Current Events > Cool maths topic

(+) Yeah! (+)

teepan95 posted...

I just finished TAing this month. My second year running, taking a small class in engineering dynamics. So motion of point masses, rigid bodies, oscillations etc. Basic stuff, but fun to teach.

Nice. I love teaching. Luckily I got to do quite a bit of it in my position as we built the business...mainly to B level execs and agents. Nothing fun like you get to teach though.

teepan95 posted...

In April, I should be starting a new research assistant job. The chair and my (future) boss have given me the OK and I've sent off all the paperwork, but I'm still waiting to sign a contract. I'll be looking at flow properties of old-school paints and how they correlate to the dried out paint on canvas. The plan is to do that for a year until I pass all my modules and can start my Master's thesis, which will be on that topic.

Flow properties of paints? Is there a practical application to that or is it to practice modeling.

Sounds like a fun niche subject but maybe paint is a "model model" for other fluids.

Romes187 posted...

Flow properties of paints? Is there a practical application to that or is it to practice modeling.

Sounds like a fun niche subject but maybe paint is a "model model" for other fluids.

The main practical purpose is art restoration. You use the rheological (flow) data to draw conclusions about what kind of paint was used, so that when you have to restore the artwork you can do so as accurately as possible.

Keep in mind, flow doesn't necessarily mean liquid paint, but also things like how much it ran as it dried, any peaks of paint on the canvas, etc. Rheology is probably my favourite part of my syllabus, and this is my chance to immerse myself in it practically. I'll probably do a write-up on it once I'm done with the CFD stuff if there's interest.

If that's not practical enough, it's my boss's doctoral thesis topic and she needs to finish by December 2021, so she needs help and of course, I get paid. Mutually beneficial!!
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teepan95 posted...

The main practical purpose is art restoration. You use the rheological (flow) data to draw conclusions about what kind of paint was used, so that when you have to restore the artwork you can do so as accurately as possible.

Keep in mind, flow doesn't necessarily mean liquid paint, but also things like how much it ran as it dried, any peaks of paint on the canvas, etc. Rheology is probably my favourite part of my syllabus, and this is my chance to immerse myself in it practically. I'll probably do a write-up on it once I'm done with the CFD stuff if there's interest.

If that's not practical enough, it's my boss's doctoral thesis topic and she needs to finish by December 2021, so she needs help and of course, I get paid. Mutually beneficial!!

That is bad ass. Are you an art lover? I wish I did something I was super passionate about haha

Would anyone actually care if I typed up something about particle physics or exoplanets or something related for this thread?

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TarElessar posted...

Would anyone actually care if I typed up something about particle physics or exoplanets or something related for this thread?

Sounds interesting to me.

Symmetry or why we actually have stuff in our universe

Before the Big Bang, we assume that there was nothing, but why do we have anything at all? If we have equal numbers of particles and anti-particles, why doesnt the anti-part of our universe just annihilate with the part were living in? How could we actually tell the difference between particles and anti-particles to begin with?

The answer to all of those questions (at least in theory) boils down to violations of symmetry in nature. Lets talk a bit about what symmetry actually means. Wikipedia, the source of all knowledge tells us that In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. Consider we have someone running along a road if we as an observer move along with them (i.e. transform the spatial coordinates), we wouldnt expect the physics to change. We can use some mathematical wizardry to show that any such symmetry corresponds to a physical quantity that shall be conserved. Typical examples you might have heard of are total energy (symmetry of time), spatial momentum (symmetry of space), angular momentum (rotational symmetry of space) and more (conservation of probability, etc).

Lets get back to particle physics: for each particle we shall define three essential symmetries, C (charge symmetry, replacing particles with anti-particles follows the same physics), P (parity, mirroring everything along the three axes still gives the same physics), and T (time, on a local scale switching past and future still gives the same physics). Not all of them seem intuitive, and some of them are indeed violated in some processes. Consider in more detail the notion of parity if we apply the parity operator (mirror operator) to any quantity, we would expect it to either stay the same or get inverted since applying it twice should return us to our original state (-1 x -1 = 1 && +1 x +1 = 1). Following from there (but omitting mathematics), we split particles into two states, left-handed and right-handed (as they can have parity components which give -1 and ones that give +1).

