An associative algebra is a vector space V with a new binary operator called multiplication written by juxtaposing two vector expressions.
The operator is required to be associative and bilinear.
If x, y and z denote arbitrary vectors then associativity means
x(yz) = (xy)z
and bilinearity means that (for any x, if Fy = xy then F is a linear operator) and conversely.
In formulae bilinearity means:
x(y+z) = xy + xz
(x+y)z = xz + yz
(αx)y = x(αy) = α(xy)
where α denotes any element of the field of the vector space.
Commutivity (xy = yx) is not required.
There may be a vector 1 such that for all x in the algebra 1x = x1 = x. This is called an algebra with one. All of the algebras at this site have one.
If the vector space is n-dimensional and ei are n basis vectors, then it suffices to know the n2 eiej’s to know all vector products. Indeed for some set of n3 field elements ckij, eiej = Σkckijek. For any such n3 field elements there is an algebra defined by this equation, but it may not be associative. The octonions are a non associative algebra. The c’s determine the algebra up to isomorphism. Of course the c’s depend on choice of basis vectors.
An arbitrary element x of V can be expressed
x = Σkxkek
where xk are n field values.
xy = ΣkxkekΣjyjej
and thus the c’s determine multiplication for all vector pairs.
An associative algebra is a division algebra whenever
for all x, if x ≠ 0 (the zero vector) then (for all z there exists a y such that xy = z).
Here x, y and z range over the associative algebra (vector space).
xy = Σi(ΣkΣjcikjxkyj)ei.
y will exist just in case the determinant of this linear equation is not 0 for each choice of non-zero x and z.
z = Σkzkek.
When x = Σkxkek
and z = Σkzkek
xy = z => Σi(ΣkΣjcikjxkyj)ei = Σiziei
Which by the nature of basis sets such as ei means that
ΣkΣjcikjxkyj = zi for each i.
If we take dij = Σkcikjxk
the determinant of d must not be 0. For some associative algebras where will be non zero vectors x for which the determinant is zero and then it is not a division algebra.