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Linear Algebra Done Wrong (2004)
- dang 2y agoRelated: Linear Algebra Done Wrong [pdf] - https://news.ycombinator.com/item?id=999494 https://news.ycombinator.com/item?id=999494 - Dec 2009 (12 comments)
- optbuild 2y agoAfter having my first Linear Algebra course I came across the online course called Linear Dynamical Systems by Prof Stephen Boyd of Convex Optimisation fame. Every lecture was so eye opening. I couldn't believe that linear algebra could be taught in such a context with such a variety of application domains. The lectures are all available online with the assignments : https://ee263.stanford.edu/archive/ https://ee263.stanford.edu/archive/
- nextos 2y agoBoyd has published a related book which is a bit more elementary but still great: https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/
- abhgh 2y agoI hadn't known about this, so thank you! Steven Brunton also has a book that covers dynamical systems (among other things) which is linear algebra adjacent, and the expositions are great [1]. [1] https://www.amazon.com/Data-Driven-Science-Engineering-Learning-Dynamical/dp/1108422098 https://www.amazon.com/Data-Driven-Science-Engineering-Learn...
- behnamoh 2y agoWe seriously need better terminology, notation, and pedagogy when it comes to linear algebra. In 2024, such old-style text books just don't cut it anymore.
- AnimalMuppet 2y agoWhat, specifically, do you think should be done better? What style of pedagogy do you think would be more appropriate? What notation? What terminology? Apart from "argument by year number", what's actually wrong with the book?
- Jtsummers 2y ago1. What's wrong with this specific book other than that it hasn't been updated in 7 years? 2. What are examples of better terminology, notation, or pedagogy that are improvements over this book? (Doesn't need to be total, could retain the same terminology and notation but just have better pedagogy for example)
- dunefox 2y agoI think interactive linear algebra could be superior to a pure textbook. So, a textbook with interactive visualisations that show a concept and allow manipulation. Something like this maybe: https://textbooks.math.gatech.edu/ila/ https://textbooks.math.gatech.edu/ila/
- Jtsummers 2y agoThat does offer slightly better presentation for students with its interactive elements, but otherwise looks pretty standard for an undergrad linear algebra textbook in overall presentation. It has many sections that could use some interactive portion with none at all. However, I'll give you that interactive texts (or texts with supplemental interactive material) are often better for some things (and some parts of linear algebra) than a plain dead tree format.
- dunefox 2y agoYeah, I didn't mean this book specifically, rather the presentation as an example.
- andrewla 2y agoAs far as I know, this notation and terminology is still very standard. In some ways I like the idea of replacing traditional indexing with something more like the Einstein summation notation, and moving away from the arbitrary feeling that you get around how matrix multiplication is structured, and provide an immediate route into tensor theory. But so many things that are very important in linear algebra, like matrix inverses, are awkward to express in this notation.
- paulpauper 2y agoWhy does linear algebra elicit so much confusion despite being an elementary topic. I think part of the problem is that the connection between matrices and matrix operations and their applications is not explained well enough. Linear algebra is typically one's first foray into abstract mathematics, and if taught poorly sets one up for failure down the road.
- rguzman 2y agoone issue that comes to mind is the nature of the determinant. when one considers the determinant defined by the recursive definition, it seems like a highly contrived object that is difficult to work with (as it is from that definition!). avoiding that confusion requires that a lot more scaffolding be built (ala Axler in the "Done Right" book). either way you have some work: either to untangle the meaning of the weird determinant or get to the place where you can understand the determinant as the product of the eigenvalues.
- cooljoseph 2y agoDo you know why introductory textbooks don't define the determinant in terms of the exterior product? This is how some "real" mathematicians I've talked to define it. It also is more intuitive (in my opinion) to define determinants as "signed volumes" than some sum of products multiplied by signs of cycles. The product of eigenvalues definition is also somewhat intuitive to me ("How much does the matrix scale vectors in each direction? Now multiply those numbers together."), but it's harder to motivate the fact that adding rows together doesn't change the determinant, which is kind of important to actually computing the determinant.
- myworkinisgood 2y agoReally? You want to teach freshman people exterior products?
- QuercusMax 2y agoI took LinAlg the same semester I took Computer Graphics, back in 2001. The first half of LinAlg was all about solving equations, transformation matrices and that sort of stuff, and the first half CG was all 2D drawing (Bresenham and such). Second half of CG was all about OpenGL, and I was able to apply all the stuff I'd just learned in LinAlg. The second half of LinAlg was a bunch of stuff about Eigenthis and Eigenthat, with no particular explanation as to what you'd use it for or why you would care. I pass the class with an A and was even recommended by the instructor to be work as a tutor for other students, but 23 years later I couldn't tell you the use of any of that stuff. T-matrices, though - I've used that stuff my whole career, working in medical imaging and related fields. I still couldn't tell you what an Eigenvalue or Eigenvector is useful for, as I never had any reason to apply that stuff. It's not particularly abstract if you can explain why and how this stuff is used - and I know it's heavily used in many fields - just not those that I've worked in.
