| ▲ | seanhunter 4 days ago |
| If you actually want to learn linear algebra, don't use this blogpost. It's real weaksauce compared to the wealth of free information and resources available online. Firstly, the real illustrated guide to linear algebra is the youtube series "The Essence of linear algebra" by 3blue1brown[1]. It has fantastic visualisations for building intuition and in general is wildly superior to this, which seems fine but extremely superficial. If you're done with 3b1b and want to take things further, then the go-to is the excellent 18.06SC course by the late and legendary Gilbert Strang. It's amazing, it's free. [2] Still want more? OK now you're talking my language. If you are serious about linear algebra (Up to graduate level, after that you need something else) then you want the book "Linear Algebra Done Right" by Sheldon Axler. It's available for free from the author's website[3] and he has made a bunch of videos to supplement the book. There's also an RTD Math full lecture series[4] that follows the book and he explains each thing in a lot of detail (because Axler goes fast, so it's beneficial to unpack the concepts a bit sometimes). [1] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ... [2] https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011... [3] https://linear.axler.net/ and https://www.youtube.com/watch?v=lkx2BJcnyxk&list=PLGAnmvB9m7... [4] https://www.youtube.com/watch?v=7eggsIan2Y4&list=PLd-yyEHYtI... |
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| ▲ | sgdpk 4 days ago | parent | next [-] |
| Second Axler's book! (probably as a second exposure after a first course, to really understand what's going on in Linear Algebra) |
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| ▲ | seanhunter 4 days ago | parent [-] | | Yeah agree. I first picked it up before I self-studied the 18.06SC course and bounced off pretty hard, then I'm going through it again now and it's an absolute joy, but is really packed. |
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| ▲ | incognito124 4 days ago | parent | prev | next [-] |
| I also recommend Robert Beezer's "A First Course in Linear Algebra". Great for self-studying. http://linear.ups.edu/ |
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| ▲ | tptacek 4 days ago | parent | prev | next [-] |
| Every time this subject comes up, or really any math subject comes up, someone recommends 3Blue1Brown. I love 3Blue1Brown. Just like when The Shawshank Redemption plays on TBS for the 389248th time, I will stop what I'm doing and rewatch any 3Blue1Brown video as soon as it appears in my feed. But I'm not sure I've ever really learned anything from one of those videos. Appreciated something more? Absolutely. And maybe, sure, that's a kind of learning. But I cringe every time eigenvalues come up and people point to the 3B1B evector video. In fact, if your goal is to actually get any kind of facility with the concept, this "weaksauce" blog post probably has a better didactic strategy than 3B1B. It strips the concept down, provides specific, minimized worked examples, and provides a useful framing for the concept (something basically at the core of 3B1B's process). I learned linear algebra from Strang's 18.06. I later did a bunch of Axler helping my daughter through UIUC linear algebra. I like both. Strang is much closer to what the median HN person probably wants. In both cases though: don't do what I did at first, and just watch the videos and read the book. If you're not doing problems, you're probably not learning anything. This blog post comes closer to "actually doing problems" than 3B1B. Ergo: its sauce is stronger, not weaker. I came to the blog post expecting to roll my eyes. No discussion of inner product spaces? Not even a mention of conjugate symmetry? I was pleasantly surprised. It's not easy to come up with a simple, accessible framing for a topic like this, and, maybe, the dot product is particularly tricky to give an intuition for (I'll go out on a limb and say that neither Strang nor Axler do a particularly great job at it --- "it" being, explaining the "why" of the dot product to someone who doesn't really even know what a vector is). The post doesn't purport to teach all of linear algebra. It's just an exercise in trying to explain one small part of it. I'm not asking you to give the author a break, so much as suggesting that you're closing yourself off from appreciating different strategies for explaining complex topics, which is a valuable skill to have. |
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| ▲ | creata 3 days ago | parent | next [-] | | But at the very least, surely a dot product explainer should talk about the two main ways of looking at a dot product! This article leaves out the "angle and norms" (||a||·||b||·cos(θ)) interpretation entirely. It's like if someone gave you the formula for complex multiplication, without also showing you how it's all about rotations. And maybe I'm being a pedant, but the dot product should be between two things that are in the same space. In the Minnesota lottery example, the probabilities should be a row vector instead. It's the exact same calculation, so again, maybe a bit too pedantic. | | |
