standard basis

In linear algebra, the standard basis for an

n

-dimensional vector space is the basis obtained by taking the

n

basis vectors

\{ e_j: 1\leq j\leq n\}

where

e_j

is the vector with a

1

in the

j

th 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. 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 Poincar-Birkhoff-Witt theorem.

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