I was able to solve this problem after much trial error! I now have to find a one-to-one linear transformation T : H - M[sub:3pakp84r]4,4[/sub:3pakp84r] (R) where H = {a +bi +cj +dk such that a,b,c,d are real numbers}. Also the transformation has the property that if g,h are in H then T(gh) = T(g)T(h). Again any tips would be appreciated. Thank you so much.
I have been trying to solve a problem where I need to find 4x4 matrices U,V and W such that U^2 = v^2 = W^2 = -I[sub:3pakp84r]4[/sub:3pakp84r]as well as :
UV = W
VU = -W
VW = U
WV = -U
WU = V
UW = -V.
I had matrices U = [0 0 0 1]
[0 0 -1 0]
[ 0 1 0 0]
[-1 0 0 0]
and V = [0 0 0 -1]
[ 0 0 1 0]
[0 -1 0 0]
[1 0 0 0]
(They are supposed to be 4x4 matrices)
both which = -I[sub:3pakp84r]4[/sub:3pakp84r] but when I did UV I get the identity matrix for W which clearly won't
work for W^2 = -I[sub:3pakp84r]4[/sub:3pakp84r] .
Does anyone know of some strategies for finding U,V,W or do you think it will be all trial and error which hasn't been going so well?
Thanks[sub:3pakp84r][/sub:3pakp84r]
I have been trying to solve a problem where I need to find 4x4 matrices U,V and W such that U^2 = v^2 = W^2 = -I[sub:3pakp84r]4[/sub:3pakp84r]as well as :
UV = W
VU = -W
VW = U
WV = -U
WU = V
UW = -V.
I had matrices U = [0 0 0 1]
[0 0 -1 0]
[ 0 1 0 0]
[-1 0 0 0]
and V = [0 0 0 -1]
[ 0 0 1 0]
[0 -1 0 0]
[1 0 0 0]
(They are supposed to be 4x4 matrices)
both which = -I[sub:3pakp84r]4[/sub:3pakp84r] but when I did UV I get the identity matrix for W which clearly won't
work for W^2 = -I[sub:3pakp84r]4[/sub:3pakp84r] .
Does anyone know of some strategies for finding U,V,W or do you think it will be all trial and error which hasn't been going so well?
Thanks[sub:3pakp84r][/sub:3pakp84r]