yellow text on green background... my eyes!
Yea this is horrendous. Not reading this
Matrices can be your friends (2002)
11–20 of 115 posts
Re: Matrices can be your friends (2002)
#12There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics that doesn't utilize matrices as tools. What exactly is a matrix? Just a grid of numbers? don't think so.
Re: Matrices can be your friends (2002)
#13People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! 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…
A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist.
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.
Re: Matrices can be your friends (2002)
#14> Mathematicians like to see their matrices laid out on paper this way (with the array indices increasing down the columns instead of across the rows as a programmer would usually write them). 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.
Not a mathematician, but programmers definitely don't agree on whether matrices should be row-major or column-major.
I'm surprised we even agree that they should be top-down.
Re: Matrices can be your friends (2002)
#15People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! 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…
When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there was a "magical formula" to rotate things and I didn't memorize it. Exam came and demanded having memorized this shitty rotation matrix. Failed the exam, changed lectures. High quality lecturing.
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.
Re: Matrices can be your friends (2002)
#16There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…
Take linear algebra
Re: Matrices can be your friends (2002)
#17People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! 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…
When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there w…
You can do rotation with a 3x3 matrix.
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]]
Re: Matrices can be your friends (2002)
#18There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…
Take linear algebra
OK, now what?
Re: Matrices can be your friends (2002)
#19People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! 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…
A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist. I'd take issue with his "most programmers are visual…
> most programmers are visual thinkers
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.
Re: Matrices can be your friends (2002)
#20There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…
A matrix is just a grid of numbers.
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.