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 : 1\leq i\leq n\}

where ei 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 {i1, i2, i3}.

Contents

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} \right) \,
e_2 = \left( {1 \over 2} , {-\sqrt 3 \over 2} \right) \,

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

where I is any set and δij 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 = (fi)

from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, 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.

See also

References