_______ __ _______
| | |.---.-..----.| |--..-----..----. | | |.-----..--.--.--..-----.
| || _ || __|| < | -__|| _| | || -__|| | | ||__ --|
|___|___||___._||____||__|__||_____||__| |__|____||_____||________||_____|
on Gopher (inofficial)
URI Visit Hacker News on the Web
COMMENT PAGE FOR:
URI Math Academy, part 1: My eigenvector embarassment
HankAsherGhost wrote 59 min ago:
I hope someone here finds this solution set of mine useful:
URI [1]: https://drive.google.com/drive/folders/1JrMp7R4j86tMzHn0Sfa_GM...
cinntaile wrote 8 hours 14 min ago:
I hope math academy keeps adding more courses because it looks very
promising, now it's a bit too basic unfortunately.
zelos wrote 8 hours 51 min 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.
barrenko wrote 8 hours 57 min 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.
yorwba wrote 8 hours 20 min 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 â R^m at a point x
is a linear map L: R^n â 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 à n univariate functions
that you do ordinary calculus with (lots of partial derivatives) but
I doubt it would be very enlightening.
barrenko wrote 4 hours 50 min ago:
Appreciate the feedback!
pdhborges wrote 8 hours 37 min ago:
MVC has depencies on LinAlg. Ex Jacobians.
barrenko wrote 4 hours 50 min ago:
Ty!
te_chris wrote 9 hours 24 min ago:
Math Academy is the best online course Iâve ever done
pinoy420 wrote 9 hours 29 min 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 doesnât 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 wrote 4 hours 12 min 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.
pinoy420 wrote 25 min 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.
trentnix wrote 11 hours 36 min 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 ( [1] ). If you
want to check out the podcast, it's here: [2] 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.
URI [1]: https://www.mathacademy.us/about
URI [2]: https://techzinglive.com/
resource0x wrote 13 hours 37 min 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.
gauge_field wrote 5 hours 46 min 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.
3abiton wrote 5 hours 22 min ago:
I really like your approach. Any resource recommendations?
gauge_field wrote 5 hours 13 min ago:
On what topic? Linear Algebra, Quantum Mechanics?
adhamsalama wrote 10 hours 5 min ago:
3blue1brown is pretty good.
raincole wrote 3 hours 14 min 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.
rahimnathwani wrote 13 hours 13 min 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.)
cgriswald wrote 11 hours 5 min ago:
I got a B in my linear algebra course which was basically only
numerical. Iâd 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 donât think any of that numerical approach helped when I read
LADR. LADR isnât about âdoing the workâ itâs about âdoing
the work to understandâ. 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. Thatâs, frankly, the only way it works with Axler, so
if you want it, youâve 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 âbeing a student is
sufferingâ but he used it to justify a bunch of bullshit. In this
case, though, Iâd agree with him. LADR is suffering d followed by
satisfaction (and rinse and repeat).
resource0x wrote 12 hours 55 min 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.
gmays wrote 13 hours 14 min 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.
resource0x wrote 13 hours 4 min 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.
Tainnor wrote 2 hours 3 min ago:
That's not a "mathematician" thing, it's a US thing. US
universities, for some reason, insist on teaching mathematics
twice, once with lots of handwaving and then at some point you
get to do a "proof-based course".
In Europe (at least in certain countries, can't speak to all of
them), maths lectures will typically be abstract and proof-based
from day 1 - at least for maths majors (but frequently for CS and
physics students too). Other majors, such as economics and maybe
engineering, may get their own lectures that tend to be more
hand-wavey because they don't necessarily need the axioms of real
numbers to take a derivative here and there.
My linear algebra course was algebra and proof based to the
extent that maybe a little bit more geometric intuition would
have helped.
galaxyLogic wrote 14 hours 0 min ago:
So you get the explanations and you get the exercises, but can you ask
questions?
rubing wrote 14 hours 53 min 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).
insane_dreamer wrote 15 hours 56 min 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.)
cribbles wrote 8 hours 48 min 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.
viraptor wrote 11 hours 8 min 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.
gmays wrote 14 hours 1 min 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: [1] The Math Academy team (including the
founders) are also active on X/Twitter: [2] And there's a Math
Academy community on X here in case you want opinions from other
users:
URI [1]: https://gmays.com/math
URI [2]: https://x.com/_MathAcademy_
URI [3]: https://x.com/i/communities/1833198423593431339
prisenco wrote 14 hours 9 min 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.
shireboy wrote 14 hours 18 min 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.
cultofmetatron wrote 14 hours 21 min 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.
