Math Academy, part 1: My eigenvector embarassment

https://frankhecker.com/2025/02/08/math-academy-part-1/

by gmays • 1 year ago

84 63 1 year ago

I cancelled my Math Academy sub because I ran out of 30 minute blocks for SAT problems I would never need. It was too remedial.

i2go• 1 year ago

he was a physics and math major and did not know eigenvectors and eigenvalues? i would like to know how is this possible. can someone explain it to me?

windows_hater_7• 1 year ago

I asked myself the same thing. The article said “learn” Linear Algebra, not “review” Linear Algebra. Do some undergrad math programs not teach Linear Algebra?

readthenotes1• 1 year ago

It was an optional senior level course at my college

mattpallissard• 1 year ago

Same boat as the author here, except I switched from physics to software after year three.

I had never heard of them until I was _years_ into software engineering. I think this is more common than you may think. I had never dealt with linear algebra in a formal setting, despite leveraging a lot of the concepts, until then.

aithrowawaycomm• 1 year ago

He is a bit older. Linear algebra is also very old, but it didn't really become the field we know today until the 1950s. I would add that in 2025 it is cheap to buy a computer that can solve large linear systems, but that certainly wasn't true in 1975, so linear algebra was less applicable in the real world.

I am not too familiar with the pedagogical history of linear algebra, but I've been reading some advanced undergraduate geometry texts from the 30s-60s and linear algebra was generally not an assumed prerequisite. There was a particular separation between the studies of "two and three dimensional vector spaces over R" (largely geometric) versus "finite dimensional vector spaces over a field" (entirely algebraic), and determinants were presented directly as volume computations. These days undergraduates mostly treat R^2 and R^3 algebraically, maybe at the expense of geometric understanding. (E.g. Euler's rotation theorem is easily proved when restated as a theorem about matrices over R^3 with determinant +1, but Euler's original statement and proof using spherical trigonometry is deeper.)

beezle• 1 year ago

I was a double major, one in physics, in the 80s. After the three semester engineering physics classes, intro QM was taught spring sophomore year. We used Liboff. In addition, it was required for all physics, chem and engineering majors to take math 20(5?) which was linear algebra.

And given that most of basic QM was formalized by 1930 and relies upon eigenvectors, hard to see any physics course taught since that time not having it.

dboreham• 1 year ago

Hmm. He'd have to be over 90 years old to have studied before the 1950s.

teunispeters• 1 year ago

Speaking as a CS/Math dual major from the late 1980s, the explanations in the textbook were ... uselessly bad where I went (SFU). So while I know the math involved, I didn't know the terms until I retook it from a teacher who actually taught worth anything, years later. (I still don't use the terms, they're not very ... well, they're awkward and while my embedded work sometimes calls for the math, the terms ... meh. There are clearer ways to put things!).

LudwigNagasena• 1 year ago

I was surprised too. I thought Linear Algebra and Real Analysis are the foundation of any math degree.

thaumasiotes• 1 year ago

Topology is also considered a staple of a modern math degree.

moregrist• 1 year ago

I was also struck by this.

I was introduced to eigenvectors in a math course on linear algebra. They seemed esoteric but I could prove theorems and stuff… cool but kind of forgettable.

Then I took quantum mechanics. That’s where I learned eigensystems. That’s where their utility and beauty were beaten into me, problem set by problem set. In quantum mechanics, eigensystems are ubiquitous: from using ladder operators to solve the harmonic oscillator in an elegant way, to what quantum numbers actually are, to the reason behind the Heisenberg uncertainty principle, and to the so many different ways to use perturbation theory to explain atomic and molecular spectra.

You can do the basics of quantum mechanics without explicit linear algebra, and many intro physical chemistry texts aren’t able to assume the math as a pre-requisite and have to do that. But it’s tedious and awkward, like trying to learn physics without calculus.

insane_dreamer• 1 year ago

is MathAcademy that much better that KhanAcademy (which also has a Linear Algebra course and covers eigenvalues of course), which is free? Considering it for my youngest kids, but my eldest (now finished college with a degree in engineering) used Kahn Academy as a high school supplement and it was quite good (this was about 10 years ago). (She didn't take the KahnAc LinAlg course -- not sure it was around at that time -- but she did take their calc course and it helped her ace her CalcBC AP test.)

tarr11• 1 year ago

yes if your kid loves math - my kid benefited from Math Academy (currently in college studying CS). Started with Math Academy as a HS freshman and finished BC Calc 2 years early with a 5 on the AP Exam. While he did Khan Academy for many years (which is a great resource) it didn’t really motivate him as much and seemed designed for a broader more generalist audience.

