Standard basis

In mathematics, the standard basis (also called natural basis or canonical basis) of the $n$-dimensional Euclidean 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 $i$th 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 {i₁, i₂, i₃}.

Properties

By definition, the standard basis is a sequence of orthogonal 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 polynomials in "n" indeterminates over a field, namely the monomials.

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 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.

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)