Through comments here I found 3blue1brown's (clearly much loved) videos. By the third video I was shouting, "why for the love of god would we be doing this"? Based on this reaction I suspect that the content neither has intrinsic appeal to me, nor does it have obvious use in my work, projects, or life.
Pre-degree maths though, I love. My A-level maths really changed how I saw the world, and I make use of it reasonably often (well, often enough to not forget it).
I think I'm writing this here because most other commenters seem to really grasp this subject, or feel that they grasp it better having seen these videos. I'm honestly happy for you. However, if anyone is reading this who doesn't feel like that, then know you're not alone :-)
Grok.
Over the years, I come to the conclusion that one of the stumbling blocks is the definition/concept of "application". Just like the definition of "theory" is different for a layman ("My theory is..." == "My guess is...") than from a scientist's definition ("My theory is" == "My logical framework which incorporates all of the available data is..."), so the definition of "application" is different between mathematicians and engineers.
I've noticed that math books with titles like "$HIGHER_ORDER_MATH with Applications" means "$HIGHER_ORDER_MATH with Exercises". What I'm looking for is something like "$HIGHER_ORDER_MATH with Real-World Uses".
I've known LA for decades but, like you, where would I use it in my life? The turning point for me was Andrew Ng's Deep Neural Network course.
I knew that a DNN is a program of matrix operations, but how do you get 5,000 images into a matrix? One way is to resize all the images to the same n x n size, take the first pixel of each picture and break them into their RGB components. You now have the first three rows of your input matrix. Repeat for all other pixels and voila! You have a 5,000 x n matrix that you can do linear algebra on! _That's_ an application; having me add two matrices together is an exercise.
Since that insight, I've used LA in my job in the hospitably sector with impressive success because now I know how to apply it. Math books and 3B1B show you the math. We engineers (or at least this one) need real world uses.
It's perfectly okay not to learn linear algebra, by the way, especially when you don't find any incentive to do so. Otherwise, you'll find linear algebra to be one of the most intuitive tools to model so many problems.
If you do want to learn linear algebra or any other higher math, I'd strongly recommend you focus on understanding concepts intuitively first, to the point that you find many exercises in a text book straight forward. Watching 3blue1brown is a good start, but do move forward with deeper treatment. The book I find very usual is David Lay's Linear Algebra and Its Applications: https://www.amazon.com/Linear-Algebra-Its-Applications-5th/d.... Lay sets up a really intuitive geometric framework to explain the intuition of linear transformation with sufficient rigor.
Much of what makes linear algebra interesting and powerful comes from more advanced topics, especially eigenvalues. This power comes when we are not looking at a single matrix in isolation, but when we repeatedly apply a matrix. For instance, consider the equation x_t = A^t x_0. It turns out we can rewrite this an equation by diagonalizing A -- i.e., A = P^{-1} Sigma P, where Sigma is diagonal; most, but not all, matrices can be written in this form. We call s_i the "eigenvalues" of A.
Then, the equation simplifies to x_t = P^{-1} Sigma^t P x_0, or equivalently (P x_t) = Sigma^t (P x_0). This equation is dramatically simpler, since Sigma is diagonal, so if Sigma = diag(s_1, ..., s_n), then Sigma^t = diag(s_1^t, ..., s_n^t). In other words, this transformation "disentangles" the different components of A into ones that act independently. Here, the transformation x -> P x is what is called a "change of basis".
These repeated matrix applications are common in physics, where they represent how a dynamical system evolves over time. The main difference is that in physics, the system evolves continuously, but similar transformations can be applied to solve these problems.
Luckily, Chromium is getting MathML support (finally) unless something weird happens, which (presumably) means that Chrome and Edge will inherit that support as well. Firefox already has MathML, so hopefully we'll soon be in a position where three of the most widely used browsers support MathML.
Still, it would be interesting to ask the Powers that Be if they'd be willing to implement MathJax here on HN....
The article here focuses on an "operational" perspective, how the numbers get added or multiplied together to turn into other numbers. However, Linear Algebra is also useful in geometry, and other situations.
This "intuitive guide" to linear algebra sets you up very nicely for figuring out how to add and multiply matricies together. But it doesn't give you any intuition about a rotation (aka quaternions) in 3d space, for example. A lot of math books make the mistake of trying to teach all the perspectives at the same time, instead of focusing on just one viewpoint until the student gains mastery.
A Quaternion is "just" a 4x4 matrix that represents rotation in 3-dimension space. Because you only move 3-ways rotationally (yaw, pitch, and roll), you're "underconstrained" with regards to the 4x4 matrix. Etc. etc. A lot of geometry intuition needs to be built here to really understand Quaternion... and none of that geometry is explored in the blogpost.
