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a, b & c is the coordinates of the normal vector of a plane and that plane's intersection with the z=1 plane is the drawn lines here. You should give some geometric intuition first, else it becomes an algebra jerk.

How do you calculate a dual?

Yeah usually a good idea to specify what category its an isomorphism in if its not clear by context... Also funny but trivial observation: since C is a Q vectorspace with a countably infinite basis its isomorphic to Q->Q with pointwise addition and scalar multiplication

What is i^-1?

what

As a pure math student, i hate this so much. This is so wrong on so many different levels, its insane.

what

Bro what happened to that playlist on geometric algebra? I started seeing it last week, after watching your video showing the power of geometric algebra. I haven't finished it yet but its incomplete I think.

Wild

You are the only one that I found talking about that❤

but what about matrices of multivector elements? is it used?

to describe a 3 dimensional object you only need 3 dimensions. in order to describe a TRANSFORMATION of a 3d object I think it is necessary to have a 4th dimension. Otherwise there is no meaningful transformation at all. Kind of like how time is considered a 4th dimension in our 3 dimensional world.

What's really nice about this definition is that you don't need the matrix representation of the linear function in order to calculate the determinant of the function.

At 38:00 when you said charge is current moving in time, my brain imploded with the spacetime diagram and worldlines on it. Thank you.

I'm really liking the proposal of this series and I understand that at this point some conditions are being left implicit so as to not clutter the explanation, but, just so I know I'm following it, if we are to take vectors as (non-empty) equivalence classes of oriented segments of same lenght and orientation (so as to allow for freely moving around a vector), in order for the half-space to not count as a space we must add the condition that all vectors have a representative starting at the origin, correct?

is the series continuing? this is fantastic

How about Z_2 span of {1,x,x^2,….}?

Nonsense gem

I really appreciate these videos as I'm learning GA. Animations are really helpful and books generally don't have them.

How do you deduce that it must be the square root?

9:18 the cross product also works in 7 dimensions 🤭

I don't know what double over is, but the first thing I would do is check if it was somehow being forced into a half angle situation.

Once you've 3b1b on board, it's only a matter of time. I'm convinced. I've always felt that angle units should be full turns, i.e. 1 unit = tau radians. It doesn't usually save me any time to know that the ratio with the circumference is already in there; as a coder, I'm mostly passing angles off to trig functions anyway, and it would be easier to type 1/4 than Math.PI/2.

I don't understand why people find it so difficult to illustrate 4D. Use an animation. Each 3D frame is a slice. You can use some gradients to give an impression of 4D locality if you wish. You can use color as a 4th coordinate, too. It's not so hard.

Any ideas about how to go about extending PGA into spacetime a la STA? It's easy enough to think of geometrically, with time-like lines forming object paths and space-like planes being full 3D volumes. However in implementation, there are a number of stumbling blocks I can anticipate, starting with how to blend the two concepts of vector. VGA/STA have vectors that indicate points/events with the coordinates equal to their components, whereas in PGA, vectors are equations that involve the coordinates, and the objects are formed from all points for which the equation equals zero. Very different concepts, and I'm not quite sure how to bridge them. Where does the metric fit, i.e. which basis vector(s) need to square negative in order to make all the transformations work? If I could fill in some of the blanks, I think it could be really interesting. You would be able to capture an object's motion through spacetime as an exponential rotor/motor, and maybe formulate forces that way.

Zero is a number. The sum of no numbers is undefined.

Love from kerala ❤

That identity means wv=uvwu-1, it reflects the vw plane around the arbitrary vector u. So two-dimensional planes all look the same?

I get really confused about the regressive product. The video talks about how the regressive product is the join and how it can be intuitively thought of as the intersection of the two sets of lines/planes, but nothing about how you can manipulate the regressive product is mentioned.

what about concave polygons

Will you expand this further to the product of four vectors and more? As an exercise, I did the product of four vectors, uvwz, and found it to contain a scalar, bivector, and a grade-4 part. Based on this, it seems the product of # n vectors produces a sum of objects that are odd if n is odd, and even if n is even. Is my thinking on the right track?

To know what each terms are in terms of geometric product, you can substitute all permutations for that.

Geometric algebra out here saying offensive things about the Jews.

As for the symmetric part: If v and w are orthogonal, you get the the classic bac - cab formula of the vector triple product. So maybe it is some kind of generalization of that with the result being in the span of all three vectors instead of just b and c?