Matrices are axes. Vectors, or points, side by side. It wasn't until graduate school, after years of math and OpenGL that What that meant really sunk in and I really got how and why matrix ops are made of dot products.
The article is right in my case, I didn't learn the intuitive understanding at first, and it could have been taught that way. It is also well written, for people that know math, but I also feel like the article describes math using more math, and that the point could be better made with a picture or two. There's something about just seeing the correspondence between matrix rows, or columns, and the axes of a space, or transform, that finally helped it all sink in for me.
3 years into a math degree and this never clicked until I watched Feynman's QED lectures.
Everyone just tells you, "oh now this can be a matrix." Nobody tells you why matrices are useful, or how we ended up with them. Just, you know matrices use them.
The problem can be that most mathematics maintain that mathematics can be done for its own sake (something which I 100% agree on). However interesting mathematics is, though, there will be plenty of students more interested in the applications to the real world (and there's nothing wrong with that) or they may even have to interest in the mathematics until they see some clever ways it can be used to solve or abstract a specific problem, at which point they take a greater interest in the math itself.
As anecdotal evidence, most math teachers I had in college were at best uninterested and at worst disdainful of applying the math to the real world beyond theorems in a blackboard. Most physics teachers, however, in introducing us to new concepts made an effort to show as soon as possible how the definitions we made were inspired by real world problems and in turn simplified or helped create new physics. This made it much easier to appreciate the pure math itself.
Just explaining the pure mathmatical roots isn't always done, or cross field relations. It drives me insane when you use cross field tools, and when you bring up, Oh so we're doing X but with Y.
The response is often, No we do X because of Z. Which often you're learning Z. So now you just feel lost and confused. Its not 2 classes later until you'll learn A maps Y to Z and X is actually an operation of A so it applies to both.
I guess it's just me but the joy of math has always been its tangled relationships to itself.
Yes, I think typical intro classes focus too much on the rows and not enough on the columns. We learn about change of basis where each element in the new vector is a dot product with a row of the matrix. But you can also look at the new vector as a weighted combination of all the columns in the matrix.
You mentioned opengl and it's a perfect example of why you need to learn both. The Model matrix is column-oriented - if Z is up in the model but Y is up in the world, you need a Y unit vector in the third column of the matrix. The View matrix is row-oriented - each row is an axis of the camera's orientation. The columns perspective also makes it easier to understand homogeneous coordinates.
The article is right in my case, I didn't learn the intuitive understanding at first, and it could have been taught that way. It is also well written, for people that know math, but I also feel like the article describes math using more math, and that the point could be better made with a picture or two. There's something about just seeing the correspondence between matrix rows, or columns, and the axes of a space, or transform, that finally helped it all sink in for me.