Which is fine. Focus is good. But when people approach Linear Algebra, its important to know that its "so useful" that there are too many ways of looking at Linear Algebra... too many different, yet equivalent, understandings of the subject.
I was recently watching videos and trying to read papers on geometric algebra and getting totally and utterly lost on actually applying it.
I've rarely found any higher math instruction which takes for the form, "so you have this specific problem X, here is how we can solve it with technique Y"[1]. But I suspect that it is because it is higher math (which presumably means 'higher order' math).
Without this, and without an inherent enjoyment of the pureness of the math, it seems somewhat esoteric for me personally. I'm not complaining, nor do I really think it should be any other way. I'm just reflecting on it really.
This also makes me think of my foray into monads: "The thing about monads is once you finally understand them you immediately lose the ability to explain what they are to others." Not saying that's the case here, just feels related.
[1] At least where I found problem X to be satisfactory. I didn't find my lecturer's problem of, "you're stood on a mountain described by this PDE, on what vector must one walk in order to stay at the same altitude" to be very applicable. I was a pretty wilful student though.
Geometric algebra seems more practical than most subjects but even its introductions suffer from this.
Well, it means that understanding it means that you understand yourself! Well, not just that, but it might eventually lead to you understanding the architecture of the universe! After that? How about God? Or is God maybe a part of the architecture?
OK, so maybe I won’t try to go into the details there since opinions on it differ. Essentially, through this language though, you can master almost anything that you want! Yes, it might seem crazy, but the things that happen around you and the things which every poet and playwright and prisoner and savior ever composed can be explained through mathematics! Through it, you can also open up entirely new and utterly interesting worlds!
Let’s say that you dropped your pen this morning. To someone with no background in physics, or mathematics, this means nothing. On the other hand, to someone like me, it means quite a bit. The slight delay and movement altered the gravitational and electromagnetic field around you and echoed on into eternity. Also, it most likely changed your days structure and composition, and shifted your life into a new line (see quantum mechanics and chaos and linear dynamics) and had a profoundly large impact on everything else around you. You may have avoided a car crash, or you might have met a person who you wouldn’t have normally encountered all because of that small change. And this small change echoed on and effected everything else in turn. And this is all very mathematical, and extremely beautiful, but most people don’t know anything about it, but it does mean something to people like me: we’re all incredibly inter-connected, and our lives are ruled by chaos. Everything that you do, and everything that you say, and all the things you see and inter-related and ohh so close, but we tend not to see it and it all has to do with not understanding the fundamental mathematics!
The above is only touching on one small aspect of it though, as it only deals with physics and chaos. There are entire branches of math which make the world incredibly interesting which have nothing to do with the fields I just mentioned! Hey, did you know that standing next to someone who might look slightly like your wife will cause you to behave in a similar manner in which you behave when she’s actually around you? Yes, neurons that fire together wire together, and it’s very mathematical underneath but yet so simple! How about the fact that E = mc squared isn’t really true? Yes, the formula has an extra term (square root of 1 minus v squared over c squared) which makes it possible for massless particles to have energy and deals with relativistic effects. Did you know that you can summarize most of modern classical physics in just a few equations? (Yup, you can find most of them here: https://www.feynmanlectures.caltech.edu/II_18.html ). How about balance? Did you know that if you were standing at arm’s length from someone and each of you had one percent more electrons than protons, the repelling force would be so incredible that the repulsion would be enough to lift a “weight” of the planet! Yup – math is full of fun surprises!
Now, modern math is sort of like a constant tease which shows you the shell of this beautiful program and this excitement, and you know the beauty is there, but it’s not easy to understand and grasp! For one, most of mathematics is filled with jargon and language that is incredibly information dense, and it looks like it’s been written by a schizophrenic C programmer who’s paranoid about losing his job, so the information tends to be lumped into these incredibly dense formulas which hide the beauty and truth, but the beauty and truth will always be there! You just need to have a bit of persistence and dedication. It also doesn’t help that most teachers tend to not make things nice. They puke out the same old standardized stuff regurgitated and taught to them, and so round and round it goes.
Hopefully though, we’ll get better at teaching it and conveying it’s structure as we learn better ways of not making things cumbersome and uninteresting to other people! Wow, I need to stop writing – sorry for the large wall of text, but I hope you get what I mean!
Now that the ideas of things like vector spaces, norms, orthogonality, rank, basis, etc are nearly second nature, the concepts are useful as I study other branches of math which would feel impenetrable otherwise.