The foundation of the classical radioactive decay we know (beta to be precise) is mitigated by something called the weak force (one of the fundamental forces). Experimental evidence (very strongly) shows that the weak force acts to different degrees on LH and RH states. That basically means that if we start with an equal amount of LH and RH states (since we have no reason to think otherwise), some particles can follow certain decays while others dont. Furthermore, mathematics shows that a total CPT symmetry (i.e. applying all three transformations at once) can never be violated. That leads to the (experimentally proven) result that particles and anti-particles participate in different decays not only can we distinguish them by which decays they follow, but it also explains the symmetry break in matter and why we have particles (instead of anti-particles) to begin with. Interestingly enough, it also means that there are some particles in the universe which can not interact with any other form of particles (or do they really exist if we cant observe them?).

Apologies for the rather lengthy read, idk if anyone is interested in these things to begin with since they have both experimental and theoretical components and people usually only like one of those exclusively.

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TarElessar posted...

Would anyone actually care if I typed up something about particle physics or exoplanets or something related for this thread?

I'm literally posting about topology and lie groups so ...YES :) (not yes that I care, but YES please do)

TarElessar posted...

Symmetry or why we actually have stuff in our universe

Awesome stuff - would love to hear some more from you. Symmetry is awesome.

https://en.wikipedia.org/wiki/Emmy_Noether

Supplemental reading since women scientists are so unknown...but this one deserves a bigger spotlight!

Tangent Spaces

https://en.wikipedia.org/wiki/Tangent_space

A very important thing in physics. Tangent spaces are basically what they sound like - spaces that are tangent to a point on a manifold (see earlier def. of manifold)

There are a few ways to define it depending on whether we want to embed the manifold in a larger space or define it intrinsically

First you need to think about a tangent vector at a point. If you have a curve that is running along a manifold, its easy to think of the tangent vector as the directional derivative at that point. But you can also think of it as something called the velocity of the curve.

Since a curve is a map from R^n to the manifold, and you can then create a second map from the curve to another R^n by a chart, we can differentiate (because we know how to differentiate maps between R^n). What this does is gives us the infinitesimal vector at a point on the manifold.

Why is this so important? Because that actually creates a basis for a "space of all possible tangent vectors" at a point on the manifold. The tangent space is a vector space which allows you to do all sorts of math on it, including calculating the Lagrangian of a system (you'll need a bit more but this is a key part) which gives you the equations of motion for a system which is what physics is all about.

Okay, so we have a tangent space at EACH point on the manifold. But we can also take the "set of all possible tangent spaces" and create what is called a tangent bundle.

Think of bundles as coordinate systems on steroids. If you're following along in this topic, try and get a good understanding of tangent spaces first because it'll help with the bundle talk later.

Tangent vectors are easy to think of, tangent spaces are somewhat easy to think of, but thinking of the set of all possible spaces "glued" together gets a bit tricky. Bundles are hard to visualize but next post i will give some basic examples which are easy (trivial bundles)

I forgot to mention

For every tangent space, you can define a cotangent space which creates a whole new set of fun things to do (including creating a cotangent bundle, etc).

That'll be another post as well. Work time.

Romes187 posted...

That is bad ass. Are you an art lover? I wish I did something I was super passionate about haha

Not generally speaking, no. Something about this project just appealed to me

I'm sure you'll get your chance to do so!!

TarElessar posted...

Not all of them seem intuitive, and some of them are indeed violated in some processes.

So say if time symmetry were to be broken, would it manifest itself in total energy being created or destroyed?

Romes187 posted...

Tangent Spaces

Sounds good, even if it took me a few tries to understand the topic haha

My next post probably won't be until tomorrow, spent 11 hours revising with a friend and my brain is mush right now
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teepan95 posted...

So say if time symmetry were to be broken, would it manifest itself in total energy being created or destroyed?

We're talking about time in a different context here, on the basis of particle processes.
Think more in terms of the flow of entropy - in a universe where we had time reversal, we would head towards a state of zero entropy (as well as most likely remember the future instead of the past and many other thought experiments).

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TarElessar posted...

We're talking about time in a different context here, on the basis of particle processes.
Think more in terms of the flow of entropy - in a universe where we had time reversal, we would head towards a state of zero entropy (as well as most likely remember the future instead of the past and many other thought experiments).

yeah but a big problem is in the early states of the universe there was super high entropy I thought. Then you have to deal with the arrow of time stuff

its been a while since ive delved deep into entropy but Sean Carroll has a decent entry level great courses on the subject of the arrow of time. not a huge fan of his stuff outside of physics, but he's a good educator at a beginners level for that and for QM

https://en.m.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Millennium_Prize_Problems

These are the 6 math problems which are still unsolved by any human. Worth at least a million dollars if someone can solve one. If you are really good at math why not check out the list and see if you can crack one

EricDraven59 posted...

https://en.m.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Millennium_Prize_Problems

These are the 6 math problems which are still unsolved by any human. Worth at least a million dollars if someone can solve one. If you are really good at math why not check out the list and see if you can crack one

I can solve the navier stokes equations

0=0

Thank you for coming to my TED talk
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EricDraven59 posted...

https://en.m.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Millennium_Prize_Problems

These are the 6 math problems which are still unsolved by any human. Worth at least a million dollars if someone can solve one. If you are really good at math why not check out the list and see if you can crack one

Going for Yang Mills Gap

Maybe one day ITT we can get to what a Yang Mills theory is but I likely wouldn't be able to give too much on it since I only vaguely know what they are. The details get VERY math-y and there's likely only a handful of people in the world that really grasp that shit. We can try though!