HPMOR wrote 14 hours 8 min 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.
prisenco wrote 14 hours 6 min ago:
I'm working through Foundations specifically to do Math for
Machine Learning and I'm very excited for that course. What were
your thoughts on it?
Also, if they ever did a Math for Computer Graphics course I'd
never cancel my subscription.
ATMLOTTOBEER wrote 13 hours 6 min ago:
Itâs just ok. Certainly better than watching some random yt
videos or passively reading a tb. For context I did ~11k pts to
finish M4ML. Itâs a little frustrating to see them chase the
money and make ML/programming courses while the core
differentiator (lesson quality) is still lacking in the later
math topics. There are persistent issues that annoyed me so
much that by the end it was like pulling teeth go grind out the
points every day.
It has you overfit on the style of questions they ask, and I
never felt like I got a good grasp of lots of the later topics
despite passing my reviews and quizzes no problem.
HPMOR wrote 13 hours 59 min ago:
I think Math for ML is __fantastic__. And based on the
curriculum Jason has published for ML, it seems very __very__
promising. I've done a fair bit of ML @ Cornell, so I've had
exposure to a lot of the material he plans on covering.
However, I glossed over a lot of the theory because some
weakness in my math. I feel like this has been remediated with
M4ML and ML should expand and solidify my understanding.
Edit: Wrt to computer graphics, going through M4ML and the ML
sequence will really help you understand what's happening
there. Convolutional Neural Nets, Gaussian Splatting, all rely
on these same principles.
raincole wrote 15 hours 24 min 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.
prisenco wrote 14 hours 4 min ago:
| it hardly teaches you how to write proofs
Proofs are coming, the site is a work in progress.
URI [1]: https://x.com/justinskycak/status/1835085776524394951
tarr11 wrote 15 hours 48 min 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 didnât 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,
i2go wrote 16 hours 46 min 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?
moregrist wrote 8 hours 52 min 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. Thatâs where I learned eigensystems.
Thatâs 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 arenât able to
assume the math as a pre-requisite and have to do that. But itâs
tedious and awkward, like trying to learn physics without calculus.
LudwigNagasena wrote 11 hours 15 min ago:
I was surprised too. I thought Linear Algebra and Real Analysis are
the foundation of any math degree.
thaumasiotes wrote 10 hours 13 min ago:
Topology is also considered a staple of a modern math degree.
mkl wrote 7 hours 33 min ago:
In my experience (maths degrees at two universities with >15000
students, one of which I now teach at) group theory and abstract
algebra more generally are much more likely to be part of a maths
degree than topology. I've never heard anyone describe topology
that way.
teunispeters wrote 13 hours 39 min 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!).
aithrowawaycomm wrote 15 hours 47 min 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.)
dboreham wrote 11 hours 54 min ago:
Hmm. He'd have to be over 90 years old to have studied before the
1950s.
beezle wrote 15 hours 28 min 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.
FranklinChen wrote 13 hours 43 min ago:
Whoa, Liboff, that book... I only vaguely remember it now (took
QM in 1988). I took "math for mathematicians" (Math 25) instead
of "math for physicists" (Math 22?), but remember my classmates
who took first year "math for physicists" got eigenvectors very
quickly right off the bat in the pre-published book they used
URI [1]: https://www.cambridge.org/core/books/course-in-mathemati...
mattpallissard wrote 16 hours 9 min 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.
windows_hater_7 wrote 16 hours 12 min ago:
I asked myself the same thing. The article said âlearnâ Linear
Algebra, not âreviewâ Linear Algebra. Do some undergrad math
programs not teach Linear Algebra?
readthenotes1 wrote 16 hours 11 min ago:
It was an optional senior level course at my college
doctorpangloss wrote 16 hours 55 min 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.
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