If I had to pick between a matrix being a row of vectors or a column of covectors, I'd pick the latter. And M[i][j] should be the element in row i column j, which is nonnegotiable.
Could a mathematician please confirm of disconfirm this?
I think that different branches of mathematics have different rules about this, which is why careful writers make it explicit.
(In many branches the idea is that you care about the abstract linear transformation and properties instead of the dirty coefficients that depend on the specific base. I don't expect a mathematician to have an strong opinion on the order. All are equivalent via isomorphism.)
Anyone who has taken linear algebra should know that (1) a rotation is a linear operation, (2) the result of a linear operation is calculated with matrix multiplication, (3) the result of a matrix multiplication is determined by what it does to the standard basis vectors, the results of which form the columns of the matrix.
This guy makes it sound like he had to come up with these concepts from scratch, and it's some sort of pure visual genius rather than math. But... it's just math.
Just imagine that everyone has equal math ability, except the model of math and representations of mathematical concepts and notation is more made for a certain type of brains than others. These kind of explanations allow bringing those people in as well.
I'd take issue with his "most programmers are visual thinkers", though. Maybe most graphics programmers are, but I doubt it's an overwhelming majority even there.
I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum.
Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.
Later in another lecture at another university, I had to rotate points around a center point again. This time found 3 3x3 matrices on wikipedia, one for each axis. Maybe making at least seemingly a little bit more sense, but I think I never got to the basis of that stuff. Never seen a good visual explanation of this stuff. I ended up implementing the 3 matrices multiplications and checked the 3D coordinates coming out of that in my head by visualizing and thinking hard about whether the coordinates could be correct.
I think visualization is the least of my problems. Most math teaching sucks though, and sometimes it is just the wrong format or not visualized at all, which makes it very hard to understand.
The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction).
Since you can combine a series of matrix multiplications by just pre-multiplying the matrix, this sets you up for doing a very efficient "move, scale, rotate" of an object using a single matrix multiplication of that pre-calculated 4x4 matrix.
If you just want to, e.g., scale and rotate the object, a 3x3 matrix suffices. Sounds like your first lecture jumped way too fast to the "here's the fully general version of this", which is much harder for building intuition for.
Sorry you had a bad intro to this stuff. It's actually kinda cool when explained well. I think they probably should have started by showing how you can use a matrix for scaling:
[[2, 0, 0],
[0, 1.5, 0],
[0, 0, 1]]
A lot of areas use use grid of numbers. And matrix theory actually incorporates every area that uses grids of numbers, and every rule in those areas.
For example the simplest difficult thing in matrix theory, matrix multiplication is an example for this IMO. It looks really weird in the context of grid of numbers, and its properties seem incidental, and the proofs are complicated. But matrix multiplication is really simple and natural in the context of linear transformations between vector spaces.
pavlov•2h ago
The age doesn't affect this matrix part, but just FYI that any specific APIs discussed will probably be out of date compared to modern GPU programming.