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I'm not the grandparent. I'm not sure what you mean by "required", but, no, I don't think so. You can represent a lot of algebraic objects and operations using

I'm not the grandparent. I'm not sure what you mean by "required", but, no, I don't think so. You can represent a lot of algebraic objects and operations using matrices and linear algebra operations. The most accessible example of that is probably complex numbers: https://en.wikipedia.org/wiki/Complex_number#Matrix_representation_of_complex_numbers https://en.wikipedia.org/wiki/Complex_number#Matrix_represen... and I wouldn't say that "matrices are still required in complex numbers" any more than I'd say "matrices are still required in geometric algebra".

BlueTemplar 7y ago
OK, after refreshing my memory on GA, it seems that one of the nice things about it is that you don't have to think about the "row" vs "column" vectors present in "Vector Algebra". Which are usually represented in the form of matrices. But you don't "need" them in GA, since you can directly work with vectors (= arrays) ! (Which makes it even more curious as to why matrices would be more efficient for 3D work ??)
- gugagore 7y ago
  I think I understand roughly what you mean when you distinguish row and column vectors. For example, let [x, y] represent a row vector and [x, y]^T represent a column vector. If you have a function f that maps [x, y]^T -> z, (you might write it z=f(x,y)), then the gradient(f) is a function [x, y]^T -> [x, y]. That is to say, the gradient is a row vector. It's a different kind of vector than the input to f. And it transforms different (c.f. https://math.stackexchange.com/a/3200912/ https://math.stackexchange.com/a/3200912/) So using that kind of thinking, the cross product of two column vectors is a row vector: https://www.youtube.com/watch?v=BaM7OCEm3G0 https://www.youtube.com/watch?v=BaM7OCEm3G0 As you say, Geometric Algebra doesn't talk about row vectors and column vectors. For example, in 3D GA,you can choose a representation in R^8. That's 1 scalar, 1 pseudo-scalar, 3 column-y components, and 3 row-y components.
  - BlueTemplar 7y ago
    lol, ok - so then it's more the opposite - in GA there are row an column "vectors" (3-components)... but they are not "enough" (for that R^8 representation), so a matrix representation might be misleading ?
- aspaceman 7y ago
  From reading a lot of comments it seems like misinformation. One thing worth keeping in mind is that game developers are highly risk averse. For good reason.

BlueTemplar 7y ago

OK, after refreshing my memory on GA, it seems that one of the nice things about it is that you don't have to think about the "row" vs "column" vectors present in "Vector Algebra". Which are usually represented in the form of matrices. But you don't "need" them in GA, since you can directly work with vectors (= arrays) ! (Which makes it even more curious as to why matrices would be more efficient for 3D work ??)

gugagore 7y ago
I think I understand roughly what you mean when you distinguish row and column vectors. For example, let [x, y] represent a row vector and [x, y]^T represent a column vector. If you have a function f that maps [x, y]^T -> z, (you might write it z=f(x,y)), then the gradient(f) is a function [x, y]^T -> [x, y]. That is to say, the gradient is a row vector. It's a different kind of vector than the input to f. And it transforms different (c.f. https://math.stackexchange.com/a/3200912/ https://math.stackexchange.com/a/3200912/) So using that kind of thinking, the cross product of two column vectors is a row vector: https://www.youtube.com/watch?v=BaM7OCEm3G0 https://www.youtube.com/watch?v=BaM7OCEm3G0 As you say, Geometric Algebra doesn't talk about row vectors and column vectors. For example, in 3D GA,you can choose a representation in R^8. That's 1 scalar, 1 pseudo-scalar, 3 column-y components, and 3 row-y components.
- BlueTemplar 7y ago
  lol, ok - so then it's more the opposite - in GA there are row an column "vectors" (3-components)... but they are not "enough" (for that R^8 representation), so a matrix representation might be misleading ?
aspaceman 7y ago
From reading a lot of comments it seems like misinformation. One thing worth keeping in mind is that game developers are highly risk averse. For good reason.