- generationP 2y agoFYI, there is a newer version at https://sites.google.com/a/brown.edu/sergei-treil-homepage/linear-algebra-done-wrong https://sites.google.com/a/brown.edu/sergei-treil-homepage/l... . Google seems to never have learnt about this site, which is weird as it is on a Google server.
- godelski 2y agoSomething seems to be seriously wrong with google. By "something" I mean "a lot". I know there are Google engineers here, possibly search. Do you guys not also feel it? I mean you dogfood, right? I know it is a hard problem but boy has it become harder to use, especially in the last 6 months.
- akira2501 2y agoLike.. the very fact I typed "javascript" or "css" or "html" means you probably don't need to apply your "did you mean" logic to my searches. If I'm searching for something highly technical, I absolutely DO NOT NEED your help, Google.
- p1esk 2y agoYou’re not the type of users they care about.
- akira2501 2y agoWell, I write software for the web, and at this point I consider Google to be an entirely negative company. So now, to the maximum extent possible, I work to undermine their interests and to not participate with their products or addons or browsers. Great strategy on their part.
- godelski 2y agoYou've over simplified and created a major error. It's a common one though, so allow me to explain. There is no single entity Google cares about. There is no "average user" (nor median). Trying to make the product best for an "average" user makes it bad for everyone. You have made the assumption that this solution space is smooth and relatively uniformly distributed. Consider this: on a uniformly distribution, if you interpolate between two points, those interpolations are representative of the distribution. But if you do so on a gaussian distribution, this is not true. This is because the density of these distributions are not evenly distributed (this is what "uniform" in uniform distribution means). For more complicated distributions you need different interpolations. Many real world distributions are the agglomeration of varying distributions (you probably have clusters and pockets). The distribution of searches (or customers) is neither normal nor uniform. To understand some further complexity, it is important to remember that the burden of success is the requirement for further nuance. Pareto. It may take few resources to get to "80%"/"good enough" but significant to increase that another 10%. Think about a Taylor Series (Fourier Approximation, or pick your preferred example), the first order approximation will only take you so far. It may be good enough to start, but the need for higher order (and this more complex calculations) approximations are increasingly necessary for precision (non-linearly). So simplify where you can, but don't forget what you've done. If you do, you'll be stuck. Or worse, you will compound your error (also very common).
- aquafox 2y agoFor a really intuitive introduction to linear algebra, I highly recommend Gilbert Strang's "Linear Algebra and Learning from Data": https://math.mit.edu/~gs/learningfromdata/ https://math.mit.edu/~gs/learningfromdata/
- godelski 2y ago2011 lectures: https://www.youtube.com/playlist?list=PL221E2BBF13BECF6C https://www.youtube.com/playlist?list=PL221E2BBF13BECF6C Also useful: https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE21EJCxwmWvvek https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE2...
- munchler 2y ago> Why should anyone read this book if it presents the subject in a wrong way? What is particularly done “wrong” in the book? Does he ever answer this question? It's posed at the start of the book, but then immediately ignored.
- Syzygies 2y agoThe title is a jab at and a response to "Linear Algebra Done Right" (https://linear.axler.net https://linear.axler.net) by Sheldon Axler. Sheldon's book is more suitable for a second course in linear algebra. It's an excellent book, that my more advanced students have enjoyed. Sheldon, however, approaches the subject with some rigid dogma: Determinants are evil. His view is they're not practical to compute at scale. It can be good to learn theory with one's determinant hand tied behind one's back, just as it can be good to learn "constructive" mathematics, but this should not be one's only approach. I used to attend an academic sponsor's day at the math institute MSRI (now SLMath). Afternoons reached to keep us busy. One year Sheldon's wife demonstrated some academic teaching software, and my group was tasked to solve a linear algebra problem that Sheldon had provided. Unaware that everyone was watching us on the big screen, I solved it instantly, using determinants. I still don't understand Sheldon's longer solution.
- munchler 2y agoThis is very informative. Thank you for the explanation.
- btilly 2y agoHis view is not just that they are not practical to compute at scale. (They actually are practical to compute. Just not with the formula. Use row reduction instead.) His view is that they are presented as an arbitrary piece of magic that people really don't understand. And, exactly because they don't really understand the determinant, they wind up not properly understanding fundamental concepts in linear algebra. Such as Jordan normal form. If your goal is actually to properly understand linear algebra, he's absolutely right. If your goal is to use linear algebra for a wide variety of practical purposes, though, determinants are often essential.
- beryilma 2y ago... and another beauty of a math book written in LaTeX is that you can fit a 286-page book into just 1.3 MB. Good luck doing that in MS Word.