In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the standard basis (also called natural basis or canonical basis) for a Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

. For example, the standard basis for the Euclidean plane are the vectors
and the standard basis for three-dimensional space

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

 are the vectors
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

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...

 for these vectors, including {e_x, e_y, e_z}, {e₁, e₂, e₃}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written with a hat

Caret

Caret usually refers to the spacing symbol ^ in ASCII and other character sets. In Unicode, however, the corresponding character is , whereas the Unicode character named caret is actually a similar but lowered symbol: ....

 to emphasize their status as unit vectors.

These vectors are a basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

 in the sense that any other vector can be expressed uniquely as a linear combination

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

 of these. For example, every vector v in three-dimensional space can be written uniquely as
the scalars

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

 v_x, v_y, v_z being the scalar components of the vector v.

In -dimensional Euclidean space, there are n different standard basis vectors
where e_i denotes the vector with a 1 in the th coordinate and 0's elsewhere.

Properties

By definition, the standard basis is a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 of orthogonal unit vectors. In other words, it is an ordered and orthonormal

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

 basis.

However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors,


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 polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s in n indeterminates over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, namely the monomial

Monomial

In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s.

All of the preceding are special cases of the family


where is any set and 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

Free module

In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...

)


of all families


from I into a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 R, which are zero except for a finite number of indices, if we interpret 1 as 1_R, the unit in R.

Other usages

The existence of other 'standard' bases has become a topic of interest in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, beginning with work of Hodge

W. V. D. Hodge

William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...

 from 1943 on Grassmannian

Grassmannian

In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

s. It is now a part of representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

 called standard monomial theory. The idea of standard basis in the universal enveloping algebra

Universal enveloping algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...

 of a Lie algebra

Lie algebra

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 algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 is established by the Poincaré-Birkhoff-Witt theorem.

Gröbner bases

Gröbner basis

In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...

are also sometimes called standard bases.