glossary
algebra :
to form any algebra, we need elements and an associative product among them.
 
Euclidian space :
(This definition is for dimension 2 but this may be generalized for any dimension).
Consider the plane R x R = {(x,y) | x,y <- R}. Introduce a linear structure by addition  (x1,y1)+(x2,y2)=(x1+x2,y1+y2) and by scaling µ(x,y) = (µx, µy), where µ <- R. The linear structure makes the plane R x R into a linear space . Take a basis (e1,e2) of , say e1=(1,0), e2=(0,1). Introduce the length of               r = xe1 + ye2 as   |r| = sqrt(x² + y²). The introduction of length makes into an Euclidian plane, also denoted by . The basis vectors e1, e2 are unit vectors,  |e1|=1, |e2|=1.
Related to length there is a scalar valued product a.b = a1b1 + a2b2 of two vectors        
a = a1e1 + a2e2,  b = b1e1 + b2e2 <- R². The scalar product is symetric, a.b = b.a.
Two vectors a and b are orthogonal if their scalar product vanishes, a.b = 0. The unit vectors e1 and e2 are orthogonal. (P. Lounesto)
 
Clifford product :
Next, we introduce an associative, but non commutative (nonsymetric) product for vectors in . If the vector r is multiplied by itself  or squared, r r = r², we require that the square of the vector equals the square of the length of the vector :  r² = |r|².
In coordinate form, we introduce a product for vectors such that : (xe1 + ye2)² = x² + y²
Using the distributive law, without assuming commutativity, gives : x²e1² +e2² + xy(e1e2 + e2e1) = x² + y²
This is satisfied if the orthogonal unit vectors e1, e2 obey the multiplication rules
 
e1²    =    e2²   = 1            wich correspond to      |e1| = |e2| = 1
e1e2 = - e2e1                                                   e1   ⊥  e2
 
Using associativity and anticommutativity we can calculate the square as (e1e2)² = -e1²e2² = -1. Since the square of the product e1e2 is negative, it folows that e1e2 is neither a scalar nor a vector. The product e1e2 is a new kind of quantity called a bivector, and represents the oriented plane area of the square with sides e1 and e2. Write for short e12 = e1e2
Define for two vectors a = a1e1 + a2e2, b = b1e1 + b2e2 the Clifford product (Geometric product)
ab = (a1b1 + a2b2) + (a1b2 - a2b1)e12, a sum of a scalar and a bivector. (P. Lounesto)
 
outer product :   a ^ b = - b ^ a  
Generalize the cross product to higher dimensions
The outer product of r vectors   a1 ^ a2 ^ ...^ ar   can be defined as the anti-symmetric part of the geometric product of the r vectors. It is called an r-blade, or a blade of grade r  Ar.
Ar ^ Bs = <A B>r+s
 
inner product :  a . b = b . a
Generalize the scalar product to higher dimentions
Ar . Bs = <A B>|r-s|
 
geometric product : a b = a . b + a ^ b    
decomposes into symmetric and anti-symmetric parts.
inner and outer product, function of geometric product :       (with A' = invol(A)  )
a ^ A = 1/2(a A + A' a)                A ^ a = 1/2(A a + a A')
a . A  = 1/2(a A - A' a)                 A . a  = 1/2(A a - a A')
 
commutator product :
A x B = 1/2(AB - BA)
A x (BC) = (A x B)C + B(A x C)
A x (B x C) = (A x B) x C + B x (A x C)      Jacobi identity
 
Scalar :  
A point (real number) - grade 0 - no geometric extent
Vector :
A directed length - grade 1 - extent in 1 direction
Bivector :  
A directed area - grade 2 - extent in 2 directions
Trivector :  
A directed volume - grade 3 - extent in 3 directions
oriented volume :
In Cln, the maximal grade of a blade is n.
In Cl3 the trivector is a pseudoscalar : j = e123, j² = -1, |j| = 1.
 

blade :  
A scalar, a vector, or the outer product of any number of vectors A = a1 ^ a2 ^ ...^ ar      <A>r = A
grade :
Blade dimension r  (= subspace dimension)  <A>r
multivector    :  
Linear combination of blades  A = <A>0 + <A>1 + ... + <A>r + ... + <A>n
 
reversion :
rev(a1 ...ar) = ar ...a1  
Reverse basis element order e0 -> e0, e1 -> e1, e2 -> e2, ..., e12 -> e21, e23 -> e32, ..., e123 -> e321
grade involution :  
invol(a1 ...ar) = (−1)**r (a1 ...ar)
Change basis element direction e0 -> e0, e1 -> -e1, e2 -> -e2, ..., e12 -> e12, e23 -> e23, ..., e123 -> -e123
Clifford conjugation :  
cj(A) = rev(invol(A) = invol(rev(A)
conjugation = reversion(involution) = involution(reversion)
parity :
A+ even if invol(A) =   A
A- odd   if invol(A) =  -A
A+ A+ = A+  -->  define even subalgebra
dual :
dual(A) = A / j  (j = e12..n, Cln)  
dual(<A>r) = <A>n-r
(a · A)  j  = a ^ (A j)
(a ^ A) j  = a ·  (A j)
 
in Cl3
pseudoscalar : j = e12..n, dual(j) = e0    
 
translation :  
Transformation wich preserve distances, direction and orientation.
reflection :
Transformation wich preserves distances and leaves a blade, the blade of reflection, unchanged.
rotation :  
Transformation wich preserves distances and leaves a line, the axis of rotation, unchanged.
meet :
Intersection of vector subspaces.
A V B = dual(A) . B      grade(A) + grade(B) >= n
projection :
PAr (x) = (x · Ar)/Ar .
rejection :
P⊥Ar (x) = (x ^ Ar)/Ar .
orthogonal decomposition :
x = PAr (x) + P⊥Ar (x)
 
automorphism :  
Invertible linear maping that preserve geometric products :   f(A B) = f(A) f(B)
anti-automorphism :  
Linear mapping that reverse the order of geometric products : g(A B) = g(B) g(A)
The main anti-automorphism of vector algebra is called reversion : rev (a1 ...ar) = ar ...a1.
involution :
An involution h is an invertible linear mapping whose composition with itself is the identity map : h(h(A)) = A
The main involution of geometric algebra is called grade involution : invol (a1 ...ar) = (−1)**r (a1...ar).
 
magnitude :  
value of a scalar, length of a vector, area of a bivector, volume of a trivector, the sum of all these in mixed multivector.
Extended from vectors to any multivector A by |A| = sqrt( sum (|Ai|,0,n))  where |Ai| = sqrt(Ai . Ai) and Ai = <A>i
scalar product :  
Extended from vectors to any multivector A and B by <A rev(B)> = <B rev(A)> = sum (<Ai rev(Bi)>,0,n)
where <...> = <...>0 denote the scalar part.
 
linear independance :  
if a1 ^ a2 ^ ...^ ar != 0
 
basis (frame) :
If a blade Ar admits the decomposition Ar = a1 ^ a2 ^ . . . ^ ar , the set of vectors {ai ; i = 1, . . . , r} is said to be a frame (or basis) for Ar  (pseudoscalar).
A dual frame {ri} is defined by the equations ri · aj = δij.
If Ar is invertible, these equations can be solved for the ri, with the result
ri = (−1)**(i+1) (a1 ^ ... ^ai-1 ^ ai+1 ^ ... ^ ar)/Ar