# Standard basis

Not to be confused with a kind of Gröbner basis introduced by Heisuke Hironaka for power series rings.
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j, and k.

In mathematics, the standard basis (also called natural basis or canonical basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system. For example, the standard basis for the Euclidean plane is formed by vectors

$\mathbf{e}_x = (1,0),\quad \mathbf{e}_y = (0,1),$

and the standard basis for three-dimensional space is formed by vectors

$\mathbf{e}_x = (1,0,0),\quad \mathbf{e}_y = (0,1,0),\quad \mathbf{e}_z=(0,0,1).$

Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for these vectors, including {exeyez}, {e1e2e3}, {ijk}, and {xyz}. These vectors are sometimes written with a hat to emphasize their status as unit vectors. Each of these vectors is sometimes referred to as the versor of the corresponding Cartesian axis.

These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as

$v_x\,\mathbf{e}_x + v_y\,\mathbf{e}_y + v_z\,\mathbf{e}_z,$

the scalars vxvyvz being the scalar components of the vector v.

In $n$-dimensional Euclidean space, the standard basis consists of n distinct vectors

$\{ \mathbf{e}_i : 1\leq i\leq n\},$

where ei denotes the vector with a 1 in the $i$th coordinate and 0's elsewhere.

## 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,

$v_1 = \left( {\sqrt 3 \over 2} , {1 \over 2} \right) \,$
$v_2 = \left( {1 \over 2} , {-\sqrt 3 \over 2} \right) \,$

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 $\delta_{ij}$ 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 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.

In physics, the standard basis vectors for a given Euclidean space are sometimes referred to as the versors of the axes of the corresponding Cartesian coordinate system.