Standard basis
From Biocrawler, the free encyclopedia.
In linear algebra, the standard basis or canonical basis for the n-dimensional coordinate space is the basis obtained by taking the n basis vectors 
where ej is the vector with a 1 in the jth coordinate and 0 elsewhere. In many senses, it is the "obvious" basis.
Standard basis are perfectly localized in the sense that all but one element of each base are zero.
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 I is any set and δij 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)
- R(I)
of all families
- f = (fi)
from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, the unit in R.
Other usages
The existence of other 'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of representation theory called standard monomial theory. The idea of standard basis in the universal enveloping algebra of a Lie algebra is established by the Poincaré-Birkhoff-Witt theorem.
