Standard basis - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the dimension of a Vector space V is the cardinality (i Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point 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

Properties

By definition, the standard basis is a sequence of orthogonal unit vectors. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length In other words, it is an ordered and orthonormal basis. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, an orthonormal basis of an Inner product space V (i

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.

Generalizations

There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the word monomial means two different things in the context of Polynomials The first meaning is a product of powers of Variables

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. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two 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 Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

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. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with William Vallance Douglas Hodge FRS ( 17 June 1903 - 7 July 1975) was a Scottish Mathematician, specifically a Geometer. In Mathematics, a Grassmannian is a space which parameterizes all Linear subspaces of a Vector space V of a given Dimension. It is now a part of representation theory called standard monomial theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of The idea of standard basis in the universal enveloping algebra of a Lie algebra is established by the Poincaré-Birkhoff-Witt theorem. In Mathematics, for any Lie algebra L one can construct its universal enveloping algebra U ( L) In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In the theory of Lie algebras the Poincaré–Birkhoff–Witt theorem (stated by Henri Poincaré (1900 and proved by Garrett Birkhoff (1937 and

Gröbner bases are also sometimes called standard bases. In Computer algebra, computational Algebraic geometry, and computational Commutative algebra, a Gröbner basis is a particular kind of generating subset

References

Ryan, Patrick J. This page lists some examples of vector spaces. See Vector space for the definitions of terms used on this page (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)

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Contents

Properties

Generalizations

Other usages

See also

References