This is great. I really appreciate visual explanations and the way you build up the motivation. I'm using a few resources to learn linear algebra right now, including "The No Bullshit Guide to Linear Algebra", which has been pretty decent so far. Does anyone have other recommendations? I've found a lot of books to be too dense or academic for what I need. My goal is to develop a practical, working understanding I can apply directly.
Ok, boy, I'm also reviewing LinAlg textbooks as we speak. Coming in with a similar interest for ML / AI.
I've done math on KA academy up to linear algebra, with other resources / textbooks / et al. depending on the topic.
People will recommend 3B1B, Strang (MIT OCW Lin Alg lessons). For me the 3B1B is too "intuitionist" for a first serious pass, and Strang can be wonderful but then go off on a tangent during a lecture that I can't follow, it's a staple resource that I use alongside others.
LADR4e is also nice but I can't follow the proofs there sadly (yet).
There is also 'Linear Algebra done wrong', as well as the Hefferon book, which all end up being proof-y quite quickly. They seem like they'll be good for a second / third pass at a linear algebra.
Side note - for a second or a third pass in LA it seems there is such a thing as 'abstract linear algebra' as a subject and the texbooks there don't seem that much harder to follow than the "basic" linear algebra ones designated for a second pass.
I've gotten off to the most of a start with ROB101 textbook (https://github.com/michiganrobotics/rob101/blob/main/Fall%20...), up until linear dependence / independence, along the MIT Strang lectures. ROB101 is nice as it deals with the coding aspect of it all, and I can follow in my head as I am used to the coding aspect of ML / AI.
I also have a couple obscure eastern european math texbook(s) for practice assignments.
Thank you very much I'll check out these resources. ROB101 looks really great.
I love the 3B1B videos, but I've noticed my attention tends to drift when watching videos. I've learned that I absorb information best through text. For me, videos work well as a supplement, but not as the main way to learn.
The OP's article though simple, still does not really explain things intuitively. The key is to understand the concept of a Vector from multiple perspectives/coordinate systems and map the operations on vectors to movements/calculations in the coordinate space (i.e. 2D/3D/n-space). Only then will Vector Spaces/Matrices/etc. become intelligible and we can begin to look at Physical problems naturally in terms of vectors/vector calculus.
The following are helpful here;
1) About Vectors by Banesh Hoffmann.
2) A History of Vector Analysis: The Evolution of the Idea of a Vectorial System by Michael Crowe.
I really like the second part of the blogpost but starting with Gaussian elimination is a little "mysterious" for lack of a better word. It seems more logical to start with a problem ("how to solve linear equations?" "how to find intersections of lines?"), show its solution graphically, and then present the computational method or algorithm that provides this solution. Doing it backwards is a little like teaching the chain rule in calculus before drawing the geometric pictures of how derivatives are like slopes.
Or something like to the tune of "what does it mean that we can eliminate", which is still unclear to me. But a lovely article, the way you (op) introduce the column perspective and really hepful for a novice such as myself.
+ there are many textbooks on LA. Not a lot of them introduce stuff in the same order or in the same manner. I think that's part of why LA is difficult to teach, and difficult to comprehend, and maybe there is no unique way to do it, so we kinda need all the perspectives we can get.
That "Bam!" thing just brought Josh Starmer to mind. Anyone remember his book with the illustrated ML stuff? I used to watch his YouTube channel too. I really dig these kinds of explainers; they make learning so much more fun.
I really like this, and I think one way to make it even more clear would be to use other variable letters to represent breads and milks, because their x’s and y’s somehow morph into the x’s and y’s that represent carbs and protein in the graph.
As much as I like posts like this I can't feel anything other than hate for the substack platform, it just sucks I'm sorry but I can't understand how people can rely on that bloated web app. I just click around and it's so slow and buggy, recently I canceled a subscription because it kepts signin me out and the signup signin experience just suck
Seems a bit premature? This is "linear algebra" in the sense of middle/high school algebra in linear equations. I suppose many more chapters are coming?
This is nice. Until I took an actual semester of it in college, linear algebra was a total mystery to me. Great job.
For those unfamiliar with vectors, it might be helpful to briefly explain how the two vectors (their magnitude and direction) represent the one bread and one milk and how vectors can be moved around and added to each other.
I don’t like these examples because IRL nobody does things this way.
Try actual problems that require you to use these tools and the inter-relationships between them, where it becomes blindingly obvious why they exist. Calculus is a prime example and it’s comical most students find Calculus hard because their LA is weak. But Calculus has extensive uses, just not for doing basic carb counting.
Honestly all these cute websites give people a false sense that they're actually learning something. The only way to learn this stuff is get one of the million good LA books out there and work through the problems. But that's hard, so people look for shortcuts.
Yeah I think when students actually hit Calculus-level related rates, a small dim light starts to glow. Obviously it only gets brighter the less you have to hold onto and the more you have to mathematically present something that you are trying to reason about that all the tools start to make sense, the relationships are asking you “is this true in my case or do I need to take a step back?” and so forth.
I don’t have an axe to grind against the site I think it’s fine, but if someone wants to learn LA, a college level course followed by an intense grind of word problems and having to work backwards and forwards and finding flaws in answers might be a better way to develop the noggin for it. Just my 2c.