We also tried some other programs like Art of Problem Solving (great program, but required very synchronous classes which were hard to fit in)

My suggestion would be try it for a few months,

raincole2• 1 year ago

Math Academy has some "gamification" (using this term loosely) to push you to do your homework. That's actually great if you care about learning efficiently.

But as far as I remember, it hardly teaches you how to write proofs, so how much it actually teaches math is a bit questionable.

prisenco3• 1 year ago

| it hardly teaches you how to write proofs

Proofs are coming, the site is a work in progress.

https://x.com/justinskycak/status/1835085776524394951

cultofmetatron2• 1 year ago

is MathAcademy that much better that KhanAcademy

I'm using mathacademy and I will unequivocally say its better and worth the money. First it uses a assessment to test your knowledge and will send you modules to fill in the gaps. I finished foundations 1 and now I'm in the middle of foundations 2.

One thing I love about math academy is that you spend very little time reading and no time watching videos on the subject. You get a short walkthrough of how to solve a problem and then it gives you a few more problems building up the complexity. A few days later it gives you the problems again to test your knowlege. the interface is not as pretty as khan academy but you're basically learning by doing and its very effective. I wish it was around when I was in university.

HPMOR3• 1 year ago

MathAcademy really is fantastic. I've done 5300 xp so far (80+ hours), and am almost finished with their Math for Machine Learning. It's remediated a lot of things I've struggled with during my University ML classes. Seriously a wonderful pedagogical experience. I can integrate multivariate functions easily, know all my derivative trig identities, and I don't get confused by continuous random variables any more.

shireboy• 1 year ago

We homeschool our son and after a lot of trouble with an online course that used Saxon math, we landed on Math Academy. I found it much easier to use than Khan both for the parent and student. I do sometimes wish there was video content and discussion of application. But my son does it first thing every weekday and really seems to do good with it. Also worth noting it is accredited and lets you print a transcript.

prisenco3• 1 year ago

Much better for anyone so motivated. There's considerably less handholding, the questions are a little "trickier" (without being unnecessarily cruel) and there's a more intense pacing to it since there's no videos. It's focused on learning-by-doing, which I love.

For many, I would recommend Khan Academy. It's a great resource, especially since it's free. But if learning math is about more than just passing a class, Math Academy is worth every penny.

gmays2• 1 year ago

Yes, Math Academy is insanely good. It's a lot more intense, focused learning and less edutainment.

I'm an adult and not a kid, but wrote about my experience after 100 days of using it daily here: https://gmays.com/math

The Math Academy team (including the founders) are also active on X/Twitter: https://x.com/_MathAcademy_

And there's a Math Academy community on X here in case you want opinions from other users: https://x.com/i/communities/1833198423593431339

viraptor• 1 year ago

If your goal is to practice and be able to pass tests perfectly - yes, it's much better. If you just want an overview of some area for a specific task, probably not. MA's approach is "you're going to learn it and you're going to learn all the foundations for it and you'll perfect the tests", which is great for many people - especially if you're actually going to be tested on things in the future. And being 99% practice, 1% reading really leans into that idea.

cribbles• 1 year ago

Yes. I use and enthusiastically endorse Math Academy. It is far and away the best self-learning educational resource I have ever tried; leagues better than Khan Academy.

I'm a self-motivated adult learner, so I don't know what it's like for kids. Though the program was originally designed for them, so I suspect their experience would broadly be similar to mine.

As other commenters have mentioned, you need to be okay with grinding through problem sets with no videos or UI pizzazz -- maybe this doesn't work for everybody. I'd compare it to the difference between trying to learn a language through scattered YouTube videos and Duolingo versus tandem and grinding on a good Anki set.

NB: I'm taking it for the Math for ML track and am currently most of the way through the Math Foundations III course. So I can only comment on the lower level courses.

rubing• 1 year ago

eigenvectors were the only tough part of the linear algebra course i took, i think that's b/c it's quite a bit to learn before you start seeing the point of it. methods like PCA are rely heavily on eigendecompositon and allow you to reduce the dimensions of data...this is useful in all sorts of ways like compression for instance (e.g. getting rid of the dimensions that aren't very meaningful).

galaxyLogic• 1 year ago

So you get the explanations and you get the exercises, but can you ask questions?

resource0x3• 1 year ago

As a math major, I scored a perfect 100 on my Linear Algebra exam in 1974. However, just two days later, I couldn't recall a single thing.