I mean we've kinda discussed YM theories with the symmetry groups and stuff but its only the surface.

Still need bundles, connections, etc to get to gauge theory

teepan95 posted...

I can solve the navier stokes equations

0=0

Thank you for coming to my TED talk

P=NP is easier. On the premise P is nonzero, divide both sides by it, and therefore N = 1.

QED.

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Every group of prime order is cyclic. (Easy to prove if you know Lagrange's theorem)

Every group of order pq where p < q are distinct primes and p does not divide q-1 is also cyclic. (Not so easy)


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FLOUR posted...

Every group of prime order is cyclic. (Easy to prove if you know Lagrange's theorem)

Every group of order pq where p < q are distinct primes and p does not divide q-1 is also cyclic. (Not so easy)

Sylow Theory also gives some cool "more advanced" versions of these types of results that are fun.

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Pure maths is definitely not my forte

Here's an unsolved problem: the Cox-Merz relation

In a nutshell, there's different kinds of rheological tests, which lead to different viscosities. Theoretically, these viscosities are all unrelated to one another

Experimentally, results show that the (dynamic) viscosity under steady shear () is equal to the absolute value of the complex viscosity under oscillatory shear |*()| when the shear rate is equal to the frequency and no one knows why
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FLOUR posted...

Every group of prime order is cyclic. (Easy to prove if you know Lagrange's theorem)

Every group of order pq where p < q are distinct primes and p does not divide q-1 is also cyclic. (Not so easy)

yay more group theory

anyone ever dig into galois theory? he is my fav mathematician. Died young in a dual, crazy revolutionary, did his math in prison, extremely ahead of his time.

my kinda math guy

teepan95 posted...

Pure maths is definitely not my forte

Here's an unsolved problem: the Cox-Merz relation

In a nutshell, there's different kinds of rheological tests, which lead to different viscosities. Theoretically, these viscosities are all unrelated to one another

Experimentally, results show that the (dynamic) viscosity under steady shear () is equal to the absolute value of the complex viscosity under oscillatory shear |*()| when the shear rate is equal to the frequency and no one knows why

What are the different kinds of tests? Or the main ones?

Here's one I can define somewhat quickly now that we know what a manifold is. Well, also need to know that if the manifold is C(infinity) differentiable its called smooth (no matter how much you zoom in, you can 'move' across the manifold with no discontinuity)

So a Lie group (pronounced 'Lee' not 'lie' as in lying) is a group that has the structure of a smooth manifold. It's continuous. Every group I mentioned on page 1 of this topic (U1, SO3, SU3, etc) are all lie groups.

Contrast that with discrete groups such as symmetries of a triangle or something.

Lie groups give rise to a lie algebra which allows you to calculate the "generators" of the group which I believe end up being your basis on the bundle that you attach them to (a bundle with lie groups as its fibers is called a principle G bundle)

Here's a sample of the hopf fibration which is just too cool looking

https://pbs.twimg.com/media/ERk6xaQUEAAUepy?format=jpg&name=large

Seeing a bunch of stuff mentioned that I've encountered in different contexts. I'll get back to those, but first off something that I learned recently. Here's a blog post that covers it in detail:
https://marctenbosch.com/quaternions/#h_1

But to sum it up, it's about rotations in 3D. This is a very central topic for 3D rendering, so any modern game engine will carry out rotations using something called quaternions. The nave way of rotating (Euler rotations, you rotate around one basis vector at a time) can run into some weird problems around certain edge cases called gimbal lock. So people figured out that this obscure generalization of complex numbers in three dimensions can actually represent a rotation around an arbitrary axis as a multiplication. It's pretty neat, and complex numbers do the same thing in 2D. BUT, this blog post argues against quaternions, since they're a special, obscure case of a geometric algebra operator called rotor. I hadn't seen linear algebra be extended into geometric algebra like this before, and it was quite neat! In the end rotors do the same thing as quaternions but they are actually intuitive and less of a black box.

