f(x) = 2x
g(x) = x + 5
g(f(x)) = (2x) + 5 = 2x + 5
f(g(x)) = 2(x + 5) = 2x + 10
You can do "multiplication" on these packs to compose them together, and even though the rules feel obtuse in abstract, they follow the logic of function composition. A matrix represents a similar function transform, only it's in 3 dimensions, and in order to handle rotation, it needs to swap around x/y/z. So an identity matrix is really saying:
f(x) = 1x + 0y + 0z + 0
f(y) = 0x + 1y + 0z + 0
f(z) = 0x + 0y + 1z + 0Or you can learn about them in school and be given neither a real-world use case nor a rationale or derivation for mmul.
Or you can encounter them in college, and there the experience depends on whether it’s a good or bad kind of a linalg class. But even there – after all the painstaking definitions and lemmas and derivations – it’s easy to end up not grokking how mmul does what it does even if you grasp all the building blocks – vector spaces, bases, how matrices can represent bases and systems of linear equations and linear operations on vectors and how they’re all kinda equivalent.
We had a week-long group project in the first year whose theme was "a 360° pong". Our group decided that it meant the paddles had to travel in a circle around the playfield and I've decided that matrices and stacked 2D transformations were the way to go. The other students gave me blank stares, I basically said "trust me, you don't need to understand them to use them" and off we went coding.
We ended up with the most impressive pong clone out of all the groups, as nearly all of them had axis-aligned rectangular paddles going around a rectangular path, whereas ours had a stretched half-circle paddle going around a circle path always facing the center of the playfield, alongside extra features like walls and a level editor.
First class the Monday morning after, the math teacher announced that the next topic was matrices. We stared at each other in the group and grinned manically.
If anyone wants to stare at an old C codebase from 10 years ago by a bunch of first year students: https://code.google.com/archive/p/pong-norris/
Saw it explained in a single slide of Andrew Ng's ML course and everything clicked perfectly.
Where I lived math was taught like fucking shit, it was all algebra and zero context as to why it could be useful in real-life scenarios, zero abstraction such as visualization or metaphor. Everyone involved in concocting that pedagogic aberration should feel terrible about it.