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:\{ e_i : 1\leq i\leq n\}
where e_j 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.

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 times the standard basis of R3 is denoted by {'''i''', '''j''', '''k'''}.


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 .


SEE ALSO