matrix b_{k,k+1} =b_{k+1,k}= k/sqrt((2k-1)(2k+1)) For nodes we need eigenvalues and for weights we need first entry of eigenvector corresponding with eigenvalue There is other method based on Newton's method and asymptotic approximations but i dont know the details Maybe video with code written from scratch

• 
  @AyushRaj-ut9mm19 күн бұрын

  6:34

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  @YumekuiNeru22 күн бұрын

  1:20 how do you prove that the two empty spaces are the same, like the size of the empty space does not change if you rearrange the pieces?

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  @MathTheBeautiful22 күн бұрын

  I **assume** that it's true

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  @jonathangreene68523 күн бұрын

  Yes this is exactly what I was looking for. You explained it great

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  @MathTheBeautiful20 күн бұрын

  Glad it was helpful!

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  @blue_lobster_23 күн бұрын

  than

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  @MathTheBeautiful20 күн бұрын

  You're welcome!

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  @GT1987324 күн бұрын

  I could be wrong but I think the dot product came from multiplication of quaternions. If you multiply 2 quaternions q1 and q2 (a good exercise; giving them generic components a bi cj dk etc.) you will find that many of the terms cancel, leaving you essentially with a dot product term and a cross product term. They can be thought of as symmetric and antisymmetric parts. The history of it began with quaternions first. Then people like Heavyside and Gibbs were proponents of extracting these parts of the quaternionic product into a dot product and cross product and the rest is history. It's an interesting period of history where even some of the greats struggled with quaternions and also with Maxwell's laws. There was a lot of organization and consolidation that took place, but by forgetting the history we expose ourselves to risk of a lot of confusion in the sea of vector calculus and namblas and hodge star duals, clifford algebras, etc etc.

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  @elements-2428 күн бұрын

  Can I switch between row and column operation while calculating elementary matrix?

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  @MathTheBeautiful27 күн бұрын

  Yes

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  @boutiquemaths28 күн бұрын

  Sublime lecture. Such illuminating examples. 🪩 I also smiled and appreciated your dislike of ∈. I'm not against it (it's e for element at least) but it's true that I always feel a bit unnecessarily pompous when I use it.

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  @Fractured_Scholar29 күн бұрын

  Why do you keep erasing the Tau term in favor or Pi? Do you not find Tau to be far more unifying?

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  @blue_lobster_Ай бұрын

  thank you, amazing explanation, may God bless you

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  @vikraal6974Ай бұрын

  This feels like a Charlie Chaplin movie after mic died.

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  @cardinalblues7121Ай бұрын

  This is amazing, thank you so much for revealing the beauty of the mathematics. why not upload videos for Calc I. I am looking forward to your precious videos

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  @julijangrajfoner1730Ай бұрын

  great explanation, thanks!

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  @cescllopisАй бұрын

  ME,I THINK IT SHOULD READ <PARAMETRIZATION>.Cf. e.g. <THE OXFORD DICTIONARY OF MATHEMATICS>. ROLF NEVANLINNA HAS WRITEN A BOOK WITH THE TITLE <UNIFORMISIERUNG> [GERMAN] SPRINGER -VERLAG ,1967 2.EDITION.

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  @MathTheBeautifulАй бұрын

  What's it about?

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  @cescllopisАй бұрын

  ME,I THINK IT SHOULD READ <PARAMETRIZATION>. Cf. e.g. THE OXFORD DICTIONARY OF MATHEMATICS.

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  @MathTheBeautifulАй бұрын

  Good to know

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  @harrisondorn7091Ай бұрын

  I didn't realize math could be so beautiful! :)

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  @MathTheBeautifulАй бұрын

  Thank you!

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  @ralvarezb78Ай бұрын

  Very clear and impressive MASTERCLASS

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  @MathTheBeautifulАй бұрын

  Thank you!

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  @ralvarezb78Ай бұрын

  ​@@MathTheBeautifulThanks to you. I'm electrical engineer, with more than 15 years of experience and I'll subscribe next Monday on Maths at "Université de la Sorbonne" (Paris). Your video transmits not only deep knowledge on what is going on, but also passion of the teacher, and this is the very important thing. Congratulations!

