Clifford (Geometric) Algebra vs Linear Algebra etc

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Any rotation is a double reflection in a pair of vectors, as illustrated in Figure 7-2 on the right. In this figure, the red vector x is reflected first in the blue vector a, and then in the blue vector b. The result is that x is rotated over an angle φ, which is twice the angle between a and b.

In geometric algebra, you can use a unit vector a as a reflection operator by employing it in a sandwiching product: ax/a reflects the vector x in the line of a.

It then follows from the figure that the geometric product of two unit vectors R = b a fully encodes the rotation as a double reflection. R is called a rotor, and it is a regular element of the real algebra. The rotation of any object O by the rotor R is performed the same way as quaternions: RO/R. Quaternions are just one special case of rotors, i.e., quaternions are rotors for a 3-D Euclidean space. The i, j, k coordinates of a quaternion are just the coordinates of the bivector part of the rotor, which encodes the rotation plane (drawn as the yellow disc in the figure on the right). In geometric algebra, the square of a real unit plane like i happens to be -1, and that is why rotors and quaternions remind us of complex numbers. Emphasizing this relationship to complex numbers (as quaternion treatments typically do) makes them needlessly hard to understand and visualize.

>>8288186
as much as i think GA is better, your explanation of "being needlessly hard" is what's ironic

>>8288195
>being needlessly hard

Where was this explanation?

>>8288179
>Do you hate Linear Algebra?
You can't do clifford algebra without linear algebra. A clifford algebra is just the tensor algebra of some quadratic vector space modulo the ideal generated by elements of the form v⊗v - Q(v).

Linear algebra is the literal best math I've ever learnt. Aside from DEs

>>8288244
For undergraduate physics and other things like 3D rendering I think GA is more intuitive. It's more for applied subjects than for useless math majors.

>>8288179
Clifford algebras are linear dumbass

>>8288179
OP, you misunderstand the word algebra in the two contexts (and from this misunderstanding stems your idiotic post). There is algebra as in "I studied algebra", and there is algebra as in "I studied algebras". The algebra in "linear algebra" is of the first type, and that in "the Clifford algebra" is of the second type.

Tbh, the word algebra is a shit precisely because of this ambiguity. I use "ring" instead of "algebra" whenever I can (and R-ring if we're talking about a ring over R (or what would normally be called an R-algebra)).

I think both are shit, someone in the not so near future is going to point out of how shit both are and a whole new system will be devised by /sci/

>hating a subject area

oh sweet summer child

The first trick to understand algebras is to know, that we are not talking about numbers as we know them from school. Complex numbers seem more like a vector than a number. Notice how different algebras consider completely different objects (matrices as an example). It is easier to think that certain rule are correct even though they seem wrong BECAUSE from the start we are not dealing with numbers as we know them from the real world.

>>8288315
>being intuitive is more important than being useful