41–47 of 47 posts
Earlier quoted context omitted.
What if I told you the derivative is a linear operator? (matrix meme style)
Of course, but there are plenty of linear operators, that property alone doesn't define it.
This article should have been titled as "Graphic Notes on Linear Algebra for Everyone", and Prof. Strang kindly suggested this big name. I was lucky that this drew this attention.
There are some other visuals I'm trying around the area. - Eigenvalues https://anagileway.com/2021/10/01/map-of-eigenvalues/
-Matrix classification https://anagileway.com/2020/09/29/matrix-world-in-linear-alg...
When I was an undergraduate, I didn't get this understanding of linear algebra... but after watching all the Prof. Strang's 18.06 classes in MIT OpenCourseWare, now I have much clear view of this area... So I really appreciate his way of teaching.
BTW, I even made a T-shirt and sent him ! https://anagileway.com/2020/06/04/prof-gilbert-strang-linear...
Earlier quoted context omitted.
The question seems to be: should integration and differentiation be taught before matrix operations, or after? IMHO, since linear algebra is largely a tool for solving differential equations, I think calculus should be taught first, as the fundamental knowledge.
> [...] integration and differentiation be taught before matrix operations [...] When worded this way, it sure doesn't sound like something worth doing. Matrix algebra (computational rules for matrix-vector and matrix-matrix products) is just an "implementation detail" of the general idea of a linear transformation. The notion of a linear transformation T(x) = y where x is an input vector and y is an output vector is…
I remember when we were introduced to abstract vector spaces in high school, and we were all pretty confused, even though this was a high school dedicated to mathematics, the foremost in the country.
Even complex vectors had us scratching our heads, which in retrospect seems absurdly trivial. It's just that we were used to thinking in very concrete terms, anything purely abstract is 10 times harder to grasp, so you probably can't teach things like linear transforms without matrices.
Derivatives and integrals on the other hand are very easy to visualise.
Also check out 3Blue1Brown's Essence of Linear Algebra https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...
And here is linear algebra as path counting [1]. This is closely related to path integrals in quantum physics. The rules for combining quantum amplitudes are the rules for combining path counts [2]. This is also how graphical linear algebra works [3]. And since we are all hackers here, if you replace the underlying number system with the min-sum semiring you get things like Dijkstra's algorithm. I really like how all…
What am I missing?
Wow this is really well done. It's like a visual-algebraic approach... I've never seen this before. I like how the author sets up a "grammar of matrix multiplications," and then reuses the same patterns in the rest of the document. For people who might not be familiar, these visual notes are inspired by and complement Prof. Strang's new book https://math.mit.edu/~gs/everyone/ and course https://ocw.mit.edu/resources/…
Yes, I am a big fan of him, and I came up with this idea during several email exchanges with him.
Earlier quoted context omitted.
The question seems to be: should integration and differentiation be taught before matrix operations, or after? IMHO, since linear algebra is largely a tool for solving differential equations, I think calculus should be taught first, as the fundamental knowledge.
> [...] integration and differentiation be taught before matrix operations [...] When worded this way, it sure doesn't sound like something worth doing. Matrix algebra (computational rules for matrix-vector and matrix-matrix products) is just an "implementation detail" of the general idea of a linear transformation. The notion of a linear transformation T(x) = y where x is an input vector and y is an output vector is…
Earlier quoted context omitted.
And here is linear algebra as path counting [1]. This is closely related to path integrals in quantum physics. The rules for combining quantum amplitudes are the rules for combining path counts [2]. This is also how graphical linear algebra works [3]. And since we are all hackers here, if you replace the underlying number system with the min-sum semiring you get things like Dijkstra's algorithm. I really like how all…
I watched the path counting one and I want my time refunded please. It had nothing to do with linear algebra. Nothing to do with quantum physics either, except that they used bracket notation for some inexplicable reason. What am I missing?