I feel like it's obligatory to also drop a link to the 3blue1brown series on linear algebra, for anyone interested in learning - it is a step up from what's in this post, but these videos are brilliant and still super accessible:
"Aside: Another solution for the above is 23 pennies. Or -4 nickels + 43 pennies."
This is where the math nerds just can't help themselves, and I'm here for it. However, these things drive me crazy at the same time. You cannot have -4 nickels. In pure math with only x and y, sure those values can be negative. But when using real world examples using physical objects, no, you cannot have a negative nickel. Maybe you owe your mate the value of 4 nickels, but that's outside the scope of this lesson. Your negative nickels are not in another dimension (because again, the math works that way). You want to help people understand math with real world concepts but then go and confuse things with pure math concepts. And these negative nickels are still not even getting into imaginary nickels territory like you have square root of -4 nickels.
about 15 years ago I started an aggregator to accumulate/sort/filter the best instruction of various topics, kinda like Reddit for learning. This is such a perfect example of the kind of thing I hoped would filter to the top. Thinking about trying to redo it. Is there a use for this sort of thing in today's world?
An easily searchable platform with curated high quality guides would be a good place to start when trying to do anything. Guides aren't something I'd want to stumble on, like YC posts, but something I would be seeking out. Probably a top feature would be a robust tagging system/search engine rather than the social Reddit elements like karma, hot page, trending subs, etc. Would be cool!
47 comments
[ 4.0 ms ] story [ 56.6 ms ] threadAlso, HT to your user name! Egon Schiele is one of my favorite artists! Loved seeing his works at the Neue in NYC.
Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares
https://web.stanford.edu/~boyd/vmls/
I've done math on KA academy up to linear algebra, with other resources / textbooks / et al. depending on the topic.
People will recommend 3B1B, Strang (MIT OCW Lin Alg lessons). For me the 3B1B is too "intuitionist" for a first serious pass, and Strang can be wonderful but then go off on a tangent during a lecture that I can't follow, it's a staple resource that I use alongside others.
LADR4e is also nice but I can't follow the proofs there sadly (yet). There is also 'Linear Algebra done wrong', as well as the Hefferon book, which all end up being proof-y quite quickly. They seem like they'll be good for a second / third pass at a linear algebra.
Side note - for a second or a third pass in LA it seems there is such a thing as 'abstract linear algebra' as a subject and the texbooks there don't seem that much harder to follow than the "basic" linear algebra ones designated for a second pass.
I've gotten off to the most of a start with ROB101 textbook (https://github.com/michiganrobotics/rob101/blob/main/Fall%20...), up until linear dependence / independence, along the MIT Strang lectures. ROB101 is nice as it deals with the coding aspect of it all, and I can follow in my head as I am used to the coding aspect of ML / AI.
I also have a couple obscure eastern european math texbook(s) for practice assignments.
Most lately I have been reviewing this course / book - https://www.math.ucdavis.edu/~linear/ (which has cool notes at https://www.math.ucdavis.edu/~linear/old), and getting a lot of mileage from https://math.berkeley.edu/~arash/54/notes/.
I love the 3B1B videos, but I've noticed my attention tends to drift when watching videos. I've learned that I absorb information best through text. For me, videos work well as a supplement, but not as the main way to learn.
Thanks again.
https://news.ycombinator.com/item?id=45110857
https://news.ycombinator.com/item?id=45088830
The OP's article though simple, still does not really explain things intuitively. The key is to understand the concept of a Vector from multiple perspectives/coordinate systems and map the operations on vectors to movements/calculations in the coordinate space (i.e. 2D/3D/n-space). Only then will Vector Spaces/Matrices/etc. become intelligible and we can begin to look at Physical problems naturally in terms of vectors/vector calculus.
The following are helpful here;
1) About Vectors by Banesh Hoffmann.
2) A History of Vector Analysis: The Evolution of the Idea of a Vectorial System by Michael Crowe.
+ there are many textbooks on LA. Not a lot of them introduce stuff in the same order or in the same manner. I think that's part of why LA is difficult to teach, and difficult to comprehend, and maybe there is no unique way to do it, so we kinda need all the perspectives we can get.
B: I miss scroll bars. I really, really miss scroll bars.
For those unfamiliar with vectors, it might be helpful to briefly explain how the two vectors (their magnitude and direction) represent the one bread and one milk and how vectors can be moved around and added to each other.
Try actual problems that require you to use these tools and the inter-relationships between them, where it becomes blindingly obvious why they exist. Calculus is a prime example and it’s comical most students find Calculus hard because their LA is weak. But Calculus has extensive uses, just not for doing basic carb counting.
I don’t have an axe to grind against the site I think it’s fine, but if someone wants to learn LA, a college level course followed by an intense grind of word problems and having to work backwards and forwards and finding flaws in answers might be a better way to develop the noggin for it. Just my 2c.
https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...
This is where the math nerds just can't help themselves, and I'm here for it. However, these things drive me crazy at the same time. You cannot have -4 nickels. In pure math with only x and y, sure those values can be negative. But when using real world examples using physical objects, no, you cannot have a negative nickel. Maybe you owe your mate the value of 4 nickels, but that's outside the scope of this lesson. Your negative nickels are not in another dimension (because again, the math works that way). You want to help people understand math with real world concepts but then go and confuse things with pure math concepts. And these negative nickels are still not even getting into imaginary nickels territory like you have square root of -4 nickels.