Standard basis

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In mathematics, the standard basis (also called natural basis or canonical basis) of the $n$ -dimensional Euclidean space Rⁿ is the basis obtained by taking the $n$ basis vectors

$\{ e_i : 1\leq i\leq n\}$

where $e i$ is the vector with a $1$ in the $i$ th coordinate and $0$ elsewhere. In many ways, it is the "obvious" basis.

For example, the standard basis for R³ is given by the three vectors

$e_1 = (1, 0, 0)\,$

$e_2 = (0, 1, 0)\,$

$e_3 = (0, 0, 1)\,$

Coordinates with respect to this basis are the usual $x y z$ -coordinates. Often the standard basis of R³ is denoted by {i, j, k} or {i₁, i₂, i₃}.

1 Properties
2 Generalizations
3 Other usages
4 See also
5 References

[edit] 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,

$e_1 = (0.866, 0.5)\,$

$e_2 = (0.5, -0.866)\,$

are orthogonal unit vectors, but the orthonormal basis they form does not meet the definition of standard basis.

[edit] 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 $δ i j$ 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.

[edit] 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.

[edit] See also

Examples of vector spaces#Generalized coordinate space

[edit] References

Ryan, Patrick J. (1986). Euclidean and non-Euclidean geometry: an analytical approach. Cambridge; New York: Cambridge University Press. ISBN 0521276357. (page 198)

Schneider, Philip J.; Eberly, David H. (2003). Geometric tools for computer graphics. Amsterdam; Boston: Morgan Kaufmann Publishers. ISBN 1558605940. (page 112)