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2:53 This also reminds of commuting diagrams from category theory

27:50 Isn't the example matrix for the weak nuclear force actually a reflection since its determinate would be -1 and thus it wouldn't be in SU(2)? I think you forgot to add a negative sign to one of the antidiagonal components.

Woo Canada mentioned! However, I feel like they'll be too preoccupied with the economy right now to focus on doing any science experiments, unfortunately

I really like the connection between (computer science) graphs and matrices. I recently learned about spectral graph theory (en.wikipedia.org/wiki/Spectral_graph_theory) and generalized eigen decomposition. It's beautiful that two things that are both so abstract and seemingly different like dots & lines and grids of numbers can have such a deep connection; it's so rewarding to be able to switch between the graph or matrix perspective to understand something since (in my opinion) graph-like relationships appear EVERYWHERE in the world--tree structures and chain-like monoids are more so just nice specialty cases.

I know there's something connection in complex number and vector, it really similar, like the notation (a+bi and ai+bj), how it work etc, that connection is ✨matrices✨ tysm for making this video

16:06 I laughed in this one, ofc it's not wrong cuz it still 1x1 matrices but idk it just funny 😭😭

I'm a little bit late to this, but greetings from Kazakhstan 🇰🇿, I hope your dad is doing great and thanks for such good quality videos

Is there a rule about the commutativity of the identity element ("1")? Specifically, does it matter whether you do 1*x or x*1 (where x is an element of the monoid and * is the binary operation)? At 18:47, it says that one can prove that the identity element is unique. Does the proof assume commutativity of the identity element? Or can it be proven that the identity element is unique even when 1*x != x*1?

11:00 They key to being able to compare apples and oranges in vectors is the concept of covectors / forms / functionals. (I haven't gotten further in your series yet so forgive me if you discuss this point regarding tensors and co-contra variance later.) The dot product of (a c) * (b d) = ab + cd is actually a shortcut for `(b d) L (a c)` where L is linear transformation (specifically a metric) that defines how to measure heterogenous things like apples and oranges in terms of one another. What the dot product usually implies is using the "default" metric transformation which is just the identity matrix, meaning that 1 apple = 1 orange. But we could imagine one where 1 orange = 10 grapes if the metric were based on something like weight. And metrics that are not just trivially the identity matrix lead to the idea of manifolds such as various maps of the earth or spacetime in general relativity.

I dont know i guess it only makes sense that the best math channel i know has to slowly be put behind a paywall. It's just a bit sad... Still love the vids keep it up ❤

Edit: Apparently I'm completely wrong and the real reason is that traces are invariant under conjugation i.e. tr(PMP^(-1)) = tr(M). I'm not sure why that's important but that's apparently the actual reason. 11:20 I'm not really sure but my mind immediately goes to lie groups/algebras. The trace isn't a homomorphism of the matrices themselves, but it is a homomorphism of the infinitesimal transformations generated by the matrices. If we consider each matrix M to be elements of a lie algebra. Then we can find the corresponding Lie Group elements as e^M. Since two matrices AB ≠ BA, then e^A*e^B ≠ e^(A+B). Instead by the Baker-Campbell-Hausdorff formula we get e^A*e^B = e^Z where Z = A + B + (AB-BA)/2 + higher order terms. However if we only consider infinitesimally small transformations generated by A and B up to first order, we can get around this. Let ϵ be an infinitesimally small positive number. We'll consider elements e^(ϵM) Then e^(ϵA)*e^(ϵB) = e^Z where Z = ϵA + ϵB + ϵ^2(AB-BA)/2 + O(ϵ^3) Ignoring higher order terms gives Z = ϵ(A+B) -> e^Z = e^(ϵ(A+B)) Then taking the determinant on each of e^(ϵA), e^(ϵB), e^Z, gives a homomorphism. det(e^(ϵA)*e^(ϵB)) = det(e^(ϵA))*det(e^(ϵB)) Then we use the identity det(e^(t*M)) = e^(tr(t*M)) = e^(t*tr(M) This gives e^(ϵ*tr(A)) * e^(ϵ*tr(B)) = e^(tr(Z)) = e^(ϵ*tr(A+B)) which is trivially true but more importantly this is also a homomorphism of e^(ϵM) We started by taking the exponential of m so now we'll take the logarithm of the result. Then ϵ*tr(A) + ϵ*tr(B) = tr(Z) = ϵ*tr(A+B) is also a homomorphism of e^(ϵM), but this time going from multiplication to addition. I could be completely off base but I spent too long writing this comment to not post it now..

