An Illustrated Introduction to Linear Algebra, Chapter 2: The Dot Product

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...

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.

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.

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...

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.

> 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.

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.