egonschieleAn Illustrated Introduction to Linear Algebra, Chapter 2: The Dot Product
https://www.ducktyped.org/p/linear-algebra-chapter-2-the-dot
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...
Vectors | Chapter 1, Essence of linear algebra
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.
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.