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An illustrated introduction to linear algebra
71–80 of 91 posts
Re: An illustrated introduction to linear algebra
#72I know this is going to be super controversial, but I genuinely find illustrations of mathematical concepts below a minimum threshold of complexity totally useless and frequently detrimental.
Below a certain level of complexity the human brain is much faster and efficient operating on abstract symbols, like 'x' and 'y'. You can solve equations and figure things out in a fraction of the time it takes you to visualize bananas, goats, coins, bread, milk, etc.
Visualizations have a role in developing intuitions about complex structures, such as what the a matrix does to a vector or what cosine similarity means, and so on.
But in recent years, everyone and the next man has suddenly assumed that visualizing the number 1 or 2 in terms of every day objects somehow helps learning. It doesn't.
Re: An illustrated introduction to linear algebra
#73I 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…
Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.
Do you have any plan to turn it into a full book—maybe called Grokking Linear Algebra ?
Re: An illustrated introduction to linear algebra
#74This is clear and useful but I wish you'd picked different example numbers. Using 1 and 2 for both bread and milk makes it harder to look at the matrix form and immediately see whether a 1 in a matrix is the bread 1 or the milk 1. If you could use 1,2,3,4 instead of 1,2,1,2 it would make things much clearer.
I agree with this critique because, with learning linear algebra, there are a lot of numbers flying around and the order of them is very important. This is why I like to use the prime sequence for my example numbers, because you can also see where they contributed to results of multiplication operations.
ooh, yeah, prime numbers is an even better idea.
Re: An illustrated introduction to linear algebra
#75This is clear and useful but I wish you'd picked different example numbers. Using 1 and 2 for both bread and milk makes it harder to look at the matrix form and immediately see whether a 1 in a matrix is the bread 1 or the milk 1. If you could use 1,2,3,4 instead of 1,2,1,2 it would make things much clearer.
Agreed, I need to make this part less confusing
Re: An illustrated introduction to linear algebra
#76Earlier quoted context omitted.
Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.
Do you have any plan to turn it into a full book—maybe called Grokking Linear Algebra ?
Lol. Maybe! I did enjoy writing Grokking Algorithms, but writing a full book is a real commitment. That one took me 3 years.
Re: An illustrated introduction to linear algebra
#77This 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…
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 lectur…
That's quite the list! How does this one compare? Anything you think is missing?
Re: An illustrated introduction to linear algebra
#78I know this is going to be super controversial, but I genuinely find illustrations of mathematical concepts below a minimum threshold of complexity totally useless and frequently detrimental. Below a certain level of complexity the human brain is much faster and efficient operating on abstract symbols, like 'x' and 'y'. You can solve equations and figure things out in a fraction of the time it takes you to visualize…
I think the milk and bread is just a helpful real world example of how an object might contain two number that need to be solved for simultaneously (carbs and protein). It's more of a why than a how.
Re: An illustrated introduction to linear algebra
#79I know this is going to be super controversial, but I genuinely find illustrations of mathematical concepts below a minimum threshold of complexity totally useless and frequently detrimental. Below a certain level of complexity the human brain is much faster and efficient operating on abstract symbols, like 'x' and 'y'. You can solve equations and figure things out in a fraction of the time it takes you to visualize…
Everyone is different! I personally find examples and visuals a very important part of teaching.
> But in recent years
Just to expand on this a bit: I have been teaching this way since at least 2016, when I published a book on algorithms called Grokking Algorithms. It is an illustrated guide to algorithms. If you didn't like this post, I imagine you won't like the book either :)
Here is an interview I did with Corey Quinn where I talk more about my teaching philosophy: https://www.youtube.com/watch?v=lZFvTTgR-V4
Re: An illustrated introduction to linear algebra
#80This is clear and useful but I wish you'd picked different example numbers. Using 1 and 2 for both bread and milk makes it harder to look at the matrix form and immediately see whether a 1 in a matrix is the bread 1 or the milk 1. If you could use 1,2,3,4 instead of 1,2,1,2 it would make things much clearer.
I agree with this critique because, with learning linear algebra, there are a lot of numbers flying around and the order of them is very important. This is why I like to use the prime sequence for my example numbers, because you can also see where they contributed to results of multiplication operations.
Agreed completely when ever I need random example sequences it is often sequences of primes or some subset like even indexed primes (meaning 2, 5, 11, ...) mixed with odd indexed (primes 3, 7, 13...) when dealing with complex numbers, or every fourth if I want two sequences of complex numbers. The only trouble is they do start going pretty large.