• Back to you guys's discussion. I encountered Hermitian matrices quite recently as I've begun to dive deeper into adaptive signal processing and optimization. Turns out they're quite important in that context as well!
• I'm glad to see tangent spaces mentioned. There are several 3D rendering techniques that make use of the tangent space. Here's a good overview: https://www.gamasutra.com/view/feature/129939/messing_with_tangent_space.php?print=1
• Also fun to see Lie groups mentioned (they're actually pronunced Lee-eh, Sophus Lie was a Norwegian mathematician)! At my uni we had one of the leading experts in the world in developing Lie group analysis for solving differential equations. Sadly he passed away about a year ago, but I took a course he developed and it was pretty interesting. Not that I understood much but still :)

I'd love to do some write-ups about the math that I'm involved with, but I'm afraid I don't know it very well, hehe. I work in an applied field and we rely heavily on stuff like the weird Abel transform, Fourier integral operators, and generally just signal processing and statistics stuff.

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Ive only ever heard it called a lee group...interesting. Sophus Lie was a beast. I believe his initia motivation was, as you say, solving diff equations. Little did he know...

never heard of a rotor, though I have heard of and seen Clifford algebras, but that is intuitive and from the wiki we get another view of what a spinor is. Nice! Clifford was a beast too.

spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.

Something you might find interesting in light of recent events. Here is how you model the spread of infectious diseases.

Let the total population be separated into three non-intersecting compartments N = S + I + R

Susceptible - Denoted by S. These individuals are those who are healthy and can contract the disease
Infective - Denoted by I. These individuals are those who have contracted the disease and can spread it to others.
Removed/Recovered - Denoted by R. These individuals have recovered and cannot contract the disease again.

We want to model how each of these compartments changes with respect to time. So we get a system of ordinary differential equations.

We will start with dS/dt. Once we have that the rest of the system of ODEs is easy to write down.

Let cN = number of contacts per unit time an infectious individual makes with the population
S/N = the probability that contact is with a susceptible individual
p = the probability that contact with a susceptible individual results in transmission of the disease

cN*S/N*p = BS (we set B = pc)

So we have S' = -BSI (the rate at which individuals leave the susceptible class and enter the infective class)

I' = BSI - aI (note that people leave the susceptible class at a rate of BSI so they must enter the infective class at a rate of BSI. aI is the rate at which people recover from the disease and leave the infective class into the recovered class).

R' = aI (the rate at which people leave the infective class and enter the recovered class).

S' = -BSI
I' = BSI - aI
R' = aI

This is called the SIR model for the spread of infectious diseases. Note that S' + I' + R' = 0. This makes sense because we assumed a constant population N = S + I + R. Which means N' must be zero.

For different diseases that spread over a longer time scale we cannot assume a constant population so we must factor in population dynamics. The basic SIR model above is still very effective for modeling seasonal diseases like the flu and is a good starting point to learn mathematical epidemiology.

The most important quantity in the model is B/a which is called the reproduction number. This number essentially tells us whether or not the disease will die out or become an epidemic.

B/a > 1 epidemic
B/a < 1 disease dies out

One of the major applications of this model is that once we have the reproduction number B/a we know how much of the population needs to be vaccinated so as to confer herd immunity onto the population. You might have seen the term herd immunity before. When a population has herd immunity it means is that enough of the population has been vaccinated so that the disease cannot spread (we essentially make the reproduction number less than 1 so the disease dies out).

Smallpox has a reproduction number of approximately 5. By having over 80% of the global population vaccinated against smallpox we had a global herd immunity against smallpox. Within a generation smallpox was wiped out and now the only place smallpox lives is in laboratories. This is the only disease for which we attained global herd immunity.

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Nice! Relevant and mathematical.

Differential equations really are all-powerful

Wonder what the B/a will be for corona

https://www.youtube.com/watch?v=ovJcsL7vyrk

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Well I think most times in English I hear people say Lee too, I just happen to know Norwegian a bit so I can step in and be the "Ackshually..." guy

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Romes187 posted...

Nice! Relevant and mathematical.

Differential equations really are all-powerful

Wonder what the B/a will be for corona

It's probably going to be pretty high. The reproduction number can often be more deadly than the actual mortality rate of the disease if that makes sense. Like most people scoff at how dangerous the flu can be because they've had it before and they are fine. The flu of 1918 that spread to a significant portion of the population was one of the most devastating events in human history.

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Bundles at a first approximation

Been talking a lot about bundles but haven't really explained what they really do. Before I get into that, we should get a trivial example in our minds to help motivate our intuition. There are a lot of types of bundles but one of the easier ones to get is the vector bundle.

