An Intuitive Guide to Linear Algebra (2012)

Viewing Linear Algebra as the study of linear operators instead of matrices makes everything so much simpler.

Of course AB != BA

Composition makes sense

Inverse makes sense

This is the book that helped me get it http://linear.axler.net/

I remember taking Advanced Algebra Honors in 10th grade. It was basically Algebra II with a few (seemingly teacher-selected based on the experience of students who had a different teacher) advanced topics thrown in. One of them was matrices, and I was completely stumped by them. I now encounter them all the time, and wish I'd been able to wrap my head around them when I was younger.

OP Here:

I just wanted to give some context to how I found this page, and why I thought it would be good to post.

I may be putting myself on the spot here: I never took a linear algebra course in undergrad. It was a heavily encouraged option, of course, but I felt I understood the basic rules enough to not really need formal study. I opted to study other areas, partially motivated by a fear I wouldn't do well and would hurt my GPA (my god was I vain, I feel I could do so much more studying full-time with my current world-view).

As time has gone on, and ML and quantum computing have simply blown up since I graduated in 2014, I quickly realized the magnitude of my mistake. I have frantically self-studied for years to try to make up the gaps in my mathematical understanding, and linear algebra has come up time and time again. I can do the processes, but they never clicked, I had no intuition.

I want to help others in my position cut to the chase, and study the highest yield, most intuition giving resources.

I actually developed the mental model shared in this guide on my own, and was positively delighted to find to this while thinking over a comment I was drafting on here. This page lays things out so clearly. The component steps are intuitive and I can commit them to memory/recall what they mean without needing to dig up my notes to self!

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I find this page gives an excellent foundation, and goes great with these resources:

* A web site which clearly shows how to do matrix multiplication in a way that's easy to recall, it makes the procedure like riding a bike:

http://matrixmultiplication.xyz/

Huge thanks to Jeremy Howard of fast.ai for mentioning this in one of his lectures, this tool is how I finally got matrix multiplication to click

* A paper named "An Introduction to Quantum Computing" (bear with me, it's superbly well written and very approachable):

https://arxiv.org/abs/0708.0261

Page 3 of that paper lays out matrix multiplication (e.g.: applying a "transformation matrix" in the spatial parlance of 3blue1brown's videos) as a traversal of a directed graph. A very useful understanding, and shows how generalizable the tools of linear algebra really are in my opinion.

* The essence of linear algebra, by 3blue1brown (fantastic for a geometric/"transformation of space" view of linear alg):

https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

20+ years ago I combined learning JavaScript and Linear Algebra https://mkaz.blog/math/javascript-linear-algebra-calculator/

Already this is so helpful, as someone with only a partial high school math background. The visual "pouring" analogy alone made it worth the read.