Slow. Clear. Concise. Brilliant.

Cool stuff. I like all the miracles.

first youtuber that's on beast mode

Non-linear isn't as bad as non-convex. Linear things are nice because knowledge about local properties becomes knowledge about global ones. For example, the derivative or slope of a flat line is the same everywhere. The next best situation is to have convex (aka concave) things because local knowledge at least provides global estimates. For example, once you know the local derivative/slope, you can figure out how the global one will behave even though it's not constant globally. Non-concave curvy things are not so nice to be able to extrapolate. This is something I learned regarding optimization but I think ideas generalize to other fields that classify things as linear or curved.

20:10 I wonder how this relates to the generalized Stokes' theorem: which written in LaTeX is ```(\int_{\partial \Omega} \omega = \int_{\Omega} d\omega)```. It essentially says that in Calculus on smooth manifolds, the boundary is the opposite of the interior, and if you know one, you can figure out the other.

I love how it's obviously MS paint from 6:00 onwards haha. But seriously, it's great that you provide so many visuals along with your detailed voice narration, even if the visuals are not super fancy. They're clear enough to explain the point but probably simple enough it allows you to produce your videos fast enough.

Are conjugates an example of adjoint pairs from category theory?

I learned there is a similar opposing pattern between structural richness and number of properties in computer science called the Boom hierarchy. Researchers showed that common data structures could satisfy several properties if they were structurally simple but had to sacrifice those properties in order to have more complex internals. For example, sets have no structure (except for membership) but are associative, commutative, idempotent and unitary respecting their construction operation. But trees can have lots of of internal structure because they've sacrificed associativity in their construction to have the nested format. Bags and lists are considered between those two extremes.

I've personally come to believe that properties like order (en.wikipedia.org/wiki/Partially_ordered_set) and connection (en.wikipedia.org/wiki/Connected_relation) are some of the most fundamental in the universe. We often think of numbers (algebraic fields, rings, etc) when we think of math but I feel like being able to compare discrete things with some sense of topological connectedness (which can be induced by an ordering) is at the heart of it at category theory and set/type theory. Eventually the idea of succession (natural numbers) and metric or measurable spaces and numbers with their symbolic operations come next, but there's something so great about lattices :)

6:46 really.. how? Lets say we want to make an angle of (1/7)*2π out of the available angles: (1/3)*2π , (1/4)*2π , (1/5)*2π So we want to find integers x,y,z such that [0]: (1/7)*2π = (1/3)*2πx + (1/4)*2πy + (1/5)*2πz Cancelling 2π gives [1]: (1/7) = (1/3)x + (1/4)y + (1/5)z Cancelling the common denominator of 3*4*5*7=420 gives [2]: (3*4*5) = (4*5*7)x + (3*5*7)y + (3*4*7)z Reducing modulo 7 gives [3]: 60 ≡ 0 (mod 7) ⇒ 4 ≡ 0 (mod 7) [3] is clearly not true. The equation [1] also fails because (via congruence relations) it implies [4]: x=3a , y=4b , z=5c for integers a,b,c ⇒ 1 = 7a + 7b + 7c ⇒ 1/7 = a + b + c [4] is clearly not true. The group of integers under addition is closed (otherwise it wouldnt be a group). If x,y,z are allowed to be rational numbers then [1] has an infinte number of rational solutins. But how is this fundamentally different from taking all rational roots of unity on the unit circle to begin with.

Thank you!

Can you please explain direct sum of eign basis part

Would it be ok to write, say, nat = ~ℕ ? This would be so that I can use traditional set symbols instead, mainly for explaining to colleagues.

Thanks a lot for filling the gaps between textbook and intuition

This is amazing content! Your channel really deserves to grow based on the quality of this video alone. I thought I finally understood what a monoid is (after pursuing the meaning of a monad from functional programming). I understood them to be flat list-like things where the associativity of their constructing/joining binary operation ensured the container object would have the nice properties of being "chunkable" to utilize fan-out-in parallel optimizations on the underlying elements (as compared to something with a nested or tree-like structure where it's no longer trivial to split the whole object and transform the smaller chunks). But the comparison to n-ary operations and how the existence of an identity element, and associativity allows the thing to span from nullary to as many parameters as you want was really eye opening! (I wonder what would be the case for an infinite arity for the operation? I guess that would require an extra property on the monoid to be closed under infinite series, aka having the limit of infinite arity operations still being similar Cauchy completeness. Maybe that steps over into analysis from algebra.)

fun fact (im greek): chi is pronounced "he", as in for example "he is an athlete", but with a thicker H: hHe!

