A 2020 Vision of Linear Algebra • Hacker News Board

Re: A 2020 Vision of Linear Algebra

#91

Post by nafizh » Tue, May 12, 2020, 1:08 PM UTC

irl_zebra wrote: post #90

I've been wanting to learn linear algebra. I had some exposure in college along with my calc classes, but never really understood it fundamentally. Like it was mentioned, I mostly did matrix transforms but didn't realize fundamentally grasp. I started doing LA on Khan academy, and checked out Linear Algebra Done Right. LADR was a little too much into the deep end for me. KA seemed to be good. One nice thing about KA…

If you want to learn LA by coding, coding the matrix by philip klein is a really good book. He even has his Coursera lecture videos (not available on coursera anymore last I checked) up on his website.

Re: A 2020 Vision of Linear Algebra

#92

Post by atrettel » Tue, May 12, 2020, 1:12 PM UTC

jacobolus wrote: post #45

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…

Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years ofte…

> Pure mathematics departments for at least the past maybe 80 years often look down on numerical analysis, statistics, engineering, and natural science, and adopt a position that education of students should be optimized in the direction of helping them prove the maximally general results using the most abstract and technical machinery, with an unfortunate emphasis on symbol twiddling vs. examining concrete examples.

I had this view when I took linear algebra as an undergraduate, but I have gradually changed on the subject over time. I took a standard "linear algebra for scientists and engineers" course but I found it too abstract at the time. The instructor rarely concentrated on examples and applications despite the more applied focus in the course title. Later I came to appreciate the abstraction, since it helped me understand more advanced mathematical topics unrelated to the "number-crunching" I originally associated the topic with. I now think the instructor had a more "unified" approach, but I didn't realize it at the time.

Re: A 2020 Vision of Linear Algebra

#93

Post by DagAgren » Tue, May 12, 2020, 1:16 PM UTC

dleslie wrote: post #86

DagAgren wrote: post #79

Earlier quoted context omitted.

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

Start with a container. M a Then add a way to put things in the container. a -> M a Then add a way to use the thing in the container. M a -> (a -> M b) -> M b

Yes, well, good example.

Re: A 2020 Vision of Linear Algebra

#94

Post by nothis » Tue, May 12, 2020, 1:19 PM UTC

pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

I guess a mathematician might look down upon sticking with 2d and 3d stuff because it leaves out all the interesting things that happen at 92382 or negative infinity. But yea, matrices are basically just a convenient way to write rows and rows of "ax + by + cz...". In linear algebra, you just do it so often, people made up their own syntax. And nothing can visualize it like transforming graphics, IMO.

You don't even have to go 3D, just starting with the points of a rectangle in 2D and asking, "how do you put the edge points of this rectangle 10px to the left, rotate them 45° and stretch them 200% vertically?" and you've applied a matrix. Even if you're not using the fancy brackets, you're using a matrix, and understanding it.

Re: A 2020 Vision of Linear Algebra

#95

Post by AlanYx » Tue, May 12, 2020, 1:23 PM UTC

enitihas wrote: post #35

Another good Linear Algebra book is "Linear Algebra Done Right", which Springer is giving for free right now. Link: https://link.springer.com/book/10.1007/978-3-319-11080-6

Thanks for this -- do you know if there's a consolidated list anywhere of other books Springer is making available for free right now?

Re: A 2020 Vision of Linear Algebra

#96

Post by cbm-vic-20 » Tue, May 12, 2020, 1:26 PM UTC

Phlogistique wrote: post #69

pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

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?"

Re: A 2020 Vision of Linear Algebra

#97

Post by gowld » Tue, May 12, 2020, 1:27 PM UTC

pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

> Wait, is linear algebra really just about heaps of multiplication and addition?

That's just one of dozens of things LA is "about"

> why the hell couldn't my textbooks explain it in these terms on page 1

Because you wouldn't have understood terms like

> orthonormal

and because it would have been unhelpful to everyone else who want in LA for the exact same reason you were.

Being obviously in retrospect doesn't mean it was obvious in forespect. You had to learn the material first.

Re: A 2020 Vision of Linear Algebra

#98

Post by jerf » Tue, May 12, 2020, 1:27 PM UTC

dleslie wrote: post #86

DagAgren wrote: post #79

Earlier quoted context omitted.

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

Start with a container. M a Then add a way to put things in the container. a -> M a Then add a way to use the thing in the container. M a -> (a -> M b) -> M b

Well, you see, that's one of the problems... monad implementations don't have to be "containers", or at least not the way most people mean. This was one of the critical errors in many of the aforementioned "tutorials". IO, the quintessential monad, is not a container, for instance.

(A nearly-exact parallel can be seen in the Iterator interface. You can describe it as "a thing that walks through a container presenting the items in order"... and yeah, that's the majority use case and where the idea came from... but it's also wrong. What it really is is just "a thing that presents items in some order". It doesn't have to be from "a container". You can have an iterator that produces integers in order, or strings in lexigraphic order, or yields bytes from a socket as they come in, or other things that have no "container" anywhere to be found. If you have "from a container" in your mental model then those things are confusing; if you understand it simply as "presenting items in order" then having an iterator that just yields integers makes perfect sense. A lot of the Monad confusion comes from adding extra clauses to what it is. Though by no means all of it.)

Re: A 2020 Vision of Linear Algebra

#99

Post by gowld » Tue, May 12, 2020, 1:30 PM UTC

scared2 wrote: post #62

knzhou wrote: post #2

Gilbert Strang's linear algebra course blew my mind back in high school, and I still use insights from it every day. Strang has a particular lecturing style where he approaches every topic several times, often beginning many lectures before the main treatment. At first I thought it was a bit confusing, but later I realized it helped build fluency, just like a language class. I'm really thankful to MIT OCW for putting…

In High school?

Yes. Linear Algebra is an extension of what is commonly called Algebra 2 or Precalculus in high school.

LA and Calculus can be studied independently in any order and then fruitfully combined later.

Re: A 2020 Vision of Linear Algebra

#100

Post by gowld » Tue, May 12, 2020, 1:33 PM UTC

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.