Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

A 2020 Vision of Linear Algebra

ocw.mit.edu

Post reply on HN
Print view View original

81–90 of 205 posts

Re: A 2020 Vision of Linear Algebra

#81

Post by jerf »

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?

The "monad" problem is worse; many of the "tutorials" were actively wrong about some critical element, often more than one. I don't think I've seen someone claim to have linear algebra "click" but be fundamentally wrong about it somehow.

One advantage of linear algebra is that it is, well, linear. Linear is nice. It means you can decompose things into their independent elements, and put them all together again, without loss. The monad interface, as simple as it is, is not linear; specific implementations of it can have levels of complexity more like a Turing machine.

Re: A 2020 Vision of Linear Algebra

#82

Post by bencw »

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?

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.

Re: A 2020 Vision of Linear Algebra

#83

Post by bencw »

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…

This is very well put. Knowledge has a hierarchical (or perhaps even cyclical!) structure and it's unrealistic to think that a body of knowledge can be taught or learned sequentially.

Re: A 2020 Vision of Linear Algebra

#85

Post by bencw »

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

This book is great and very much complementary to Strang's approach in that it leans more towards "abstract" linear algebra.

Re: A 2020 Vision of Linear Algebra

#86

Post by dleslie »

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.

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

Re: A 2020 Vision of Linear Algebra

#87

Post by Myrmornis »

dntbnmpls wrote: post #51
brmgb wrote: post #36

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

> After watching this and having read the comment, I am quite puzzled by the approach American seems to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? The US is a very big place. I doubt there is an american approach to linear algebra. We really don't have a single approach to anything. Different schools and majors probably approach the topic differently. My college had a linear a…

> Different strokes for different folks. If it worked out for you that's all that matters.

That sort of relativism is trite, specious, and insincere. If you don’t think it matters then why are you participating in the discussion?

Re: A 2020 Vision of Linear Algebra

#88

Post by ktta »

anandrm wrote: post #84

Just curious .. what really are the usecases where of Linear Algebra is applied ? Any domain of software development ?

Any fields that have anything to do with video, image, audio, games, machine learning.

Just to have a taste of use cases: compression, filters(image filters for de-noising, HP & LP filters for audio), encoding/decoding, computer vision techniques, cryptography, neural nets, computer graphics (this is where most people learn how to use it in real computer programs)

Re: A 2020 Vision of Linear Algebra

#89

Post by jcmoyer »

vertak wrote: post #70

Can anyone grow why in the first 5 minutes of part 1 he shows a 3 by 3 matrix multiples by a 1 by 3 vector yet verbally he pulls out of no where this idea that if you have _two_ 1 by 3 vectors that pass through the origin then their linear combinations can be represented by a plane? The jump from the 3D to the 2D has me lost and I gave up

The same concept applies in 2D, which might help you build the intuition to understand it in 3D.

If you have a vector v=(1,0) that points to the right, you can scale this vector infinitely in that direction by multiplying it by a positive scalar.

5v = (5,0)

62.1v = (62.1,0)

Similarly, you can scale that vector infinitely in the opposite direction (i.e. left) by multiplying it by a negative scalar:

-987v = (-987,0)

If we call this scalar c, the expression cv allows us to represent any point along the X axis simply by varying c, meaning that cv defines a line along that axis.

Similarly, we can do the same for a vector w=(0,1) along the Y axis, scaling it by d.

Now we have a method for moving to any point on the XY plane simply by varying c and d in the linear combination: cv + dw, meaning that we've defined a plane using two vectors.

Two caveats:

- this won't work if v and w are parallel; for example, if v = -w (and neither are zero) then we can only move along a line instead of a plane

- it also won't work if either of the vectors are zero, because no matter what you multiply by, a zero vector can only represent a single point

Re: A 2020 Vision of Linear Algebra

#90

Post by irl_zebra »

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 is that when I didn't quite remember something (i.e. how exactly to multiply a matrix) I could just go to an earlier pre-LA lesson, pick it up, and then go back to LA where I left off. I'm a few lessons in.

What do you all recommend for someone like me?

Post reply on HN