Standard basis

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

${\displaystyle \mathbf {e} _{x}=(1,0),\quad \mathbf {e} _{y}=(0,1),}$

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

${\displaystyle \mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).}$

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

${\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},}$

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

In ${\displaystyle n}$-dimensional Euclidean space, the standard basis consists of n distinct vectors

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

where ei denotes the vector with a 1 in the ${\displaystyle i}$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,

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

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

${\displaystyle {(e_{i})}_{i\in I}=((\delta _{ij})_{j\in I})_{i\in I}}$

where ${\displaystyle I}$ is any set and ${\displaystyle \delta _{ij}}$ 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)

${\displaystyle R^{(I)}}$

of all families

${\displaystyle f=(f_{i})}$

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.