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 consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system. For example, the standard basis for the Euclidean plane are the vectors

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

and the standard basis for three-dimensional space are the vectors

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

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}, {e₁, e₂, e₃}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors.

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 v_x, v_y, v_z 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 e_i 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 1_R, the unit in R.

In the context of geometric algebra with quadratic form Q : V → R, a standard basis refers to an orthogonal basis {e_i} of the generating vector space V for which each element is normalized so that Q(e_i) ∈ {−1, 0, +1}.

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

• Canonical units
• Examples of vector spaces#Generalized coordinate space

References

• Ryan, Patrick J. (1986). Euclidean and non-Euclidean geometry: an analytical approach. Cambridge; New York: Cambridge University Press. ISBN 0-521-27635-7.  (page 198)

• Schneider, Philip J.; Eberly, David H. (2003). Geometric tools for computer graphics. Amsterdam; Boston: Morgan Kaufmann Publishers. ISBN 1-55860-594-0.  (page 112)