| ▲ | tptacek 3 days ago | parent [-] | | So does Strang! (I just checked, Linear Algebra & Applications 4E). Also: sir, this is a blog post. It's wild seeing people say "if you really want to understand this topic, pick up Axler". I mean, yeah, also if you were serious you could just enroll in your local community college's Linear Algebra course. My feeling is that a lot of the critique here is really signaling. For whatever reason, linear algebra is super high-status in this community, and people want to communicate that they've done something serious with it. (I'm sure I'm guilty of that too.) | | |
| ▲ | creata 3 days ago | parent | next [-] | | > It's wild seeing people say "if you really want to understand this topic, pick up Axler" I agree. I don't think "Linear Algebra Done Right" is a good fit for most people. It's way too dry, and I don't think his crusade against determinants helps the book. I don't know what a good book suggestion would be, though. Maybe just nab the course notes/slides/exercises off some university's website? > this is a blog post Blog posts can be and often are amazing. > My feeling is that a lot of the critique here is really signaling. Weird accusation, so just to be clear, I haven't done anything serious with anything, ever. | | |
| ▲ | tptacek 3 days ago | parent [-] | | I wasn't referring to anybody in particular. But, like: what is the point of calling out a blog post for not presenting the angle interpretation of the dot product? How would that have fit with the goals of this post? You presented it as a defect, but that logic also suggests Strang's explanation is defective. | | |
| ▲ | creata 3 days ago | parent [-] | | > that logic also suggests Strang's explanation is defective I haven't read Strang's book, so I can't comment on that. But yeah, if it never mentions the formula ||a||·||b||·cos(θ) or at least talks about how the dot product relates to parallel and orthogonal vectors, I would consider that a big hole in an introduction to the dot product. > How would that have fit with the goals of this post? Because the post is titled "an introduction to linear algebra... the dot product", and this is something that I believe should be in anything that considers itself an introduction to the dot product. You seem to disagree, and I'd like to ask: why? I think this a fundamental aspect of the dot product, again, just as fundamental as the relationship between complex multiplication and rotation. I think my view is common. > calling out a blog post I didn't intend to do anything as strong as "call out" the blog post. I just wanted to express surprise at someone so strongly praising ("its sauce is stronger [than 3B1B's video series]") an alright post. | | |
| ▲ | tptacek 3 days ago | parent [-] | | Well, he's one of the most famous and best-respected educators of linear algebra, and this is an unusually basic piece of linear algebra to be taking aim at him for, so one answer is: if you have to ask whether his approach is defective, you should first evaluate how strong your own understanding is. That's argument from authority, but then: Strang is an authority. The typical push-pull on a message board is between Strang and Axler, and, if you want to find out if Axler is going to save your argument, flip to 6.A in LADR. The direct response to your question is: the centrality of the angular interpretation of the inner product is the kind of thing I feel like you'd say if your primary purpose for learning linear algebra is to program video games. Linear algebra isn't "about" geometry, and, in particular, Strang's teaching goal centers vector spaces and the relationship between spaces. You need inner products to apply linear transformations with matrices. You basically don't need the angular interpretation... at all? if your goal is to teach linear algebra as a tool for data manipulation and, as Strang did in his last course, as a vehicle for deep learning. (You need orthogonality, but you didn't say `a⟂b=0`, you said `|a||b|cosθ`). I'm not rating this blog post "higher" than 3B1B; the comparison doesn't even make sense. The blog post and the video series simply have different objectives. | | |
| ▲ | ndriscoll 3 days ago | parent | next [-] | | You don't need inner products for linear transformations. You just need the idea of a basis and linearity. You define your transformation on a basis (which is all a matrix is: the list of where the map sends each basis element), and it is automatically defined everywhere else via linearity. The textbook my undergrad class used (Curtis) doesn't define inner products until after linear transformations and matrices, for example. The angular interpretation and geometry are basically the entire point of inner products (inner products are how you define a large chunk of geometry). Angles and projections are the entire intuition behind talking about orthogonality, which is super important practically to basically every field. | | |
