An Illustrated Introduction to Linear Algebra, Chapter 2: The Dot Product
43 comments
·November 3, 2025egonschiele
tptacek
I try to only teach as much as is necessary to get the student to the next point, which is matrix multiplication
Preemptively noting: this is also Strang's strategy.
WrongOnInternet
When I see the word "illustrated," I expect to see graphs or something that would help me visualize how linear algebra works. The only thing "illustrated" about this post is that he hand drew some table which could have been easily with some basic HTML+CSS.
tptacek
What graphical illustration do you think this is missing? How would that make things better? Have you ever seen http://matrixmultiplication.xyz/? Great graphical illustration. Also: a really unhelpful way to understand matrix multiplication.
This is part 2 of a series, all under the same name; the first part is extensively illustrated (and I'm not sure the part 1 illustrations are all that helpful).
griffzhowl
Illustrating the dot product using the projection of one vector on another conveys the geometric idea. Then it's transparent why "orthogonal" means dot product = 0.
The author seems to be taking a different tack though, and maybe doesn't want to be too tied to this particular geometric picture
drdec
I don't understand the down votes, I had the same reaction. Other posters have suggested some better resources, check those out
vixen99
That's your preference. However "To illustrate is to make something more clear or visible. Children's books are illustrated with pictures. An example can illustrate an abstract idea. "illustrate" comes from the Latin illustrare 'to light up or enlighten.'"
0_____0
It's extremely obvious that the sense of "illustrated" meant here is "containing illustrations."
sarchertech
In the context of books or internet books illustrated almost exclusively means “with pictures”.
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seanhunter
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...
sgdpk
Second Axler's book! (probably as a second exposure after a first course, to really understand what's going on in Linear Algebra)
seanhunter
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.
tptacek
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
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
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.)
incognito124
I also recommend Robert Beezer's "A First Course in Linear Algebra". Great for self-studying.
barrenko
Thank you, that's a new one.
beklein
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
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
The only worthwhile content of dot products is geometry?
photochemsyn
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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griffzhowl
A great resource that isn't mentioned often is the linear algebra chapters in Birkhoff and Mac Lane's Survey of Modern Algebra. Chapters 7,8,9, and 10 (in the 4th and 5th editions anyway) are a self-contained book-within-a-book of about 200 pages on both the computational and theoretical aspects of vector spaces, matrices, linear transformations, and determinants.
Many times I've been puzzled by a concept just to go there and find it made simple and obvious. It's a real golden nuggett... Plus if you then want to go further into groups, rings, fields, and Galois theory, that's also there.
cultofmetatron
mathacademy has a course on linear algebra. currently working my way back up from nothign to get to it. easily the best resource for learning math on the internet.
tptacek
I do love Math Academy (I signed up 9 months ago in the hopes of replacing my NYT Crossword habit with something more productive, and 9 months later I'm gearing up for multivariable calc, which is neat given that except for linear algebra, which I self-studied out of necessity for cryptography work, all my math education stopped in sophomore year of high school).
It has a very different purpose than a post like this though! Also: there's probably more effort at exposition in this blog post than in all of Math Academy's coverage (that's not a dunk on Math Academy).
cultofmetatron
> here's probably more effort at exposition in this blog post than in all of Math Academy's coverage (that's not a dunk on Math Academy).
haha definitely agree. a lot of these blog posts are great if you want to read about math. mathacademy is pretty much all the exposition chopped out and you spend 90% of your time doing math. I can see how some wouldn't like it but I think the problem solving aspect makes it was more useful for bruteforcing your way towards building intuition
tptacek
I like comparing it to Lingua Latina Familia Romana, a book that teaches Latin basically without any English; it opens in Latin and just keeps going that way and somehow you're able to follow along. Both are kind of trippy experiences.
bsoles
> Summary: A dot product is a weighted sum of two vectors.
Nope. This is incorrect. The dot product is a weighted sum of a vector's elements, where the weights are the elements of the other vector. Weighted sum of two vectors would require a third entity to provide the weights.
tptacek
It's an interesting callout; if you go Google "weighted sum of two vectors", it's not too hard to find more authoritative sources (nothing as authoritative as Axler or Strang, of course) describing either a dot product or a linear combination in those terms.
photochemsyn
Some hand-written (not AI-generated) prompts to consider:
"An expert in university-level linear algebra, including solving systems of equations, matrices, determinants, eigenvalues and eigenvectors, symmetry calculations, etc. - is asked the following question by a student: "This is all great, professor, and linearity is also at the heart of calculus, eg the derivative as a linear transformation, but I would now like you to explain what distinguishes linear from non-linear algebra."
"What kind of trouble can the student of physics and engineering and computation get into if they start assuming that their linear models are exact representations of reality?"
"A student new to the machine learning field states confidently, 'machine learning is based on linear models' - but is that statement correct in general? Where do these models fail?"
The point is that even though it takes a lot of time and effort to grasp the inner workings of linear models and the tools and techniques of linear algebra used to build such models, understanding their failure modes and limits is even more important. Many historical engineering disasters (and economic collapses, ahem) were due to over-extrapolation of and excessive faith in linear models.
Hey everyone, I'm the author. I'm seeing a lot of the same comments here, so I want to address them.
I teach math by leading with examples. I try to show the intuition behind an idea, and why it is interesting. For this series, my reader is someone who knows algebra, and likes learning new things, especially when a teacher shows what is interesting about a topic.
## You didn't cover x about the dot product.
I try to only teach as much as is necessary to get the student to the next point, which is matrix multiplication. I usually end up cutting a lot of material out of my chapters to keep them simple. In this case, I cut out a whole section on the properties of a dot product, as well as a discussion about inner and outer products, because those weren't necessary to get to matrix multiplication. I think this context was lost while posting to HN.
## 3B1B already has a series on this.
I love 3B1B, but his style of teaching and mine are quite different. Even though we both teach visually, his videos are densely packed with information and his expectation is that you will watch the video a few times till you understand the topic. He also leads with math more than I do. My posts are written more like stories. My goal is they should be easy to get into, and by the time you have finished reading, you should understand more about the topic. I don't expect readers to read through multiple times. I personally learned linear algebra through Strang's videos and textbook, and those videos are awesome, but can be confusing. If you found the Strang or 3b1b videos confusing, hopefully my posts will make it easier for you to follow them. I think comment is spot on: https://news.ycombinator.com/item?id=45800657
If these ideas resonate with you, I think you'll like this post, and if not, there are plenty of guides that go the more traditional route. You can also read the first post in the series and see if you like it: https://www.ducktyped.org/p/an-illustrated-introduction-to-l...
For another example of my writing, see my series on AWS: https://www.ducktyped.org/p/a-mini-book-on-aws-networking-in...