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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 o

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=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=1 https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ... [2] https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/ https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011... [3] https://linear.axler.net/ https://linear.axler.net/ and https://www.youtube.com/watch?v=lkx2BJcnyxk&list=PLGAnmvB9m7zOBVCZBUUmSinFV0wEir2Vw&index=1 https://www.youtube.com/watch?v=lkx2BJcnyxk&list=PLGAnmvB9m7... [4] https://www.youtube.com/watch?v=7eggsIan2Y4&list=PLd-yyEHYtIhKJG70DJmAKvHA_chG2kYRH https://www.youtube.com/watch?v=7eggsIan2Y4&list=PLd-yyEHYtI...

sgdpk 1y ago
Second Axler's book! (probably as a second exposure after a first course, to really understand what's going on in Linear Algebra)
- seanhunter 1y ago
  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.
deleted 1y ago
[deleted]
incognito124 1y ago
I also recommend Robert Beezer's "A First Course in Linear Algebra". Great for self-studying. http://linear.ups.edu/ http://linear.ups.edu/
- barrenko 1y ago
  Thank you, that's a new one.
selimthegrim 1y ago
Strang is still very much alive AFAIK
- tptacek 1y ago
  Yep. He retired, is all.
tptacek 1y ago
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.
- creata 1y ago
  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.
photochemsyn 1y ago
The Axler text was discussed here (631 pts 295 comments): https://news.ycombinator.com/item?id=38060159 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-modern-introduction-by-david-poole.665360/ https://www.physicsforums.com/threads/linear-algebra-a-moder...
beklein 1y ago
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.
- nh23423fefe 1y ago
  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 1y ago
    The only worthwhile content of dot products is geometry?
    - nh23423fefe 1y ago
      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^2 this 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.
bee_rider 1y ago
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.
the-mitr 1y ago
Thanks for these resources

sgdpk 1y ago

Second Axler's book! (probably as a second exposure after a first course, to really understand what's going on in Linear Algebra)

seanhunter 1y ago
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.

deleted 1y ago

[deleted]

incognito124 1y ago

I also recommend Robert Beezer's "A First Course in Linear Algebra". Great for self-studying. http://linear.ups.edu/ http://linear.ups.edu/

barrenko 1y ago
Thank you, that's a new one.

selimthegrim 1y ago

Strang is still very much alive AFAIK

tptacek 1y ago
Yep. He retired, is all.

tptacek 1y ago

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.

creata 1y ago
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.

photochemsyn 1y ago

The Axler text was discussed here (631 pts 295 comments): https://news.ycombinator.com/item?id=38060159 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-modern-introduction-by-david-poole.665360/ https://www.physicsforums.com/threads/linear-algebra-a-moder...

beklein 1y ago

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.

nh23423fefe 1y ago
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 1y ago
  The only worthwhile content of dot products is geometry?
  - nh23423fefe 1y ago
    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^2 this 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.

bee_rider 1y ago

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.

the-mitr 1y ago

Thanks for these resources