A 2020 Vision of Linear Algebra • Hacker News Board

Re: A 2020 Vision of Linear Algebra

#171

Post by eximius » Tue, May 12, 2020, 6:26 PM UTC

hprotagonist wrote: post #168

eximius wrote: post #161

Earlier quoted context omitted.

That's a rant I'd like to hear.

"look guys, at end of class, exam is always long list of silly second order system of differential equations. Well, every time, we can make so-called "guess" that solution looks like e*rt. Why? We know that because professors will only give well-behaved systems on final exams because it's hard to grade the other kind. So we know characteristic polynomials look like so (because of course they do, you can just memorize…

Minus the matrix bits, that's basically how I slogged through DiffEq. Showing up to class was pointless because the prof would make up an equation to exploratively solve and inevitably it would be poorly behaved and the lecture would end with "... And and and for this kind of problem we have to use numerical methods".

Re: A 2020 Vision of Linear Algebra

#172

Post by dleslie » Tue, May 12, 2020, 6:33 PM UTC

Koshkin wrote: post #143

dleslie wrote: post #109

Earlier quoted context omitted.

Containers make sense. Abstract computer science doesn't. Part of why Haskell appears like such an implacable curmudgeon is the predilection of its community to believe that users must grasp type and logic theory to use it. They don't. Just like they don't need to have a mental model of their computer to write software for it.

In my experience, not having a mental model of the computer you are going to run your software on will bite you on the ass sooner or later.

Countless mobile apps and web sites have been made with nigh-zero understanding of the VMs, rendering engines, or underlaying machine architecture.

It's not the 80s anymore.

Re: A 2020 Vision of Linear Algebra

#173

Post by ivansavz » Tue, May 12, 2020, 6:35 PM UTC

For anyone who is already familiar with the Prof. Strang's lectures from previous years, the main new thing in this five-lecture mini-series is he tries to condense the material even further—maximum intuition and power-ideas, instead of the full-length in-class lecture format with derivations. This makes the material difficult to understand for beginners, but makes a great second source in addition to or after a regular LA class.

One of the interesting new ways of thinking in these lectures is the A = CR decomposition for any matrix A, where C is a matrix that contains a basis for the column space of A, while R contains the non-zero rows in RREF(A) — in other words a basis for the row space, see https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-li...

Example you can play with: https://live.sympy.org/?evaluate=C%20%3D%20Matrix(%5B%5B1%2C...

Thinking of A as CR might be a little intense as first-contact with linear algebra, but I think it contains the "essence" of what is going on, and could potentially set the stage for when these concepts are explained (normally much later in a linear algebra course). Also, I think the "A=CR picture" is a nice justification for where RREF(A) comes about... otherwise students always complain that the first few chapters on Gauss-Jordan elimination is "mind-numbing arithmetic" (which is kind of true...) but maybe if we present the algorithm as "finding the CR-decomposition which will help you understand dozens of other concepts in the remainder of the course" it would motivate more people to learn about RREFs and the G-J algo.

Re: A 2020 Vision of Linear Algebra

#174

Post by ivansavz » Tue, May 12, 2020, 6:53 PM UTC

ivansavz wrote: post #173

For anyone who is already familiar with the Prof. Strang's lectures from previous years, the main new thing in this five-lecture mini-series is he tries to condense the material even further—maximum intuition and power-ideas, instead of the full-length in-class lecture format with derivations. This makes the material difficult to understand for beginners, but makes a great second source in addition to or after a regu…

Since y'all are code-literate, here the SymPy function for finding the CR-decomposition of any matrix A:

  def crd(A):
      """
      Computes the CR decomposition of the matrix A.
      """
      rrefA, licols = A.rref()  # compute RREF(A)
      C = A[:, licols]          # linearly indep. cols of A
      r = len(licols)           # = rank(A)
      R = rrefA[0:r, :]         # non-zero rows in RREF(A)
      return C, R

Test to check it works: https://live.sympy.org/?evaluate=A%20%3D%20Matrix(%5B%0A%20%...

Re: A 2020 Vision of Linear Algebra

#175

Post by dxbydt » Tue, May 12, 2020, 6:55 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…

> Maybe my understanding is all wrong, feel free to correct me, as my understanding on this subject is entirely self-taught.

I wouldn't say it is all wrong. Just that the stuff you are talking about is a very tiny fraction of LA. I took a graduate class in LA, based on Strang's book. I have the book right here in front of me. So the stuff you allude to, i.e. rotation matrix, reflection matrix & projection matrix, is on p130 of Chapter 2. We got to that in the 1st month of the semester, & it got about 1 hour of classtime total. That's it. An LA class is like 4 months, or 50 hours. If the point of LA to derive those matrices so one can do 3D computer graphics with scaling, rotation & projection ? No, that stuff is too basic. We got 1 homework problem on that, that's it.

The stuff that most of the class struggled with ( & still struggle with, because Strang goes over it rather quickly in his book), is function spaces ( chapter 3, p182), Gram Schmidt for functions ( p184), FFTs, (p195), fibonacci & lucas numbers (p255), the whole stability of differential equations chapter ( he gives these hard and fast rules like a Differential Equation is stable if trace is negative & determinant is positive, but its not too clear why. ), quadratic forms & minimum principles - that whole 6th chapter glosses over too much material imo.

