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Great! I wish I had your skill explaining ideas ❤ If you want to see some realtime sims, come have a look at my latest stuff, you might be interested to see the emergent properties if these arrangements

A representation of an algebra A is an A-Module 🤗

i love your content, and am very excited to binge these playlists! i noticed something feels off for me about what you're saying around 5:00, about how divsion can be seen as repeated subtraction. you explain this by counting how many times you can subtract a number, but there's no [counting how many times we can do something] involved when we're saying that multiplication is repeated addition. you could just as well say then that division is repeated addition, because you can count how many times you could add a number to itself before you reach a threshold. if you subtract 5, 2 times, you simply get -10 right? wouldn't it make much more sense to say multiplication with a negative number is like repeated subtraction? similarly, i would not expect it to be pretty controversial to say that repeated division is like raising a number to a negative power. this is not the same as taking a root or a logarithm, but it exists and it's well-defined. i wonder if you agree with this, or whether someone can point out a flaw in this logic :)

...When the differential equations have a physical manifestation: Represent! What's your vector victor.

Just watched the previous episode! Very glad to watch this one. Take care

I guess we should add category theory and make more arrows :D

Serious shit. I'm very curious to see some representations of the dihedral groups, apart from the obvious "geometric" one as orthogonal 2x2 matrices or as permutation matrices

Complex number multiplication looks like an enigma. Multiplication of coordinates of two 2-dimensional points AND multiplication of distances of those two points and concatenating their angles BOTH yield the same third point. These two operations seem completely unrelated. Is there some intelligible visual geometric proof that relates these coordinates, distances and angles? Both google and openai GPTs weren't very helpful as well as the rest of TH-cam.

I always wondered why the "i" matrix had the negative sign in the top left instead of the bottom right, this explains it perfectly

Absolute gem of a video!

Thank you! Excellent series, the highest quality explanations. I signed up for the Patreon. Keep up the incredible work. What a tremendous gift to the world!

This is rather good so far

Thank you!

Thank you!

Thank you!

I'm slightly confused by the wording. I thought we were to read matrix products (and hence, transformations) from right to left. Would FTF-1 then imply a different transformation from what you were talking about? Rather than forward-transforming the basis with F, we are transforming to the basis prescribed by F-1 (reading from the right) first, and then applying the transformation, no? Does this mean that when we are changing bases in the expression for FTF-1, we are to read the matrix multiplication from left to right rather than right to left?

This is easily the best explanation of conjugacy that I've ever seen. Thanks :)

Enlighten and delightful are the words. Excellent video. Keep going.

I have to say, the many rotations explanation is probably the best I've heard!

What is “number samba”?

Here is an interesting proof that does not involve eigenvalues/eigenvectors that the trace is invariant under a change of basis. It's really not that geometric anymore after several failed attempts, but it's interesting to see. The following identity is taken from MathOverflow, in the answer by Yemon Choi on the post "Geometric Intepretation of Trace". If we consider the trace as a projection of the transformed orthonormal basis vectors onto the corresponding orthonormal basis vector, we can therefore calculate the trace of A as the sum of the inner product between these transformed basis vectors and the original basis vectors. For an orthonormal basis e_1, ..., e_n, we write Tr(A) = ∑ j=1,...,n ⟨ Ae_j , e_j ⟩ This means we can write Tr(A) = n ∫x∈B ⟨Ax,x⟩ dm(x) Where: - B is the Euclidean unit sphere - m is the uniform measure on B normalized to have total mass 1 - n is the dimension of the space To see why this works: a) Let x = (x1, ..., xn) be a point on the unit sphere. Then: ⟨Ax,x⟩ = ∑i,j aij xi xj b) When we average this over the unit sphere, most terms vanish due to symmetry. The only surviving terms are those where i = j: ∫B xi xj dm(x) = 0 if i ≠ j ∫B xi^2 dm(x) = 1/n for all i c) This is because the uniform measure on the sphere treats all directions equally, and xi xj is odd, while xi^2 is even, meaning only those don't cancel when integrating over the unit sphere. Now, we consider the trace of the similar matrix: Tr(PAP^-1) = n ∫x∈B ⟨PAP^-1 x,x⟩ dm(x) Let's focus on the inner product term ⟨PAP^-1 x,x⟩. We can apply a substitution x := Py, which gives ⟨PAy,Py⟩ = ⟨Ay,y⟩ = ⟨AP^-1 x,P^-1 x⟩. The integral becomes Tr(PAP^-1) = n ∫x∈B ⟨AP^-1 x,P^-1 x⟩ dm(x) Now, if we apply the substitution y = P^-1 x on the integral, because the measure was uniform, the determinant factors cancel: ∫x∈B f(x) dm(x) = ∫y∈P^(-1)B f(Py) |det(P)| dm(y) dm(x) = |det(P^(-1))| dm(y) = (1/|det(P)|) dm(y) This is because the volume of P^(-1)B is |det(P^(-1))| = 1/|det(P)| times the volume of B. But then, we use the same symmetry argument over the ellipsoid P^(-1)B, meaning all factors cancel except for y_i^2. Tr(PAP^-1) = n ∫y∈P^(-1)B ⟨Ay,y⟩ dm(y) = Tr(A)

