- Standard basis
In
mathematics , the standard basis (also called natural basis or canonical basis) of the n-dimensionalEuclidean space Rn is the basis obtained by taking the n basis vectors:e_i : 1leq ileq n}where e_i is the vector with a 1 in the ithcoordinate 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 {i1, i2, i3}.
Properties
By definition, the standard basis is a
sequence oforthogonal unit vectors . In other words, it is an ordered and orthonormal basis.However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors,
:e_1 = left( {sqrt 3 over 2} , {1 over 2} ight) ,:e_2 = left( {1 over 2} , {-sqrt 3 over 2} ight) ,
are orthogonal unit vectors, but the orthonormal basis they form does not meet the definition of standard basis.
Generalizations
There is a "standard" basis also for the ring of
polynomial s in "n" indeterminates over a field, namely themonomial s.All of the preceding are special cases of the family
:e_i)}_{iin I}={({(delta_{ij})}_{jin I})}_{iin 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 onGrassmannian s. It is now a part ofrepresentation theory called "standard monomial theory". The idea of standard basis in theuniversal enveloping algebra of aLie algebra is established by thePoincaré-Birkhoff-Witt theorem .Gröbner bases are also sometimes called standard bases.
ee also
*
Examples of vector spaces#Generalized coordinate space References
*cite book
last = Ryan
first = Patrick J.
title = Euclidean and non-Euclidean geometry: an analytical approach
publisher = Cambridge; New York: Cambridge University Press
date = 1986
pages =
isbn = 0521276357 (page 198)*cite book
last = Schneider
first = Philip J.
coauthors = Eberly, David H.
title = Geometric tools for computer graphics
publisher = Amsterdam; Boston: Morgan Kaufmann Publishers
date = 2003
pages =
isbn = 1558605940 (page 112)
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