Gnome KatIf functions can be represented as vectors and you can do the dot product between them...
What happens if you try to use the wedge product between two functions?
This question just popped into my brain
#geometricalgebra #math
ƧƿѦςɛ♏ѦਹѤʞ@gnomekat
There must be something in the air - I was thinking recently what use an infinite dimensional Clifford algebra might be put to. Not quite sure yet. Still mulling that one. LOL
#maths #GeometricAlgebra
There must be something in the air - I was thinking recently what use an infinite dimensional Clifford algebra might be put to. Not quite sure yet. Still mulling that one. LOL
#maths #GeometricAlgebra
Just trying to think about it my brain is just generating more questions.....
Would the result be some kinda bivector in an infinite dimensional space? What does that even mean? Would it still be a plane in that space? Does it split the space of functions in two? Where some functions lie on the plane and some are on either side? Is it a rotation from one function to another? I'm so confused...
@spacemagick
Wait now I have even more questions, if it was a rotation could you rotate some arbitrary 3rd function by that rotation? This is getting weird...
Wait now I have even more questions, if it was a rotation could you rotate some arbitrary 3rd function by that rotation? This is getting weird...
@gnomekat
This wasn't on my list of things to think about this Sunday but it seems to be now :-)
To begin with there's the matter of becoming accustomed to the idea of an *uncountably* infinite dimensional vector (assuming our function's domain is the reals). Then there's the matter of getting one's head around multiplying uncountably infinite terms each by another uncountably infinitude of terms. After that it gets tricky.
#maths #GeometricAlgebra
...
This wasn't on my list of things to think about this Sunday but it seems to be now :-)
To begin with there's the matter of becoming accustomed to the idea of an *uncountably* infinite dimensional vector (assuming our function's domain is the reals). Then there's the matter of getting one's head around multiplying uncountably infinite terms each by another uncountably infinitude of terms. After that it gets tricky.
#maths #GeometricAlgebra
...
@gnomekat
I think taking the geometric product is going to produce an infinitude of terms of every vector-degree - so that's quite interesting for a start. Then there's the question of whether the number of different vector-degrees is also uncountable (whatever that means). I'm really going to have to sleep on that one. #maths #GeometricAlgebra
I think taking the geometric product is going to produce an infinitude of terms of every vector-degree - so that's quite interesting for a start. Then there's the question of whether the number of different vector-degrees is also uncountable (whatever that means). I'm really going to have to sleep on that one. #maths #GeometricAlgebra
@spacemagick thinking about it I think it would just produce a scalar + infinite dim bivector right? its still just grade 1 * grade 1
VVilliam 
@gnomekat @spacemagick I thought the cross product was only a binary operation in 3 dimensions and 8? Does it also extend to the continuum?
Well first of all ther are plenty of binary operations you can define in 3 dimensions. The dot product, cross product, wedge product, geometric product... Ect.
Also the cross product can be defined in terms of the wedge and geometric product
(u^v)I
(u wedge v) times the psudo scalar basis I for the space you are in. Which is equivalent to cross in 3d but extends to any dimension it just doesn't always return a vector.
But yeah wedge product works in any dimension. Is associative and anticommutative. Sums the grade of objects (two vectors turn into a bivector).. ect...