This series looks really promising! It's great! I would suggest to speed up a little bit the narrative, I had to listen it at 1.5 to keep me engaged

Amazing keep it up!!

Thank you!

Thank you!

The little hole you punched in the middle is why phi doesn't start at zero.

Real is dual to imaginary -- complex numbers are dual. "Always two there are" -- Yoda.

mind blown! subscribed! (:

Thank you!

Thank you!

Thank you!

Thank you!

Thank you!

Thank you!

Thank you!

Thank you!

Thank you!

In computational physics this is a very, very useful tool.

Que ne lo paguen que estiy jasta el cñ moño de abajo😊

11:13 “diagonal matrices have the effect of stretching everything away from the origin”. Not so, dear sir. These matrices that you considered are scalar multiples of the identity; more specifically, they are real scalar multiples of the identity. If a is a real number, then the linear transformation represented by a times the identity matrix does not stretch anything as a linear transformation of the plane. If it is positive and greater than one, then yes, such a scalar matrix does represent a linear transformation that stretches the plane away from the origin. However, if a is negative and greater than one in absolute value, then there is stretching going on, but also a reflection this is not purely stretching something. If a is between minus one and zero, then rather than stretching the plane, the corresponding linear transformation reflects and compresses the plane, and if a is between zero and one, then the corresponding transformation merely compresses the plane. Your explanation in such a brief sentence is no explanation at all because it leaves out way too much nuance.

I learned about complex numbers while programming a Mandelbrot set generator in Pascal on my parents' PC, based on an algorithm published in Scientific American. I remember getting pretty deep into complex numbers, and proving what I later knew to be famous basic theorems. Namely, if an equation is true among complex numbers, then it is also true among their complex conjugates; and that if you multiply two complex numbers represented as vectors, the rotation of the product is the sum of the rotations of the factors, and the length of the product is the product of the lengths. Of course now I know my way around the complex plane in terms of r* e ^ (i * theta), which makes these facts self evident, but I remember proving them just using algebra.

The sandwich product or change of basis = Duality! Domains are dual to codomains -- Group theory. Points are dual to lines -- the principle of duality in geometry. Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality. Subgroups are dual to subfields -- the Galois correspondence!

Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality.

Congrats! Mathemaniac already unveiled the transposition operation as the dual form for matrices

Zo moet Fourier het gezien hebben.

Please note that i, sin and cos are not variables. This is why one should not write them cursive. This topic is so based. I had a lot of representation matrices stuff in a group theory course.

All angles never disappoints 🙌🏻

Is it possible to represent signed numbers in terms of just unsigned numbers in vector form?

if you like that, you're gonna love APL

This is great. I've been trying to understand capacitance and inductance without resorting to to the square root of -1, which makes the math work but is far from intuitive. This is taking me into a good direction.