| ▲ | tptacek 3 days ago | parent [-] | | Re the more abstract approach to transformations, fair point, and I feel like that describes Axler well too. I'd soften my argument to just that being Strang's approach to bringing in the subject. I agree orthogonality is important. But Strang doesn't get to `a⟂b=0` by means of `cosθ`. You're halfway into the book before he's even defined the Euclidean norm. He derives orthogonality mostly algebraicaly; the only angle he talks about is π/2. | | |
| ▲ | ndriscoll 3 days ago | parent [-] | | Eh, I don't even think it's more abstract. Like if you're in intro engineering/physics math, and you have R^3 with unit vectors i,j,k, and R^2 with unit vectors I,J, then a function f: R^3 -> R^2 is linear exactly when you can calculate all f(v) = f(ai + bj + ck) = af(i) + bf(j) + cf(k). Then you can define f by "what is f(i)? (Some AI+BJ). What is f(j)? (Some CI+DJ). What is f(k)? (Some EI+FJ)", and then a matrix is just a tabulation of those things. Basically, linear functions let you pick out just n points to define them everywhere, kind of like polynomials. Matrices are that information. Perfectly concrete even at the super intro level. Matrix "multiplication" becomes automatic and trivial; it's just function composition tabulated. Actually it seems way more concrete to me than mystery rules for "multiplying" arrays. |
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| ▲ | creata 3 days ago | parent | prev [-] | | > That's argument from authority, but then: Strang is an authority. Yes, it's an argument from authority. > The typical push-pull on a message board is between Strang and Axler We're not playing Pokemon with linear algebra textbooks here... > the centrality of the angular interpretation It's not central. It's one of the ways to think of it. I think 3B1B actually did a good job emphasizing this in his video series: there are many ways to look at vectors, and all of them are sometimes useful. They can be arrows in space, or sequences of numbers, or black boxes that obey the vector space axioms, or polynomials, and so on. > if your primary purpose for learning linear algebra is to program video games. What an odd guess. Seriously, I would be surprised if most math teachers didn't mention angles and norms, scalar projections, etc. An important part of math is being able to see things from multiple angles (no pun intended), and this is a useful angle to view the dot product from. > Linear algebra isn't "about" geometry Sort of. Linear algebra proper, the study of vector spaces and linear maps between them, is not about geometry, but geometry comes in almost precisely once you equip the vector space with an inner product, such as the dot product. A bare vector space has almost no geometric content, but the inner product gives you lengths, angles, isometries, orthogonality, and all that jazz. > Strang's teaching goal centers vector spaces and the relationship between spaces. That's a good goal. > You need inner products to apply linear transformations with matrices. I think you have a fundamental misunderstanding of linear algebra here. You do not need an inner product to "apply linear transformations with matrices". > You basically don't need the angular interpretation... at all? if your goal is to teach linear algebra as a tool for data manipulation Off the top of my head, cosine similarity? It's not uncommonly used. But if you're teaching linear algebra for data manipulation, you rarely need the dot product, either. Most "dot products" in data manipulation, like the ones in this article, would be better expressed as row-vector-column-vector matrix products. > (You need orthogonality, but you didn't say `a⟂b=0`, you said `|a||b|cosθ`). I said "||a||·||b||·cos(θ) or at least talks about how the dot product relates to parallel and orthogonal vectors". Anyway, it's hard for a person not to interpret orthogonality as a statement about angles, so I'm not sure what distinction you're trying to draw here. > I'm not rating this blog post "higher" than 3B1B I guess I misinterpreted "its sauce is stronger", then. Sorry for any mistakes; this took way too long to type. | | |
| ▲ | tptacek 3 days ago | parent [-] | | We're indeed not playing Pokemon with textbooks here since neither of the two textbooks we're discussing (2 of the 3 most famous for the subject) agree with you. I haven't read Lay; is that the one you read, and does it support this argument? | | |
| ▲ | creata 3 days ago | parent [-] | | I didn't read a linear algebra textbook; I studied pure math at university. |
