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:\{ e_i : 1\leq i\leq n\}
where e_i is the vector with a 1 in the ith Coordinate and 0 elsewhere. In many ways, it is the "obvious" basis.

For example, the standard basis for R3 is given by the three vectors
:e_1 = (1,0,0)\,
:e_2 = (0,1,0)\,
:e_3 = (0,0,1)\,
Coordinates with respect to this basis are the usual xyz-coordinates. Often the standard basis of R3 is denoted by {'''i''', '''j''', '''k'''} or {'''i'''1, '''i'''2, '''i'''3}.


GENERALIZATIONS


There is a ''standard'' basis also for the ring of Polynomial s in ''n'' indeterminates over a Field , namely the Monomial s.

All of the preceding are special cases of the family

:{(e_i)}_{i\in I}={({(\delta_{ij})}_{j\in I})}_{i\in I}

where I is any set and \delta_{ij} 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=(f_i)

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''.


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 Grassmannian s. 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 .

Gröbner Bases are also sometimes called standard bases.


SEE ALSO



REFERENCES