A few years ago, with ample free time, I decided to refresh my (nonexistent) memory by watching online linear algebra lectures from various professors. I was surprised by their poor quality. They lacked motivation and intuition. Khan Academy offered no improvement. Then, someone recommended Linear Algebra Done Right (LADR). I read it three times, and by the third iteration, I finally began to appreciate the beauty of the theory. Linear algebra is a purely algebraic theory; visual aids are of limited help. In short, if you have the time, I recommend reading LADR. Otherwise, don't bother.

gmays2• 1 year ago

Have you tried Math Academy? I think the difference is that it's actually made by a team of mathematicians creating the content manually.

resource0x3• 1 year ago

Not yet, but now I will, just out of curiosity. There's a problem with mathematicians teaching the subject. After all, the youtube lectures were also given by mathematicians. In attempt to make things "accessible", they de-emphasize the algebraic part of the subject and replace it with... I don't know what. The common theme is to consider only R^n. That's not what it's about. Maybe Math Academy course is different though.

rahimnathwani• 1 year ago

I don't know whether LADR is good for someone who is new to linear algebra. I've seen it recommended so many times, so ~12 years ago when I was living in Beijing I bought two copies (one in English for me, and one in Chinese in case I needed to ask a colleague for help).

It took me time to study each page, to understand the examples, and then to attempt the exercises. It seemed very beautiful.

Then one day I came to a part I couldn't understand: I didn't see how something Axler said followed from the earlier stuff on the page actually followed. I scratched my head for a couple of hours, which is much longer than I'd spent on any previous page.

Eventually I asked a colleague for help. I showed him the page. He asked me to explain what I didn't understand. I started to explain what I knew, and how I didn't understand how this thing followed. As I was explaining it, that part suddenly clicked.

But I got stuck a few more times and didn't persevere.

I wonder whether it would have been better for me to have studied some numerical approach to linear algebra (like Strang's videos) first, rather than going straight into a book that's so abstract and proof-based.

I suppose it depends on your mathematical background.

(Your comment made me think about those folks who were once fit and muscular, then years later they are out of shape, and then they decide to get in shape say how easy it was to get back in shape. They don't realize that part of what made it easy is that they were once in shape, and they still more muscle cells or whatever.)

resource0x3• 1 year ago

Very true. But the same applies to teaching. Mathematicians don't know where even to begin - for some of them, it's all too obvious. But the same happens with any subject. Someone proposes a certain design - but after many (20... 30... 40) years in business, you feel the design won't ever work, and try to explain, and fail because you don't know where to begin.

cgriswald• 1 year ago

I got a B in my linear algebra course which was basically only numerical. I’d have gotten an A but the professor thought mountains of homework was teaching and I refused to do it all. Suffice it to say I aced every test and all the homework I actually did. None of it helped in understanding and like the grandparent I remembered none of it at the end and turned to LADR.

I don’t think any of that numerical approach helped when I read LADR. LADR isn’t about “doing the work” it’s about “doing the work to understand”. Similar to your experience I remember reading the first chapter and then among the first chapter questions I saw questions that looked like they had no basis whatsoever in what I thought I had just learned. Then, eventually, it clicked. That’s, frankly, the only way it works with Axler, so if you want it, you’ve got to do it.

My advice is to not waste time with the numerical approach and just do it.

I had a professor who used to say “being a student is suffering” but he used it to justify a bunch of bullshit. In this case, though, I’d agree with him. LADR is suffering d followed by satisfaction (and rinse and repeat).

adhamsalama• 1 year ago

3blue1brown is pretty good.

raincole2• 1 year ago

3blue1brown's linear algebra series is very different from what GP is talking about.

If you think linear algebra is something geometric, like "a 3x3 transform matrix is rotation and scaling; an eigenvector is something after transformation and parallel to its old self..." you will be surprised at how little LADR talks about these.

On the contrary, the most important part (imo) of 3b1b is that it helps you intuitively get these geometric interpretations.

gauge_field3• 1 year ago

For me, the main solution was to apply it to another problem that uses Linear Algebra as Application, which in my case was Introductory Quantum Course and implementing BLAS using Rust and C. That way you keep thinking and using this info. Otherwise, information in vacuum seems to abstract to care about.

3abiton2• 1 year ago

I really like your approach. Any resource recommendations?

trentnix• 1 year ago

Jason Roberts, the founder (and primary coder) of Math Academy, has been podcasting for over 15 years and has been talking about Math Academy and its inspiration, origins, business fundamentals, financial realities, and ambitions on the podcast for many years. A lot of that discussion is distilled in the Math Academy about page (https://www.mathacademy.us/about). If you want to check out the podcast, it's here: https://techzinglive.com/

Jason also coined the term "Luck Surface Area" which has since been popularized by a number of others.