Just helped me in my group theory course, thank you

15:10

La mejor y más clara explicación que he visto. Cuando vi un poco de teoria de grupos en la universidad quedó como un tema totalmente oscuro para mí. Ahora todo hace sentido. Excelente video!!! Me encanta esta serie

I love this video!

15:43 well, all those group/ring/commutativity is terms directly from number theory, so it's weird claim they aren't related.

9:08 well, that was confusing. at 1:40 you said it was homomorphism, i.e. function for mappping is character, but here you pick one one permutations and call it a character.

amazing, if i become a patreon in some time, will the video still be available? ik zag trouwens laatst in je Q&A dat je vlaams bent, en ik had me al afgevraagd waar je accent vandaan komt, want vaak en zo ook nu kan ik een vlaams accent niet herkennen in het engels en daar had ik het toevallig die dag zelf nog met iemand over gehad. maar goed, ik kijk al je video's zodra ze uitkomen! hopelijk kan ik binnenkort eraan bijdragen :) als je nog eens een Q&A doet ben ik benieuwd wat voor muziek je luistert

Very cool!

4:12-;

goldmine, have been trying to understand the mobius function for the last few days, absolutely beautiful explanation, thanks

1:11-;6:46-6:56;8:10-8:22;9:03-9:15;

Amazing video!! A really great introduction to some core ideas in representation theory :)

Before going deep into the math of these symetries and their applications. Where others failed, you succeeded. Deep explanations and intuivie. Really good content. Thanks !

Honestly, the looks/animations of the videos are just a bonus - I stick around because your explanations are the most clear and elucidating I've ever seen, often exceeding 3b1b

a division by zero is a contradiction in most cases, and everything follows from a contradiction

I AHVENT' WATCHED THE VIDEO DID YOU COOK IT UP SHOULD I FINSIH IT

Fourier was great video

I'm also a software dev, undergraduate in computer science and ending a Physics one. Your teaching amazing and passing is just as good to have us thinking the subject until the next one comes up. I think division by zero is a zone of trasition, it's a degenerate state, where either you lose dimensionality information as in linear algebra projection transformations or have an ambiguity as in geometry with unknown polar/azimuthal angles at the origin/poles, also in algebra they're related of to a curious pythagorean triple involving dual numbers, imaginaries and reals, the division by zero is not possible in scalar or row vector algebra, by an incompletude, but you can work around it using dual numbers matrix representation or poles and residue theorems from complex calculus, with some clever tricks. Spoilers: you arrive at differentiation operations. Borrowing again from LA, zero is the kernel of any morphism in math, so itself acts as a pivot, that makes any sum, product or exponential with it meaningless since you get the original operand, zero or one. The unique place where it changes anything is when divided by itself since 0 = k0, so k = 0/0, where k is literally anything you can put in an equation involving inverses, be it number, vectors, matrices, tensors, shapes, sets, etc.

The preview resembles a thing I did with pixel shaders, while exploring various fractals(Mandelbrot, Julia sets, Ducky fractal etc) I repeatedly applied projection matrix, using homogenous coordinates (x,y,z,1), somehow assigning z value(probably with a parabolic function z=const*(x^2+y^2)) so it was something like: take a picture, project it on parabola, take a picture again, project on parabola and so on(about a hundred times in total). If the projection plane was tilted(or shifted) a bit, it looked very similar

> why div by 0 allows this obisously, because it's uncertain and following the definition of division you can grab any result from it making reasonable equations into useless math. Wonder if category theorist have category for those.

Thought you used manim as well.

One fairly obvious (but not deep) reason for division by 0 being problematic is that multiplication by 0 is many-to-one, whereas multiplication by k for k != 0 is one-to-one. So if we define the function mul_k(x) = kx then the inverse function div_k(x) = x/k exists for all k except zero k since mul_0(x) = 0 for all x Because of this, an implication like: a*(x-y)=b*(x-y) => a = b is false if x=y (since the multiply is non-invertible), and if you can hide such a step in a proof then you can apparently prove nonsense like 2=1

RUclips as a learning resource is *hugely* under-appreciated. I’m glad you’ve jumped into this as a content creator.