YMMV, and if you can learn from condensed materials go for it, but I might be too dumb for it work lol. I think the real benefit accrues to the author who had to work out how to teach these concepts to others.
Also, the concepts "mature" in the brain. I remember sleepless nights in the first year of undergrad spent on understanding the details of the proof of the Jordan decomposition and a few years later (when studying algebraic groups) it all felt trivial.
There's no shortcut to understanding maths, just a lot of time spent in solitude trying to make sense of all these abstract concepts (and they DO make sense).
I'm not sure I understand your point. Are you just saying that this blog post isn't an adequate substitute for taking a course in linear algebra? (Of course it isn't. But who said it was?)
In my university, undergraduates have admitted that they have done fewer than 50 questions throughout the entirety of my math course. Their grades obviously reflect that, but they will do the same next semester.
In a lot of content that teaches machine learning/AI, the linear algebra substrate of it is given short shrift.
The consumers of that content infer that the backing mathematics is easy or unimportant, while the creators of the content are not actually implying that, but just want to move on to what the audience came for.
I'm not an expert in machine learning so I can't say whether you can get by without a strong understanding of linear algebra, but my intuition answer is no you can't. Beside that, I enjoy the math for its own sake so I'm happy to trudge through the textbooks.
And no, I don't think the blog post makes any claims that it is a subsitute for taking a course in linear algebra.
In college I learned more math when I was trying to build software for teaching math compared to when I was trying to learn math.
Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.
I... disagree. Some of linear algebra is about that. And it's probably a good way to view it that way when learning.
But some of my current work (coding theory) involves linear algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really at no point are we viewing what we're doing as transforming a vector space, we're just solving equations with unknowns.
- A linear transformation
- A basis set of column vectors
- A set of equations (rows) to be solved
- (your example: parity equations for coding theory)
- The covariance of elements in a vector space
- The Hessian of a function for numerical optimization
- The adjacency representation of a graph
- Just a 2D image (compression algorithms)
... (I'm sure there are plenty of others)
> I... disagree.
This is literally the definition of the term "linear algebra".
> really at no point are we viewing what we're doing as transforming a vector space, we're just solving equations with unknowns.
You may not see what you're doing as transforming vector spaces with linear operators, but that is what you're doing. It's worth pointing out that the definition of vector spaces allows any field, including finite ones, though it's true that the intuition won't be exactly the same.
Another way to say this: if you're working on a problem without thinking about the connection to linear transformations, then it's not correct to say it's a linear algebra problem without obvious connection to linear transformations; instead, it's not a linear algebra problem at all, by definition.
However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra).
W. Wesley Peterson and E. J. Weldon, Jr., Error-Correcting Codes, Second Edition, The MIT Press.
-- gee, people are still studying/learning that?
The prof knew the material really well, but to up my game in the finite field theory from other courses, I used
Oscar Zariski and Pierre Samuel, Commutative Algebra, Volume I, Van Nostand, Princeton.
which did have a lot more than I needed!
My 50,000 foot overview of linear algebra is that the subject still rests on the apparently very old problem of the numerical solution of systems of simultaneous (same unknowns) linear equations, e.g., via Gauss elimination (it's really easy, intuitive, powerful, and clever, surprisingly stable numerically, and is fast and easy to program; someone might want to type in, say, just an English language description!). Since such the subject of linear equations significantly pre-dates matrix theory, the start of matrix theory was maybe just easier notation for working with systems of linear equations. In principle, everything done with matrix theory could have been with just systems of linear equations although often at a price of a mess notationally. In particular, as I outline below, now there are lots of generalizations of systems of linear equations that use different notation and not much matrix theory.
What's amazing are the generalizations, all the way to linear systems (e.g., their ringing) in mechanical engineering, radio astronomy, molecular spectroscopy, frequencies in radio broadcasting, stochastic processes, music, mixing animal feed, linear programming, oil refinery operation optimization, min-cost network flows, non-linear optimization, Fourier theory, Banach space, oil prospecting, phased array sonar, radar, and radio astronomy, seismology, quantum mechanics, yes, error correcting codes, linear ordinary and partial differential equations, ..., and then
Nelson Dunford and Jacob T. Schwartz, Linear Operators Part I: General Theory, ISBN 0-470-22605-6, Interscience, New York.
The thing is that the field over which the space is defined can be quite arbitrary (finite, infinite, not algebraically closed etc.) which has immense consequences on the behavior of such objects.
When one drops the assumption on finite number of dimensions, the story becomes wild (and is known as functional analysis, beautiful and extremely useful branch of mathematics).
snicker7 said it very succinctly:
> However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra).
