[vcxy]

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

[xyzzyz]

(the exchange between you two is a perfect example of the disconnect between how mathematicians see linear algebra, and how physicists do)

[vcxy]

Haha, maybe so. I'm a mathematician, but based on some other comments I seem to be playing the role of physicist I think? Not sure.

[tagrun]

My favorite colors are 1, 2, 3 and 4 in some 8-bit grayscale color space. Here's a matrix

  [ 1 2 ]
  [ 3 4 ]

This represents some linear transformation, and is a matrix. Are you trying to argue otherwise?

FYI, matrices transform between vectors within a vector space.

[vcxy]

You have a list of labels you've conveniently decided to conflate with numbers. It's fine to use an enum for your data, but no, this data structure is not meant to convey a transformation.

You bring up a good point though, if this were meant to be a transformation, then we're talking about modules (Z/2^8Z being the underlying ring) and not vector spaces, which is fine. I was needlessly narrow when I said "vector spaces" earlier.

[tagrun]

You can have matrices of anything including rings, quaternions, octonions, dual numbers, matrices, vectors, bras, kets, etc etc, as long as you can multiply and add those objects. If you prefer real numbers, use the corresponding wavelength of the light, or voltage values in a photosensor, it doesn't matter. It still is a matrix.

Why do you mention things like data structures, enums and labels? This is math, not C++.

Anyway, the point is, any arbitrary 2D arrangement of numbers can be a valid matrix, whatever those numbers may represent.

[vcxy]

Sure, I can get rid of the word enum. The symbols in the example are numerals for sure, but you explicitly say they're representing colors. I don't know how to multiply colors. What I'm saying is you merely have labels there, the numbers those numerals usually represent have nothing to do with the explicitly stated meaning. If I saw that array without context, yeah sure, I'd assume it's a matrix. But there is context that tells me interpreting this as a transformation makes no sense.

I'm genuinely claiming they aren't numbers due to the explicit context. The entries are colors with numerals as labels.

Now, you've said something slightly different where I think we can agree. If the entries are from a set closed under a notion of multiplication and addition, then it represents a matrix even if it isn't being used in that way. But I still say that if you're merely using the array as a place to keep some data then I won't be using the word matrix, I'll just call it an array.

Anyway. I started this by saying it was interesting that we use the word matrix slightly differently. I still think that's interesting and think it's totally fine if you want to call an array a matrix even if the entries don't come from something where a transformation makes sense. I wouldn't "correct" someone's usage. I just think it's interesting how we use it different. Anyway, I feel like you just think I'm dumb, so I'm ending this here. You might not, but it's hard to read the vibe via text. If you have something to add you think will get me to change my word choices, feel free to respond and I'll read it, but I'm not replying on this thread anymore.

[twangist]

> FYI, matrices transform between vectors within a vector space.

Square ones do, but m x n ones represent linear maps from an n-dimensional to m-dimensional vector space (over the/a field containing the elements of the matrix).