I haven't used Math Academy myself (although it's something I intend to try one of these days), but I can safely vouch that Math Academy isn't a fly-by-night shallow edtech grift. They've spent a small fortune and thousands of hours developing and refining content and curriculum. Math Academy is a thoughtful, intentional, well-manicured solution.

pinoy4202• 1 year ago

Skip to last chapter.

The most notable of these are the synthetic division method for polynomials, the various trigonometric identities, and differentiation of products and quotients of functions.

So he learned nothing you already know at 15. Or younger in Asia.

I think he forgot his goals because it doesn’t even mention eigenvectors.

I am surprised because it is not a difficult thing to understand? It is a vector that when multiplied to a matrix (which in almost all cases would change the direction of the vector), in fact only scales it - and does not change its direction.

The scale factor is its eigenvalue.

So if you hav [[2,0],[0,3]] this should when multiplied to a vector give you [2x,3y]. But if you supply the vector [1,0] or [0,1] you see that the result multiplies that vector by two. So any multiple of these eigenvectors (e.g. [10,0]) will result in a doubling of the vector.

This is not a difficult concept. By any means.

fransje26• 1 year ago

Hooray! You managed to explain it as dryly and as poorly as any other linear algebra book or course out there.

Now, do the part with the explanation of what it actually means for a (physical) system to have eigenvalues, and what it tells you about the response of such a system to external or intensive inputs, or how to change such a system to targeted a certain response.

pinoy4202• 1 year ago

If you find that anything besides a poor dry explanation, that is on you. I am sorry this enrages you.

Eigenvalues in an oscillating system describe its resonant frequencies. Its eigenvectors can describe motion at that certain resonant frequency. Imagine a bridge. If wind or traffic match a resonant frequency (eigenvalue) it would be dangerous. Engineers can redesign it to change the corresponding eigenvector and shift the eigenvalue (its resonant frequency for that mode of oscillation) to a safer range. See that bridge in London.

te_chris• 1 year ago

Math Academy is the best online course I’ve ever done

barrenko3• 1 year ago

To be honest, I just read the introductory post, and it'stated there that the author wants to do MVC after finishing LinAlg, which is stated as their goal for end of 2025.

As someone that has the same end goal (but probably 2026 for me) - isn't it maybe wiser to do MVC before LinAlg?

Read the whole thing now, slightly disappointed OP doesn't try to tell us what an eigenvector is, based on his current progress.

pdhborges• 1 year ago

MVC has depencies on LinAlg. Ex Jacobians.

barrenko3• 1 year ago

Ty!

yorwba• 1 year ago

I assume by MVC expands to multivariate calculus?

In which case, I don't think it makes sense to do multivariate calculus before linear algebra.

The derivative of a multivariate function f: R^n → R^m at a point x is a linear map L: R^n → R^m so that f(x + v) = f(x) + Lv + o(|v|) for small v.

That means that multivariate calculus is about approximating nonlinear functions using linear ones in a small neighborhood, which enables you to apply tools from linear algebra to it.

You can kind of do multivariate calculus without linear algebra by essentially treating f as a collection of m × n univariate functions that you do ordinary calculus with (lots of partial derivatives) but I doubt it would be very enlightening.

barrenko3• 1 year ago

Appreciate the feedback!

zelos• 1 year ago

I have a fairly similar story to the OP. I have an engineering degree, but that was 25 years ago. I started reading a lot of 'proper' maths a few years back (abstract algebra, topology etc) and made decent progress, but it never quite stuck. The lack of decent problem sets with answers in so many textbooks is really limiting.

Going back through the foundations courses on Mathacademy (I started halfway through Math Foundations II, currently nearing the end of III) has been great. It's been surprising how much I've forgotten, but also reassuring how quickly it comes back. My plan is to move on to the more advanced courses with firmer foundations.

The focus on answering questions constantly helps me focus, although the multiple choice structure is kind of limiting, if inevitable. It's frustrating to have it throw a whole load more questions at you because you missed a minus sign, where a proper teacher would have seen your working and been able to tailor their feedback.

cinntaile• 1 year ago

I hope math academy keeps adding more courses because it looks very promising, now it's a bit too basic unfortunately.

HankAsherGhost• 1 year ago

I hope someone here finds this solution set of mine useful:

https://drive.google.com/drive/folders/1JrMp7R4j86tMzHn0Sfa_...

Math Academy, part 1: My eigenvector embarassment

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