If you're point is that I failed to comprehend matrices, then I don't think you have enough data to make that claim since I don't really talk about matrices. I kind of address that in my other comment [2].
I don't follow your point around "a wide array of problems and models are the same thing". That's a very vague general statement that I certainly comprehend (not sure how you inferred otherwise). Specifically, I don't see how that point relates at all to the claim I made about linear algebra.
[0] https://news.ycombinator.com/item?id=22419018
Why is that? Has anyone studied it, or is there even a solid anecdotal explanation? The best one I can imagine is many of these professors simply don't care much for teaching and are more focused on their research, which is still infuriating but at least an explanation.
I ended up with an undergrad in applied math, though I'm a software engineer now. I like math, but I feel like I never got to be all that great at it. I suspect I would've enjoyed it more and achieved more with explanations like these.
For example the author uses the word function liberally in the explanation. However, when studying functions in school in math it was super complicated for me. It was only after I started programming, learning the programming language meaning of function, and then when I was reintroduced to mathematical functions through functional programming that I truly grasped mathematical functions and all the stuff I was taught in school.
I’m not arguing that school level math does not need to be improved. I just think we should be cautious that because an explanation that seems intuitive to us now after having gained a complete introduction to all math concepts as well as programming (esp on HN) may not necessarily be as intuitive when students who are only exposed to a very small subset of mathematical concepts encounter them.
Now, CoM is a classic, a real great book, but it is useful only to people who have reached a certain level of mathematical maturity. That, presumably, is not the questioner.
A version of this is that I also see people who write, "I didn't understand this topic when I took the course but now years later, I see it is all actually very easy."
(I call this Second Book Syndrome because I don't know of a common name for it. I understand this to be what Zen people mean by the "Gateless Gate," that after struggling with something at great length a person can come to see that there is no real difficulty. But I've never heard anyone else apply that name to this phenomenon and I'm not Zen trained so I'm not sure that is right either.)
It wasn't until I tried teaching people that I realized how odd the notation can seem.
I taught Astronomy to the people in the years below me (a school tradition because it was optional), and it was absolutely exhausting trying to plan good lessons that didn't involve getting them to memorise stuff by wrote. I ""derived"" Kepler's laws for a bunch of 14-15 year olds who didn't even know logarithms yet and it was pretty brutal intellectually.
Also, I think teaching mathematics "intuitively" requires a bit of cooperation from the student - not in the sense of intelligence, just that for for every guy (or girl) who watches 3Blue1Brown (and looks deeper into the pure mathematics) there's another who's just along for the ride. I think the frequencies of those personalities are a product of how they were taught, but it's very difficult to convince people in "teaching time" as opposed to naturally (I was in the lowest maths group for years until I picked up a calculus book on a whim, but no teacher could've convinced old me that mathematics can be beautiful)
Quick Edit: To quote Tim Minchin, "Be a teacher". I might come across as moaning in the above but it's really interesting to explain things (in my experience at least - I love writing documentation!), and a good bullshit-test on yourself
Rant: If society gave a shit teacher would be the highest paid job with the highest standards, both in knowledge and in teaching skill. I mean literally 500k/year for the most noble calling on earth. The kind of progress humanity would make with a generation taught by the best, and teaching itself revolutionized, would be crazy.
https://www.amazon.de/Deutsche-Schulzeit-Erinnerungen-Erz%C3...
This creates the not-so-obvious problem that the easier you learn something the harder it might be to teach it to someone who doesn’t get it.
If you on the other hand struggle hard to find a way to understand a complex concept, you might be good at transfering that knowledge forward.
Of course AB != BA
Composition makes sense
Inverse makes sense
This is the book that helped me get it http://linear.axler.net/
There's still stuff he said that I'm unpacking today... that was a dense class.
When I see A(x) = ax, I'm not entirely sure how to read it.
Is A meant to be a function that accepts x? If so, why is the equivalent expression a * x? Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value? Is this notation specific to multiplication, to this expression, or what?
Positing that something is 'intuitive' when it depends so much on additional contextual knowledge seems ever so slightly disingenuous as best, and slightly harmful at worst; it can make the reader feel as though they must be dumb for not understanding this 'intuitive' material.
I do acknowledge that this is linear algebra, and if one doesn't have a really solid grasp of notation of regular algebra it is likely to go over their heads, but the practical explanations (such as the slope rise/run example) are quite clear and relatively simple to follow; it follows that a simple explanation of the notation might be helpful too.
F(a * x) = a * F(x)
This is showing the relationship between two uses of the same function.