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  @sihonglai9059Ай бұрын

  What textbook do you use please?

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  @MathTheBeautifulАй бұрын

  Writing one. It will soon be on grinfeld.org

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  @sihonglai9059Ай бұрын

  Taylor's expansion or finding power series representations for functions is just taking derivatives to find simplest core of the function, such as straight line acceleration creating curve speed and then leading to quadratic location function. the Taylor's series are such representation with initial conditions at every level of differentiation. Seeing it in the perspectives of control system engineering or Dynamics , that is how Equations of Motion and Euler-Lagrange Equations work.

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  @sihonglai9059Ай бұрын

  Mathematics is inherently and should be inherently beautiful and elegant. Its fundamental core is simple and unsophisticated, yet it possesses a magical beauty akin to art. Similar to art, mathematical concepts can be explained, appreciated, and embraced from various perspectives. This diversity evolves into the realm of philosophy, where not only one beautiful interpretation exists, but where limitless possibilities unfold. When all mathematical concepts converge and their interpretations harmonize into unity and none. then we will find the heaven, the kingdom of God, the Brahman. Thank you for teaching the math the way it is supposed to be taught !

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  @MathTheBeautifulАй бұрын

  Thank you - your comments are much appreciated!

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  @jamesnapier3802Ай бұрын

  I find it odd that you claim the ear is "one sensor", while with the eye, you count each rod and cone. There is more than one hair cell in the cochlea, you know....

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  @MathTheBeautifulАй бұрын

  You're exactly right. I did not know that when I was making the video. However, ther overall point is valid: there's still a single signal going into the ear carrying all of the information in that one signal. With visual data, however, every point we see sends its own signal into our eye.

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  @ehguygАй бұрын

  [[x,0] [x] = [x^2] [0,x] [1] = [x]

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  @FranzBiscuitАй бұрын

  Well done, I think I actually get it now! The presentation was wonderful, very enjoyable indeed. Cheers. SUBSCRIBED

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  @MathTheBeautifulАй бұрын

  Thank you!

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  @wagsman9999Ай бұрын

  Gorgeous stuff.

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  @MathTheBeautifulАй бұрын

  Thank you!

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  @joeheafner2495Ай бұрын

  BOOM! This may very well be the best way to bring the metric tensor into introductory physics.

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  @MathTheBeautifulАй бұрын

  Agreed

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  @user-rb9yy3ov5tАй бұрын

  Pasha, you are more of an artist than a mathematician!. Marvelous the presentation.

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  @MathTheBeautifulАй бұрын

  Haha, thank you for the compliment!

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  @BUY_YOUTUB_VIEWS.321Ай бұрын

  Your video is like a mini-masterclass. So valuable!

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  @tomholroyd7519Ай бұрын

  oh did you mention to your students the identity matrix hidden in the metric? 'cuz you know

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  @MathTheBeautifulАй бұрын

  I did not, but will at another time.

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  @tomholroyd7519Ай бұрын

  Ooooh Kayyyy Gram Matrix from now on

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  @MathTheBeautifulАй бұрын

  Cancel culture catches up with Schmidt

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  @SimchaWaldmanАй бұрын

  In the thumbnail, you can use LaTeX coding *\imath* and *\jmath* to get a dotless *i* and *j* respectively.

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  @MathTheBeautifulАй бұрын

  Thank you! Yes, I think that would have been better!

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  @johnathancorgan3994Ай бұрын

  After decades of using linear algebra in professional work, I still find these treatments of the basics quite refreshing and sometimes even enlightening. There is an elegance to the part of the world that can be modeled and predicted by linear maps.

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  @MathTheBeautifulАй бұрын

  Me too!

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  @PluralistАй бұрын

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  @camilagonzalez1859Ай бұрын

  Thank you 🙏 a😭😭

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  @MathTheBeautifulАй бұрын

  Glad you enjoyed!

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  @KaiseruSozeАй бұрын

  Excellent as usual. How long is a line? Twice as long as a half of a line. Which one you choose as a reference length is arbitrary.

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  @theoremusАй бұрын

  The radian is related to arclength of the circle. Hence, to do angle measurement in radians, one needs differential geometry.