I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head.<p>Now that the ideas of things like vector spaces, norms, orthogonality, rank, basis, etc are nearly second nature, the concepts are useful as I study other branches of math which would feel impenetrable otherwise.<p>YMMV, and if you can learn from condensed materials go for it, but I might be too dumb for it work lol. I think the real benefit accrues to the author who had to work out how to teach these concepts to others.
While this explanation is certainly much clearer than what I remember of high school maths, I still have a pretty tough time following the formula examples.<p>When I see A(x) = ax, I'm not entirely sure how to read it.<p>Is A meant to be a function that accepts x? If so, why is the equivalent expression a * x? Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value? Is this notation specific to multiplication, to this expression, or what?<p>Positing that something is 'intuitive' when it depends so much on additional contextual knowledge seems ever so slightly disingenuous as best, and slightly harmful at worst; it can make the reader feel as though they must be dumb for not understanding this 'intuitive' material.<p>I do acknowledge that this is linear algebra, and if one doesn't have a really solid grasp of notation of regular algebra it is likely to go over their heads, but the practical explanations (such as the slope rise/run example) are quite clear and relatively simple to follow; it follows that a simple explanation of the notation might be helpful too.
This is ok but nothing is as intuitive as 3B1B's series on YouTube that has been posted hundreds of times on HN [0].<p>Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.<p>[0] <a href="https://www.youtube.com/watch?v=fNk_zzaMoSs" rel="nofollow">https://www.youtube.com/watch?v=fNk_zzaMoSs</a>
I... I still really struggle with this. I'm a smart person, I've got a bachelors of engineering, I've been a professional software developer for around 14 years now, and I've built a house. But there is something about degree-level maths and beyond that I find deeply unintuitive in a way that software development isn't.<p>Through comments here I found 3blue1brown's (clearly much loved) videos. By the third video I was shouting, "why for the love of god would we be doing this"? Based on this reaction I suspect that the content neither has intrinsic appeal to me, nor does it have obvious use in my work, projects, or life.<p>Pre-degree maths though, I love. My A-level maths really changed how I saw the world, and I make use of it reasonably often (well, often enough to not forget it).<p>I think I'm writing this here because most other commenters seem to really grasp this subject, or feel that they grasp it better having seen these videos. I'm honestly happy for you. However, if anyone is reading this who doesn't feel like that, then know you're not alone :-)
If this interests you, you should check out the bivector community <a href="https://bivector.net/" rel="nofollow">https://bivector.net/</a>.<p>Join the discord <a href="https://discord.gg/vGY6pPk" rel="nofollow">https://discord.gg/vGY6pPk</a>.<p>Check out a demo <a href="https://observablehq.com/@enkimute/animated-orbits" rel="nofollow">https://observablehq.com/@enkimute/animated-orbits</a><p>Also at the end of February, there is geometric algebra event in Belgium. <a href="https://bivector.net/game2020.html" rel="nofollow">https://bivector.net/game2020.html</a> All the big names in the field will be there.
This seems to be "what are matrices and how do you work with them" and not linear algebra.
I mean that can be useful sometimes but seems more like something you would teach in a numerics course instead.<p>Actually, I think this way of explaining and motivating things (linear map==matrix) will get really, really confusing once you try to understand changes of bases or eigenvalue decomposition. A linear map is something that takes vectors and spits out vectors while preserving the vector structure (i.e. addition and scalar multiplication on the input give you addition and scalar multiplication of the output).
A thread from 2015: <a href="https://news.ycombinator.com/item?id=8920638" rel="nofollow">https://news.ycombinator.com/item?id=8920638</a><p>Discussed at the time: <a href="https://news.ycombinator.com/item?id=4633662" rel="nofollow">https://news.ycombinator.com/item?id=4633662</a>
Viewing Linear Algebra as the study of linear operators instead of matrices makes everything so much simpler.<p>Of course AB != BA<p>Composition makes sense<p>Inverse makes sense<p>This is the book that helped me get it <a href="http://linear.axler.net/" rel="nofollow">http://linear.axler.net/</a>
Goes to show how different people have different tastes. I find this type of exposition very confusing and very unenlightening. Give me a Landau-style "minimalistic"/"focused" explanation any day. Not to mention, it tries so hard to simplify things to a simple analogy (the spreadsheet thing) that it ends up being plain misleading. In other words: "Make things as simple as then can be, but no simpler."
I love explainers like this, but it frankly makes me a little angry that the vast majority of the math teachers I had in highschool and college taught in the awful way described in the setup to the piece.<p>Why is that? Has anyone studied it, or is there even a solid anecdotal explanation? The best one I can imagine is many of these professors simply don't care much for teaching and are more focused on their research, which is still infuriating but at least an explanation.<p>I ended up with an undergrad in applied math, though I'm a software engineer now. I like math, but I feel like I never got to be all that great at it. I suspect I would've enjoyed it more and achieved more with explanations like these.
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.
This is a nice resource - I wrote one myself as well which is mostly based on the series by 3Blue1Brown, as well as other resources which I found useful and which used a visual approach to introducing linear algebra.<p>You can find my guide here:<p><a href="https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals/blob/master/PDF/An%20Intuitive%20Overview%20of%20Linear%20Algebra%20Fundamentals.pdf" rel="nofollow">https://github.com/photonlines/Intuitive-Overview-of-Linear-...</a>
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.
Thank you for writing this up, or reposting.<p>I agree that it is better to understand math, and computer science, intuitively first. Learning the basics instead of learning how to think in them forces memorization and is frankly in a time gone by.<p>If only I could've been taught this way when I was younger, then I'd actually be any good at any advanced math.
The only way to test if you've 'understood' something is to apply it to real-world problems.<p>If it works - then you understood it.
If it doesn't - then you didn't.<p>"Understanding" without way of external verification seems no different to dopamine-chasing.<p>Measuring on output and all that...
OP Here:<p>I just wanted to give some context to how I found this page, and why I thought it would be good to post.<p>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).<p>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.<p>I want to help others in my position cut to the chase, and study the highest yield, most intuition giving resources.<p>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!<p>===<p>I find this page gives an excellent foundation, and goes great with these resources:<p>* 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:<p><a href="http://matrixmultiplication.xyz/" rel="nofollow">http://matrixmultiplication.xyz/</a><p><i>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</i><p>* A paper named "An Introduction to Quantum Computing" (bear with me, it's superbly well written and very approachable):<p><a href="https://arxiv.org/abs/0708.0261" rel="nofollow">https://arxiv.org/abs/0708.0261</a><p>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.<p>* The essence of linear algebra, by 3blue1brown (fantastic for a geometric/"transformation of space" view of linear alg):<p><a href="https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab" rel="nofollow">https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...</a>
<a href="http://immersivemath.com/ila/index.html" rel="nofollow">http://immersivemath.com/ila/index.html</a><p>An interactive Linear Algebra course to complement the article.
Speaking for Linear Algebra, I learnt more reading for a few hours the appendix of "The Design of Rijndael: AES - The Advanced Encryption Standard" than I did in 6 months of theoretical university teaching full of useless technical terms and solutions in search of problems...