Standard basis

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In mathematics, the standard basis or natural basis or canonical basis for the $n$ -dimensional coordinate space is the basis obtained by taking the $n$ basis vectors

$\{ e_i : 1\leq i\leq n\}$

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.

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 $x y z$ -coordinates. Often the standard basis of R³ is denoted by {i, j, k}.

[edit] 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

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

where $I$ is any set and $δ i j$ 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.

[edit] 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.