Standard basis
|
|
In linear algebra, the standard basis or canonical basis for the <math>n<math>-dimensional coordinate space is the basis obtained by taking the <math>n<math> basis vectors <math>\{ e_j: 1\leq j\leq n\}<math> where <math>e_j<math> is the vector with a <math>1<math> in the <math>j<math>th coordinate and <math>0<math> 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
- <math>{(e_i)}_{i\in I}={({(\delta_{ij})}_{j\in I})}_{i\in I}<math>
where <math>I<math> is any set and <math>\delta_{ij}<math> 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)
- <math>R^{(I)}<math>
of all families
- <math>f=(f_i)<math>
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.