Then, further along, we find:
"So, what types of functions are actually linear? Plain-old scaling by a constant, or functions that look like: F(x)=ax In our roof example, a=1/3"
I think in this second situation, F(x)=ax is not a relationship but rather a DEFINITION of the function F(x).
In programming terms:
function F(x: real) : real;
begin
Result := x * (1/3);
> Is A meant to be a function that accepts x?
Yes.
> If so, why is the equivalent expression a * x?
Because that how A(x) is defined.
>Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value?
Yes, it's an unspecified constant (a, b, c... are used to denote constants by convention), so you can really calculate A(x) for supplied values of x yet until the constant 'a' is specified.
> Is this notation specific to multiplication, to this expression, or what?
No. Functions might be defined using any expression. For example
A(x) = b^x
If it works - then you understood it. If it doesn't - then you didn't.
"Understanding" without way of external verification seems no different to dopamine-chasing.
Measuring on output and all that...
How do I falsify the expert's claims about my understanding?
This sounds like it was meant to be pejorative, but it's what (applied) linear algebra, and applied mathematics more generally, is. Anyone can learn about a certain mathematical topic upon realising it's the one relevant to the problem they're facing—and learn it way more quickly, due to motivation and focus, than they would in a general-purpose course on the topic; the art is in recognising what mathematics is relevant, and you can't do that if you've never heard of it before. Having a library of (conceptual, not cookbook) solution hooks on which to hang your problems is how you get to be good at using mathematics.
Anyhow, I disagree with that top-down approach, which seems to be very... European. I much prefer to follow a more logical path where the problem preclude the introduction to the solution.
Does this guy have a patreon?
https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVl...
This is 3Blue1Brown's video series on Differential Equations. Do support him on Patreon if this really helps you.
Join the discord https://discord.gg/vGY6pPk.
Check out a demo https://observablehq.com/@enkimute/animated-orbits
Also at the end of February, there is geometric algebra event in Belgium. https://bivector.net/game2020.html All the big names in the field will be there.
Actually, I think this way of explaining and motivating things (linear map==matrix) will get really, really confusing once you try to understand changes of bases or eigenvalue decomposition. A linear map is something that takes vectors and spits out vectors while preserving the vector structure (i.e. addition and scalar multiplication on the input give you addition and scalar multiplication of the output).
Discussed at the time: https://news.ycombinator.com/item?id=4633662
You can find my guide here:
https://github.com/photonlines/Intuitive-Overview-of-Linear-...
I agree that it is better to understand math, and computer science, intuitively first. Learning the basics instead of learning how to think in them forces memorization and is frankly in a time gone by.
If only I could've been taught this way when I was younger, then I'd actually be any good at any advanced math.
I just wanted to give some context to how I found this page, and why I thought it would be good to post.
I may be putting myself on the spot here: I never took a linear algebra course in undergrad. It was a heavily encouraged option, of course, but I felt I understood the basic rules enough to not really need formal study. I opted to study other areas, partially motivated by a fear I wouldn't do well and would hurt my GPA (my god was I vain, I feel I could do so much more studying full-time with my current world-view).
As time has gone on, and ML and quantum computing have simply blown up since I graduated in 2014, I quickly realized the magnitude of my mistake. I have frantically self-studied for years to try to make up the gaps in my mathematical understanding, and linear algebra has come up time and time again. I can do the processes, but they never clicked, I had no intuition.
I want to help others in my position cut to the chase, and study the highest yield, most intuition giving resources.
I actually developed the mental model shared in this guide on my own, and was positively delighted to find to this while thinking over a comment I was drafting on here. This page lays things out so clearly. The component steps are intuitive and I can commit them to memory/recall what they mean without needing to dig up my notes to self!
===
I find this page gives an excellent foundation, and goes great with these resources:
* A web site which clearly shows how to do matrix multiplication in a way that's easy to recall, it makes the procedure like riding a bike:
http://matrixmultiplication.xyz/
Huge thanks to Jeremy Howard of fast.ai for mentioning this in one of his lectures, this tool is how I finally got matrix multiplication to click
* A paper named "An Introduction to Quantum Computing" (bear with me, it's superbly well written and very approachable):
https://arxiv.org/abs/0708.0261
Page 3 of that paper lays out matrix multiplication (e.g.: applying a "transformation matrix" in the spatial parlance of 3blue1brown's videos) as a traversal of a directed graph. A very useful understanding, and shows how generalizable the tools of linear algebra really are in my opinion.
* The essence of linear algebra, by 3blue1brown (fantastic for a geometric/"transformation of space" view of linear alg):
https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
An interactive Linear Algebra course to complement the article.