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This is ok but nothing is as intuitive as 3B1B's series on YouTube that has been posted hundreds of times on HN [0]. Linear algebra is really about linear tran

by vecter 7y ago

This is ok but nothing is as intuitive as 3B1B's series on YouTube that has been posted hundreds of times on HN [0]. Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post. [0] https://www.youtube.com/watch?v=fNk_zzaMoSs https://www.youtube.com/watch?v=fNk_zzaMoSs

foxx-boxx 7y ago
Often these articles are written by students who either failed their exam or scared to fail it. People who actually know the subject get paid either for lectures or for real work.
sandov 7y ago
Playlist link: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
OrwellianChild 7y ago
3B1B does great work explaining these concepts, but I can't help but ask "why not both?" when it comes to explaining these concepts. Turns out, linear algebra is great for working with matrices, vector space, approximating non-linear systems, and more... Let's embrace multiple ways of teaching it and gaining intuitions rather than keeping score, eh?
nightcracker 7y ago
> Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post. I... disagree. Some of linear algebra is about that. And it's probably a good way to view it that way when learning. But some of my current work (coding theory) involves linear algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really at no point are we viewing what we're doing as transforming a vector space, we're just solving equations with unknowns.
- xscott 7y ago
  I think this is spot on. Depending on what you're doing, a matrix can be: - A linear transformation - A basis set of column vectors - A set of equations (rows) to be solved - (your example: parity equations for coding theory) - The covariance of elements in a vector space - The Hessian of a function for numerical optimization - The adjacency representation of a graph - Just a 2D image (compression algorithms) ... (I'm sure there are plenty of others) For some of these, the matrix is really just a high dimensional number. You (rarely?) never think of covariance in a Kalman filter as a linear transform, but you still need to take its Eigen vectors if you want to draw ellipses.
  - vecter 7y ago
    Great points. I wrote my comment in response to the article claiming to be an intuitive guide to linear algebra, not an intuitive guide to matrices. According to wikipedia: > Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces through matrices. [0] The Venn Diagram of Linear Algebra and Matrices definitely has a lot of non-overlap, of which your list covers some. This article should be renamed to be about matrices and not linear algebra, because it's not. [0] https://en.wikipedia.org/wiki/Linear_algebra https://en.wikipedia.org/wiki/Linear_algebra
  - JadeNB 7y ago
    > - A basis set of column vectors Let's leave the word 'basis' out, since the column vectors may well be linearly dependent.
    - FabHK 7y ago
      Or not span the space, for that matter.
      - JadeNB 7y ago
        Well, it depends on what "the space" is. Every set of vectors (in a common ambient space) spans some space—often called the column space of a matrix, if they are the column vectors of the matrix.
gitgudnubs 7y ago
Then you failed to comprehend the subject. The point is that a wide array of problems and models are really the same thing.
- vecter 7y ago
  Failed to comprehend which subject, linear algebra? I would argue no, and other people who are more on the pure mathematics side would agree [0][1]. snicker7 said it very succinctly: > However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra). If you're point is that I failed to comprehend matrices, then I don't think you have enough data to make that claim since I don't really talk about matrices. I kind of address that in my other comment [2]. I don't follow your point around "a wide array of problems and models are the same thing". That's a very vague general statement that I certainly comprehend (not sure how you inferred otherwise). Specifically, I don't see how that point relates at all to the claim I made about linear algebra. [0] https://news.ycombinator.com/item?id=22419018 https://news.ycombinator.com/item?id=22419018 [1] https://news.ycombinator.com/item?id=22417764 https://news.ycombinator.com/item?id=22417764 [2] https://news.ycombinator.com/item?id=22417595 https://news.ycombinator.com/item?id=22417595

by vecter 7y ago

foxx-boxx 7y ago

Often these articles are written by students who either failed their exam or scared to fail it. People who actually know the subject get paid either for lectures or for real work.

sandov 7y ago

Playlist link: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

OrwellianChild 7y ago

3B1B does great work explaining these concepts, but I can't help but ask "why not both?" when it comes to explaining these concepts. Turns out, linear algebra is great for working with matrices, vector space, approximating non-linear systems, and more... Let's embrace multiple ways of teaching it and gaining intuitions rather than keeping score, eh?

nightcracker 7y ago

> Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post. I... disagree. Some of linear algebra is about that. And it's probably a good way to view it that way when learning. But some of my current work (coding theory) involves linear algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really at no point are we viewing what we're doing as transforming a vector space, we're just solving equations with unknowns.

xscott 7y ago
I think this is spot on. Depending on what you're doing, a matrix can be: - A linear transformation - A basis set of column vectors - A set of equations (rows) to be solved - (your example: parity equations for coding theory) - The covariance of elements in a vector space - The Hessian of a function for numerical optimization - The adjacency representation of a graph - Just a 2D image (compression algorithms) ... (I'm sure there are plenty of others) For some of these, the matrix is really just a high dimensional number. You (rarely?) never think of covariance in a Kalman filter as a linear transform, but you still need to take its Eigen vectors if you want to draw ellipses.
- vecter 7y ago
  Great points. I wrote my comment in response to the article claiming to be an intuitive guide to linear algebra, not an intuitive guide to matrices. According to wikipedia: > Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces through matrices. [0] The Venn Diagram of Linear Algebra and Matrices definitely has a lot of non-overlap, of which your list covers some. This article should be renamed to be about matrices and not linear algebra, because it's not. [0] https://en.wikipedia.org/wiki/Linear_algebra https://en.wikipedia.org/wiki/Linear_algebra
- JadeNB 7y ago
  > - A basis set of column vectors Let's leave the word 'basis' out, since the column vectors may well be linearly dependent.
  - FabHK 7y ago
    Or not span the space, for that matter.
    - JadeNB 7y ago
      Well, it depends on what "the space" is. Every set of vectors (in a common ambient space) spans some space—often called the column space of a matrix, if they are the column vectors of the matrix.

gitgudnubs 7y ago

Then you failed to comprehend the subject. The point is that a wide array of problems and models are really the same thing.

vecter 7y ago
Failed to comprehend which subject, linear algebra? I would argue no, and other people who are more on the pure mathematics side would agree [0][1]. snicker7 said it very succinctly: > However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra). If you're point is that I failed to comprehend matrices, then I don't think you have enough data to make that claim since I don't really talk about matrices. I kind of address that in my other comment [2]. I don't follow your point around "a wide array of problems and models are the same thing". That's a very vague general statement that I certainly comprehend (not sure how you inferred otherwise). Specifically, I don't see how that point relates at all to the claim I made about linear algebra. [0] https://news.ycombinator.com/item?id=22419018 https://news.ycombinator.com/item?id=22419018 [1] https://news.ycombinator.com/item?id=22417764 https://news.ycombinator.com/item?id=22417764 [2] https://news.ycombinator.com/item?id=22417595 https://news.ycombinator.com/item?id=22417595