Overall, Strang's book is a solid A+ on how to get stuff done, but maybe a B- on why stuff works the way it works. Like, why should I find Rayleigh quotient if I want to minimize one quadratic divided by another ? Strang just says, do it & you'll get the minimum. How to find a quadratic over [-1,1] that is the least distance away from a cubic in that same space ? Again, Strang gives a method but the why part of it is quite mysterious.

Re: A 2020 Vision of Linear Algebra

#176

Post by abecode » Tue, May 12, 2020, 6:56 PM UTC

Two things really made linear algebra click for me: representing camera projections in a computer vision class and spectral graph theory, which basically connects graphs with linear algebra. In both of these, it seems like linear algebra was taken from the electrical engineering domain into computer science, which better fit my perspective.

Re: A 2020 Vision of Linear Algebra

#177

Post by crispyambulance » Tue, May 12, 2020, 7:07 PM UTC

dunefox wrote: post #165

crispyambulance wrote: post #137

Earlier quoted context omitted.

I think that whether or not students do well, there's a common theme in university math curricula for non-math majors. Basically, math gets taught as a kind of "toolbox" of techniques. Unless there's a strong follow-up in subject matter courses (for example in engineering coursework), those math skills effectively evaporate. Some places use a rigorous "proof-theoretic" approach in math curricula. It's much harder and…

> More CS, engineering and science students, IMHO, should dabble in math department courses beyond the the usual "required" sequence for their majors That was absolutely not allowed at my faculty (admittely computational linguistics, but I would have massively benefited from math courses). No courses other than the predefined ones, no matter how relevant. Now I have to learn so much afterwards, it's not even funny.

> ...have to learn so much afterwards, it's not even funny.

It's true.

The sad thing is these problems start well before university when high schools pressure students into "advanced" math coursework without demonstrating mastery of previous topics. It builds a shaky foundation and sets the student up for a lot of needless difficulty later on.

Much better to slow down, focus on fundamentals early on and then build breadth in university coursework.

Re: A 2020 Vision of Linear Algebra

#178

Post by heymijo » Tue, May 12, 2020, 7:16 PM UTC

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…

Durable and flexible knowledge...

After reading your comment and ansible's reply [0] I wanted to pause and comment on this.

The United States Air Force Academy found that its cadets who took their first calculus class with a professor who focused on conceptual understanding helped those cadets create a durable and flexible understanding of the math [1].

The kicker is that the cadets got worse scores in Calculus I and gave professors who taught in this way worse ratings.

Ansible's anecdotal reply is what a lot of students experience. A feeling of initial success with the material, but they later find that their knowledge of it was fleeting and inflexible. What the Air Force Academy study found was that professors who taught in the manner ansible described, that resulted in fleeting and inflexible knowledge, were rated higher by their students. Those students got better initial scores in Calculus I, but went on to do worse in later calculus courses and related courses.

I encourage you to read the study. It is as good of a study design and execution you can get in the social sciences.

David Epstein also discusses the study in Chapter 4 of his book, Range [2].

[0] https://news.ycombinator.com/item?id=23154241 [1] http://faculty.econ.ucdavis.edu/faculty/scarrell/profqual2.p... [2] https://www.goodreads.com/book/show/41795733-range

Re: A 2020 Vision of Linear Algebra

#179

Post by srean » Tue, May 12, 2020, 7:33 PM UTC

mturmon wrote: post #167

mturmon wrote: post #136

Earlier quoted context omitted.

I like your bibliography! Ben Noble's book was my entry to LA. I was an undergraduate and involved in a research activity that demanded a lot of knowledge of the eigenvalue problem. The concrete approach in that book helped a lot. It was only later on that I took a class based on G&vL (implementing a bunch of basic LA factorizations in Matlab), and in my spare time read Halmos's book. I understand the coordinate-free…

Self-reply: Here's a comparative review of Noble+Daniel vs. Strang: https://pdf.sciencedirectassets.com/271586/1-s2.0-S002437950...

Hey thanks. I am quite surprised to meet a fellow Nobler. I thought I was the only one. I self studied the material and no one in my social circle had read it from Noble.

Re: A 2020 Vision of Linear Algebra

#180

Post by blablabla123 » Tue, May 12, 2020, 7:52 PM UTC

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'm no expert, although I think I remember that a Monad is basically just like allowing a sequence of statements to be executed. Like executing a code file ;)

Functional languages are really weird, for instance it's possible to switch line order of statements and the compiler will still figure out how to stitch that together. I think even JS in parts has or at least had that behaviour. (Actually that's useful when having mathematical formulas that are interdependent and you're too lazy to order them topologically by dependence)

On the other hand, just executing a sequence of commands in order to do I/O is only a normal thing to do since recently as far as I understand. The sweet spot for FP is IMHO something like React where state is strictly separated from the functions. (Imagine writing Hello World using Normal Maths)

(Please correct me if I'm wrong, which is probably quite likely ;))