I am an old guy with a lot of formal technical education behind me. I have found minimally three qualities in great teachers: knowledge of the subject matter, ability to communicate at several levels but most importantly enthusiasm. All three qualities are in this series.
Finally after 3-4 re-watches and practice it made sense. You were explaining it very simply i don't know why i confused myself. Great lecture actually! Thank you
For someone who may have hard time understanding how (3a-b) in B was converted in (3a-b) in standard basis (I'll call standard basis set as E {e1, e2, e3}). Let me give a simple algebraic equation. Now initially a = [7, 2, 0]^T (^T to denote that its column) Or you can say a = 7*e1 + 2*e2 + 3*e3 (e_i is a column vector)....... (Equation 1) And when we wrote a in B we got a = [0,1,0]^T Or you can say a = 0*b1 + 1*b2 + 0*b3 (again b_i is column vector as the prof. Wrote in the video).......... (Equation 2) Now since both a are numerically same, we can say from equation 1 and 2 : a = 7*e1 + 2*e2 + 3*e3 = 0*b1 + 1*b2 + 0*b3 Or in matrix term: E * [7,2,0]^T = B * [0,1,0]^T Here E is standard basis matrix and B is new basis matrix So it becomes very clear that multiplying B with a in B (which was [0,1,0]^T) will return a in standard basis I know it was very obvious, the whole reason i did that in matrix form was to make it apparent that it only works of E is standard basis. If E is some other matrix (not Indentity i mean) then we will have to multiply with the E inverse to get the original coordinate (i.e. coordinates of vector a in previous E). I don't know if it's worth exploring or not.
In all the 200 lectures that I watched, this was super confusing. I mean what if v is not in the column space of X inverse, how can you take X_inverse * u = v as a generalised form, this doesn't feel right, what am I missing? That means X can't be arbitrary right? But in the beginning you said take arbitrary X that is invertible. It's very confusing.
Finished deterninant series, been binge watching your lectures. Series 2 was simple and complete. Does this part cover everything in linear algebra or did you exclude some topics?
where can I find the proof for that? I searched on the internet and i can't find a general proof for the sum of rows as a constant implying it is an eigenvalue. Thank you in advance.
Whenever Dr. Grinfeld speaks on a topic, you know your search for explanations is at an end. One of the few educators I know to shed light on the seemingly trivial underpinnings behind ideas. The type of teacher whose students are always different (in a good way), you feel you are actually learning the ideas behind mathematics itself.
Such a shame that this channel is not popular, every video on this channel is worth so much, I can't thank you enough, I wish I had money to help support this channel
Thanks for the great lectures. I have a question w.r.t covariant derivative of a Tensor with one index, w.r.t alpha. In addition to the two usual terms, does it have a term with normal vector component? Since Christoffel symbol resolves vectors only in tangential plane.
"The word proof is not something I'm fond of..." - gosh it means so much to hear a mathematician say that. Thank you for these videos, came back a second time because the title was funny 😂
Thank you. So let’s see if I have it correct. Re a 2D curvilinear coordinate system. The Christoffel symbol Gamna with 3 indices. A and BC. A describes a choice of two vectors. Each of them being velocity vectors based on the point of intersection of longitudinal or latitude for sake of description. B and C describe the same velocity of vectors, each of which could be longitudinal or latitudinal. So with that I can now describe the Christoffel symbol as the rate of change of B with a small change in C in the A direction. An example of a Christoffel symbol has two values that describe the vector caused by parallel transport between B and C. Therefore, if this new vector pointed directly longitudinally, it would have zero for the latitude, no value. Does that sound correct? Thank you for your help!
This new matrix isn’t unitary, but still represents the same exact transformation. So if I take a vector V in the old basis and use its components with the old matrix, and the same vector V in this new basis, the output will have different components but represent the same vector. So therefore length is preserved with respect to the new basis, but the matrix isn’t unitary. I guess the question is, if the matrix isn’t unitary, then it can’t preserve length with respect to an orthonormal basis. Please see if I’m on the right path here thank you
You're asking the right questions but some things you're saying are not right. For instance, there's something wrong with the sentence "length is preserved with respect to the new basis" since the length is independent of the basis. Try separate the actual vectors and their geometric characteristics such as length from the component space representations.
@@MathTheBeautiful I meant to say the length is preserved with respect to the components of the new basis. Hence if I was to take the dot product it would be invariant since the components transform plus the basis. Is what I’m saying about the unitary part correct?
@@matthewsarsam8920 Still not quite clear to be honest. Tell me if this is helpful: the length of a vector in *not* given by √x₁y₁+x₂y₂. when the basis is not orthonormal.
@@MathTheBeautiful I understand that. It would involve the metric for that new basis. I’m pretty much saying when u take the dot product in the new coordinates, you end up with an invariant expression since the coordinates and vectors scale oppositely. Im just trying to see how the unitary part comes in
If you add the damping term, resonance does indeed occur even when the driving frequency is not exactly the same as the natural frequency of the system. Your equation models a system with "NO damping", which is called "very high Q" system. Indeed, higher the Q of your system, the closer your driving frequency has to be to cause resonance. If you plot the graph of steady state amplitude for the solution of damped equation, you can clearly observe the effect on resonance due to different damping parameters. This graph is quite common in physics classes or experiments.
Meet Calcea Johnson and Ne'Kiya Jackson. These two young Black students are mathematical prodigies who attended St. Mary's Academy in New Orleans. They are history-making teens who solved and showed proof of the age-old math giant, the Pythagorean Theorem ( a² + b² = c²).May 6, 2024
So can I say a vector is a tensor with order 0, and the components are tensors with order 1? People keep telling me the vector has a rank of 1, which is confusing
the best lecture series out there for people starting to learn linear algebra. great concept building for a solid foundation to understand more complex problems. way better than the vast collection of calculation tutorials that just throw algorithm after algorithm at you. great work.
Thanks a lot dear professor , I just watched this lecture and I am struggling with sth I couldn't come up with, How can we prove that the RCT is related to the intrinsic properties of the surface?
Earlier in the series you mentioned that some of the geometry problems "become a joke" when we start solving solving them with vectors. Can we expect examples of that in the near future?
@@MathTheBeautiful I meant that if someone has read history of geometry, linear algebra, calculus of variation and then watch your lectures he will appreciate the depth of these lectures.
