7 ms·

Math Academy, part 1: My eigenvector embarassment

84 points by gmays 2y ago

HankAsherGhost 2y ago
I hope someone here finds this solution set of mine useful: https://drive.google.com/drive/folders/1JrMp7R4j86tMzHn0Sfa_GMl9ZVXd0Wyj https://drive.google.com/drive/folders/1JrMp7R4j86tMzHn0Sfa_...
doctorpangloss 2y 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 2y 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 2y 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 2y ago
    It was an optional senior level course at my college
- mattpallissard 2y 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 2y 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 2y 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 2y 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 https://www.cambridge.org/core/books/course-in-mathematics-for-students-of-physics/462A6E58CE373CBB411B68B73CAEA543 https://www.cambridge.org/core/books/course-in-mathematics-f...
  - deleted 2y ago
    [deleted]
- teunispeters 2y 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 2y ago
  I was surprised too. I thought Linear Algebra and Real Analysis are the foundation of any math degree.
  - thaumasiotes 2y ago
    Topology is also considered a staple of a modern math degree.
    - mkl 2y 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.
- moregrist 2y 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 2y 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 2y 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,
- raincole 2y 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.
  - emseetech 2y 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 https://x.com/justinskycak/status/1835085776524394951
- cultofmetatron 2y 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.
deleted 2y ago
[deleted]
rubing 2y 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 2y ago
So you get the explanations and you get the exercises, but can you ask questions?
resource0x 2y 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.
- gmays 2y 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 2y 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 2y 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.
trentnix 2y 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 https://www.mathacademy.us/about). If you want to check out the podcast, it's here: https://techzinglive.com/ 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.
pinoy420 2y 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 2y 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 2y 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 2y ago
Math Academy is the best online course I’ve ever done
barrenko 2y 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 2y ago
  MVC has depencies on LinAlg. Ex Jacobians.
  - barrenko 2y ago
    Ty!
- yorwba 2y 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.
  - barrenko 2y ago
    Appreciate the feedback!
zelos 2y 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 2y ago
I hope math academy keeps adding more courses because it looks very promising, now it's a bit too basic unfortunately.

84 points by gmays 2y ago

I hope someone here finds this solution set of mine useful: https://drive.google.com/drive/folders/1JrMp7R4j86tMzHn0Sfa_GMl9ZVXd0Wyj https://drive.google.com/drive/folders/1JrMp7R4j86tMzHn0Sfa_...

doctorpangloss 2y 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 2y 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 2y 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 2y ago
  It was an optional senior level course at my college
mattpallissard 2y 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 2y 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 2y 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 2y 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 https://www.cambridge.org/core/books/course-in-mathematics-for-students-of-physics/462A6E58CE373CBB411B68B73CAEA543 https://www.cambridge.org/core/books/course-in-mathematics-f...
- deleted 2y ago
  [deleted]
teunispeters 2y 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 2y ago
I was surprised too. I thought Linear Algebra and Real Analysis are the foundation of any math degree.
- thaumasiotes 2y ago
  Topology is also considered a staple of a modern math degree.
  - mkl 2y 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.
moregrist 2y 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 2y 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 2y 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,
raincole 2y 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.
- emseetech 2y 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 https://x.com/justinskycak/status/1835085776524394951
cultofmetatron 2y 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.

deleted 2y ago

[deleted]

rubing 2y 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 2y ago

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

resource0x 2y 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.

gmays 2y 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 2y 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 2y 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.

trentnix 2y 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 https://www.mathacademy.us/about). If you want to check out the podcast, it's here: https://techzinglive.com/ 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.

pinoy420 2y 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 2y 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 2y 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 2y ago

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

barrenko 2y 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 2y ago
MVC has depencies on LinAlg. Ex Jacobians.
- barrenko 2y ago
  Ty!
yorwba 2y 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.
- barrenko 2y ago
  Appreciate the feedback!

zelos 2y 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 2y ago

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