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A 2020 Vision of Linear Algebra

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Re: A 2020 Vision of Linear Algebra

#121

Post by madhadron »

bencw wrote: post #82
Phlogistique wrote: post #69

Earlier quoted context omitted.

Is it the same issue as the infamous "monads tutorials" problem, where the understanding takes a lot of time to infuse but looks obvious in retrospect when it finally clicks?

Maybe I'm biased, but I really don't think so. Monads are quite a bit more abstract than the concepts in linear algebra. Linear algebra is both geometric and algorithmic and therefore very intuitive. Most of the difficulty people have learning linear algebra can be attributed to poor teaching methods.

That depends on the part of linear algebra. In an abstract function space when you start calculating dimensions of kernels and the like and get ready to make the jump to infinite dimensions, Banach spaces, and Hilbert spaces, it's about as abstract as monads.

Re: A 2020 Vision of Linear Algebra

#122

Post by madhadron »

DagAgren wrote: post #79
Phlogistique wrote: post #69

Earlier quoted context omitted.

Is it the same issue as the infamous "monads tutorials" problem, where the understanding takes a lot of time to infuse but looks obvious in retrospect when it finally clicks?

I am convinced that monads induce a very specific kind of brain damage that makes a person incapable of ever explaining monads.

I've had good luck with explaining it as a characteristic of a programming language. In a language consisting of sequences of statements with bindings and function calls, we expect that

f(x)

is the same as

a = x; f(a)

and the same as

g = f; g(x);

That's the monad laws. Whatever craziness you want to put in the semantics, those are properties you probably would like to preserve in your language.

Re: A 2020 Vision of Linear Algebra

#123

Post by sh-run »

katzgrau wrote: post #72

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…

I took linear algebra through a community college and had one of those rare, really awesome CC instructors. He had spent most of his career at Cray and later Raytheon and then semi-retired as a community college instructor. He took time to make really great interactive Jupyter notebooks. That combined with 3 brown 1 blue videos really made linear algebra click for me.

My only regret is that I took the class as a six week short course. I think my recall would be better if I had taken the full semester. We covered all the material, but missed out on the longer spaced repetition. Linear Algebra was by far my favorite pure math course, I hope to revisit it soon. Maybe Strang's lectures are the way to do that.

Re: A 2020 Vision of Linear Algebra

#124

Post by Koshkin »

madhadron wrote: post #121
bencw wrote: post #82

Earlier quoted context omitted.

Maybe I'm biased, but I really don't think so. Monads are quite a bit more abstract than the concepts in linear algebra. Linear algebra is both geometric and algorithmic and therefore very intuitive. Most of the difficulty people have learning linear algebra can be attributed to poor teaching methods.

That depends on the part of linear algebra. In an abstract function space when you start calculating dimensions of kernels and the like and get ready to make the jump to infinite dimensions, Banach spaces, and Hilbert spaces, it's about as abstract as monads.

Well, to be fair, functional analysis is not part of linear algebra proper. (If you want to get more abstract, you go to rings and modules and from there to category theory.)

Re: A 2020 Vision of Linear Algebra

#125

Post by cat199 »

rcthompson wrote: post #120
katzgrau wrote: post #72

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…

I had a very similar situation in my linear algebra course: in hindsight, I would literally have been better off teaching myself the material than listening to the professor. To this day it's still the main weak spot in my math/stats knowledge base. I'm really interested to check out these lectures.

haven't watched but based on summary these seem more about pedagogy than the subject - that said there is a full course worth of videos taught by same professor that are pretty good

Re: A 2020 Vision of Linear Algebra

#126

Post by sandyarmstrong »

ansible wrote: post #111
katzgrau wrote: post #72

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…

> I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. I had a similar, though sort of opposite experience. In high school, I breezed through the material, and started teaching myself calculus during the summer to prepare for…

Oh man, Differential Equations. After doing well in Calc 1-3 I thought it would be no big deal. I paid attention in class and barely did the homework because it all seemed so straightforward but it was boring and I was not engaged.

I came in for the first exam, sat there for maybe 15 minutes reading the questions, and realized I had no idea how to solve any of them.

Luckily it was before the drop date! That was a turning point where I decided to only take classes that seemed fun. For me that was discrete math, number theory, abstract algebra, etc.

Re: A 2020 Vision of Linear Algebra

#127

Post by bo1024 »

gowld wrote: post #100
auggierose wrote: post #23

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transforma…

This is silly. LA is incredibly rich, broad, and feel area of study. You can't just grab one part of it and say it's the most important and first thing. And it's silly to say that whatever is most important should be first -- the central ideas depend on prepaeatt.

I’m not sure I agree when that one thing is the word linear (half the name). It just feels wrong not to ground the subject in the idea of a linear transformation.

Re: A 2020 Vision of Linear Algebra

#128

Post by Koshkin »

cbm-vic-20 wrote: post #96
Phlogistique wrote: post #69

Earlier quoted context omitted.

Is it the same issue as the infamous "monads tutorials" problem, where the understanding takes a lot of time to infuse but looks obvious in retrospect when it finally clicks?

"A monad is just a monoid in the category of endofunctors, what's the problem?"

I don't think this definition is correct. (A monad is an endofunctor.)

Re: A 2020 Vision of Linear Algebra

#129

Post by exabyte »

sh-run wrote: post #123
katzgrau wrote: post #72

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…

I took linear algebra through a community college and had one of those rare, really awesome CC instructors. He had spent most of his career at Cray and later Raytheon and then semi-retired as a community college instructor. He took time to make really great interactive Jupyter notebooks. That combined with 3 brown 1 blue videos really made linear algebra click for me. My only regret is that I took the class as a six…

There is a linear algebra series on Udemy called "Complete Linear Algebra: : theory and implementation" by Mike Cohen that I really enjoyed doing because he walks you through Matlab demonstrations (code included for Matlab and python. I adapt to Julia using the PyPlot wrapper for Julia).

I particularly like his videos because he breaks them down into small bites that are easy to work into your day and he's a great teacher.

https://www.udemy.com/share/101XOWAkYTd19WTQ==/

Re: A 2020 Vision of Linear Algebra

#130

Post by madhadron »

Koshkin wrote: post #124
madhadron wrote: post #121

Earlier quoted context omitted.

That depends on the part of linear algebra. In an abstract function space when you start calculating dimensions of kernels and the like and get ready to make the jump to infinite dimensions, Banach spaces, and Hilbert spaces, it's about as abstract as monads.

Well, to be fair, functional analysis is not part of linear algebra proper. (If you want to get more abstract, you go to rings and modules and from there to category theory.)

That's fair. I may have a bias coming from physics because quantum mechanics demands Hilbert Space Now! from the students.
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