Mathematicians sometime talk about algebra and geometry being dual to each other. One way to formalise this is by talking about opposite categories. If the objects of a category act like algebras, then in the opposite category they act like spaces.
But the category of finite dimensional vector spaces is its own opposite! This suggests that linear algebra is in some sense the place where algebra and geometry meet. Perhaps that explains why it's so tractable and efficacious.
I was excited to present a new workshop for teachers tonight titled "The Geometry of Statistics"! I love that I am continually learning as a teacher, and I'm fortunate to have opportunities to share what I learn with other teachers.
In 10 minutes I learned more about tensor products than I did in grad school. In particular, I see more of the connection between the way math uses tensor products and how physicist types think of them, as higher-dimensional matrices.
This kind of thing really riles me up. It's the kind of thing that makes me want to go back to teaching, because it hits the exact personality/psychological trait that makes me like teaching: I learn about some cool math thing and I want to charge into a classroom and say "GUYS! I just figured this out! It's so cool and interesting! OMG tensor products are the best, let me show you why!"
It bothers me that I was taught about tensor products in such a bad way. I want students to get something better than I did.
(OTOH, maybe I'm just not very smart, or was a bad student. That's a real possibility. But how great would it be for even dim bulbs and lazy students to get something out of learning about a topic?)
Hello fedinerds, got any recommendations for someone with dyscalculia (I can do theory and concepts, not numbers without analogies first) for understanding #entropy (and entropy in nature), #linearalgebra, #statistics, or any other topics of note? Any good YouTube lectures, podcasts, reads, etc.?
Looking over some highlights from my teaching year. This was a great moment! And it became a part of several conversations (and little research projects) afterwards.
"Introduced projection matrices & asked students to find something interesting about the matrix A. One student had a brilliant observation: She noticed you can write A like this, which shows how the projection is the average of the vector and its reflection!" (From Twitter)
Yesterday, Gil Strang gave his final lecture at MIT. It was monumental enough to be live streamed with guests coming in to talk about his influence. The fact that it was a normal class lecture is so appropriate. It is the end of an era in #teaching.
I only visited MIT once, for an ASA conference, but Strang has affected my life deeply. His teaching of linear algebra is the best I have ever seen, full stop.
It’s the Golub-Reinsch algorithm, as improved for EISPACK. (I have a printed copy of that volume with the ALGOL in it, BTW, though you can get the preprint online. But the FORTRAN code has numerical-stability improvements.)