# Standard basis

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

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

Here the vector **e**_{x} points in the *x* direction, the vector **e**_{y} points in the *y* direction, and the vector **e**_{z} points in the *z* direction. There are several common notations for these vectors, including {**e**_{x}, **e**_{y}, **e**_{z}}, {**e**_{1}, **e**_{2}, **e**_{3}}, {**i**, **j**, **k**}, and {**x**, **y**, **z**}. 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

the scalars *v*_{x}, *v*_{y}, *v*_{z} being the scalar components of the vector **v**.

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

where **e**_{i} denotes the vector with a 1 in the 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,

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

where is any set and 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)

of all families

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.

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.

## See also

## References

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