Standard basis - Definition and Overview

In linear algebra, the standard basis for an <math>n<math>-dimensional vector 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. 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.

Example Usage of Standard