Wikipedia definition is:

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

Translation: We can create supercharged coordinate systems that are parameterized by a space.

A bundle comes in three parts. You have a base space which is just a blob in your mind...can be any shape. And at each point of that space you "attach" what is called a fiber. This can be a different type of space. The two of those combined create what is called the total space. And there is a MAP from the base space to the fiber

This map is not a bijection as mentioned previously in the manifold section. It's onto but NOT one-to-one because each point on the base space maps to every point on that fiber.

An example helps clarify.

Let's take a circle as our base space, and our fiber will be R1 (or the real number line) from [0,1]. So at each point on the circle (S1) you have a line sticking up and down and this goes all the way around the circle. What do you have?

A cylinder. Simple enough and this is generalizing the S1 x R1 product. However, we could take that cylinder, cut it and flip it so that there is a point that maps to -1 and maintain the maps continuity around the circle.

That completely changes it to a mobius strip. So the bundle allows us to have it locally be a cross product of two manifolds even though the total space is not.

We'll extend this definition a bit later when we start attaching lie groups as our fibers to give us frame bundles.

Bundles bundles and more bundles.

Time to discuss sections: https://en.wikipedia.org/wiki/Section_(fiber_bundle)#/media/File:Bundle_section.svg

The picture really helps out but a section is a slice of the total space mapped up to the total space from the base space (btw the post above has that mixed up. The section is from base space to total space. The bundle map is from the total space to the base space which is what the post above should say)

If E = Total Space
B = base space
pi = map

so if our map pi: E -> B is the bundle map, then a section is phi(pi(x)) = x where phi: B -> E

So we start in the total space, map down to the base space, then map back up to the fiber. So we take pi(x) which takes the point on the fiber down to the base space, then we take phi of that point which maps back to the same point.

Believe it or not this is an abstract version of what it means to be a graph. See the wiki article for a detailed proof on that abstraction.

The cool part is this allows you to extend the intuition to taking sections of shapes that are not as easily visualized or many dimensional.

Sections will change depending on what type of bundle you are working with

So if you are working with a vector bundle, then a section is simply a vector field at that point.
If you are working with a cotangent bundle, then the section is a one-form

This make working with really tricky manifolds easier

Forgot to mention: you can only (well "usually") define sections locally...usually there is some kind of twist in the fiber bundle which doesn't allow for a nice section globally unless its a trivial bundle.

https://twitter.com/3Dmattias/status/1233509531214327811?s=20

cool animation

fwiw, I haven't forgotten this topic, but life has caught up with me

I plan on handling conservation of mass on Monday!
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https://en.wikipedia.org/wiki/Symbol_of_a_differential_operator

Dunno if anyone can help me wrap my head around that...

Trying to learn something else and thats a component.

https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem

Apparently the index theorem there is heavy duty lifting stuff. But man its pretty abstract. I can kind of follow along despite not getting some of the terms but usually I can see what the "machinery" is doing in my head....not so far with this one.

Bump

One more bump. Busy today so might not get a post in but will tomorrow at the latest.

https://www.youtube.com/watch?v=PFDu9oVAE-g&t=2s

Not gonna have time again today to do a write up so instead here's a really eye opening video on what eigenvectors and values are

What a busy week

we got some giant accounts sold so no time for maths

bump :)

500000000 / 325000000 = $1 million for every american

maths

Here's something I cooked up.

Prove that the sum of n consecutive positive integers is divisible by n if and only if n is odd. (Not too difficult)

Prove that the product of n consecutive positive integers is divisible by n! (A bit trickier)

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FLOUR posted...

Here's something I cooked up.

Prove that the sum of n consecutive positive integers is divisible by n if and only if n is odd. (Not too difficult)

Prove that the product of n consecutive positive integers is divisible by n! (A bit trickier)

i actually find the second one easier than the first one

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The 2nd one is more intuitive but trickier to prove.

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I suck at proofs
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teepan95 posted...

I suck at proofs

Coming up with the proof, or writing the proof despite knowing how the proof goes? Both can be a right bitch sometimes.

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More so the first, I feel
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https://youtu.be/Kas0tIxDvrg

God I love me some 3Blue1Brown.

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He did a great podcast with Lexi Fridman about teaching math.

his vids helped me understand what linear algebra is, especially eigenvalues and determinants

Romes187 posted...

He did a great podcast with Lexi Fridman about teaching math.

his vids helped me understand what linear algebra is, especially eigenvalues and determinants

God yes, his vids on the topic gave me a more intuitive grasp of the subject than multiple courses in college did.

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