Standard basis

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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 ith coordinate and 0's elsewhere.

Properties[edit]

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[edit]

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[edit]

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[edit]

References[edit]

  • 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)