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| ▲ | seanhunter 3 days ago | parent | prev | next [-] | | I take exception to the idea that I was somehow signalling, and frankly that's a pretty weird thing to say. It happens that I love maths and am studying it part-time alongside my work. I posted some links to things that I have found useful on my journey so far in the hope that they are useful to others. | | |
| ▲ | tptacek 3 days ago | parent [-] | | I don't think "taking exception" is helpful. You opened with "If you actually want to learn linear algebra, don't use this blogpost." Do you feel like that really needed to be said? |
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| ▲ | gnubison 3 days ago | parent | prev [-] | | Strang does include that. I just checked the fourth edition, like you say you did. Scrolling down two pages to get to the first page of the table of contents, I see the heading “cosines and projections onto lines”. I navigate to that section and it explains all the logic, proof, and intuition behind the connection between angles and dot products. Please don’t spread misinformation… | | |
| ▲ | tptacek 3 days ago | parent [-] | | That's page 171, 13 dense pages into the section on orthogonality, long after he's introduced the dot product, in a section where he says, in italics, "the orthogonal case is the most important"; he gives cosines about half a page before going back to a perp b. Nobody said Strang never mentioned cos θ --- that would be weird --- only that his sequencing doesn't treat the angle formula as fundamental, even in the section introducing it. And nobody has ever read Strang and thought projections didn't matter. |
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| ▲ | commandlinefan 3 days ago | parent | prev [-] | | When I was in school, my teachers and professors all insisted that the only way to learn a subject was to _practice_ it. If you want to learn math, work the problems. I was skeptical when I was young, but I have yet to find any other way to learn math (or anything else). |
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| ▲ | selimthegrim 4 days ago | parent | prev | next [-] |
| Strang is still very much alive AFAIK |
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| ▲ | photochemsyn 4 days ago | parent | prev | next [-] |
| The Axler text was discussed here (631 pts 295 comments): https://news.ycombinator.com/item?id=38060159 For most people going into science and engineering as opposed to pure mathematics, Poole's "Linear Algebra: A Modern Introduction" is probably more suitable as it's heavy on applications, such as Markov chains, error-correcting codes, spatiel orientation in robotics, GPS calculations, etc. https://www.physicsforums.com/threads/linear-algebra-a-moder... |
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| ▲ | beklein 3 days ago | parent | prev | next [-] |
| Comparing this blog post to a 500-page book or a multi-hour course and calling it “weaksauce” misses the point. This post is meant as an introduction to the dot product, and it does that really well. The formal definition (6.1) and explanation in Axler’s book wouldn’t make a good starting point for most people, it isn't even a good next step in my opinion. It’s great that you’re passionate about the topic, really, but helping more people discover math means meeting them where they are and appreciating content like this for what it’s trying to do. |
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| ▲ | nh23423fefe 3 days ago | parent [-] | | The post contains no geometry. Which is the only worthwhile content of dot products. Explaining the dot product by its implementation over R^n is pointless. Conflating 1-forms and vectors is pointless. | | |
| ▲ | tptacek 3 days ago | parent [-] | | The only worthwhile content of dot products is geometry? | | |
| ▲ | nh23423fefe 3 days ago | parent [-] | | of course. dot products are a symmetric form on vector spaces. they let you compute the spheres of radius r. given the sphere of radius r, for any pair of vectors v,w in the sphere -r^2 <= dot(v,w)=dot(w,v) <= r^2
as w varies from v to -v the value moves from r^2 through 0 to -r^2this is how we define parallel perpendicular and antiparallel. the dot product is only meaningful in a geometric context. by definition it projects vectors down to scalars. fixing the scalar value finds the spheres, and for a sphere we can vary the vectors to compute cosines. |
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| ▲ | the-mitr 3 days ago | parent | prev | next [-] |
| Thanks for these resources |
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| ▲ | bee_rider 3 days ago | parent | prev | next [-] |
| Find a copy of Yousef Saad’s books (Iterative Methods, and the Eigenvalue Problems). Find a problem you want to solve, and implement one of the solutions he describes. If you don’t understand something, that’s what the first chapter is for. |
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| ▲ | 4 days ago | parent | prev [-] |
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