frame a b c ...  linearily independant if the hypervolume a^b^c^... not = 0
 
F G linear functions
 
F(aa + bb) = aF(a) + bF(b)
F(a ^ b) = F(a) ^ F(b)  (outermorphism)
F(aA + bB) = aF(A) + bF(B)     (A B multivectors)
F(G(aa + bb))  = aF(G(a)) + bF(G(b))
F(G(aA + bB))  = aF(G(A)) + bF(G(B))
 
adjoint  :
 
definition :  a.adj(F(b)) = F(a).b
 
adj(F(a)) = ri a.F(fi)       ([r1,r2,r3] = reciprocal frame of [f1,f2,f3])
if orthonormal frame e1,e2,e3 (ri = ei) :
adj(F(a)) = e1 a.F(e1) + e2 a.F(e2) + e3 a.F(e3)
 
adj(adj(F(a)) = F(a)
adj(F(G(a)) = adj(G(adj(F(a)))        (adj FG --> adj G(adj F)       reverse the order
F symetrical function if adj(F(a)) = F(a)
F(adj(F(a)) and adj(F(F(a)) are symetrical functions
adj is a linear function and extend to arbitrary multivectors
 
determinant :
 
with I3  the unit pseudoscalar                         (e123 for   Cl3 orthonormal frame)
definition : F(I3) = det(F) I3
det( FG(I3)) = det(F) det(G) I3                      FG short script for F(G())
det(F) = det(adj(F))
 
inverse :
 
inv(F(A))         = I3 adj(F(inv(I3) A)) det(inv(F))
inv(adj(F(A)))  = I3      F(inv(I3) A)   det(inv(F))
 
eigenvectors and eigenblades  :
 
F has an eigen vector  e   if   F(e) = ae  with eigenvalue a (scalar)
det(F - aI3) = 0 wich define a polynomial equation for a
 
we extend the notion of eigenvector to that of an eigenblade :
F(Ak) = aAk      Ak is grade-k blade and a is real.
Each eigenblade determines an invariant subspace of the transformation.
    linear transformation
 

                           
APPLICATIONS 
exercices
mechanics
geometry
electromagnetism
Lorentz transf. APS
Lorentz transf. STA
exterior algebra
linear transf.
quaternions
spinors
STA
APS
conformal model
hogeneous model