Standard Basis - Generalizations

Generalizations

There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials.

All of the preceding are special cases of the family

where is any set and is the Kronecker delta, equal to zero whenever i≠j and equal to 1 if i=j. This family is the canonical basis of the R-module (free module)

of all families

from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1_R, the unit in R.

In the context of geometric algebra with quadratic form Q : V → R, a standard basis refers to an orthogonal basis {e_i} of the generating vector space V for which each element is normalized so that Q(e_i) ∈ {−1, 0, +1}.