In mathematics, the standard basis (also called natural basis or canonical basis) of a _{''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 _{''i''} denotes the vector with a 1 in the th coordinate and 0's elsewhere.
Standard bases can be defined for other

_{''R''}, the unit in ''R''.

coordinate vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...

(such as $\backslash mathbb^n$ or $\backslash mathbb^n$) is the set of vectors whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point (element of the plane), which includes affine notions of ...

$\backslash mathbb^2$ formed by the pairs of real numbers, the standard basis is formed by the vectors
:$\backslash mathbf\_x\; =\; (1,0),\backslash quad\; \backslash mathbf\_y\; =\; (0,1).$
Similarly, the standard basis for the three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

$\backslash mathbb^3$ is formed by vectors
:$\backslash mathbf\_x\; =\; (1,0,0),\backslash quad\; \backslash mathbf\_y\; =\; (0,1,0),\backslash quad\; \backslash mathbf\_z=(0,0,1).$
Here the vector enotations
''Notations'' is a book that was edited and compiled by American avant-garde composer John Cage (1912–1992) with Alison Knowles and first published in 1969 by Something Else Press. The book is made up of a large collection of graphical scores ...

for standard-basis vectors, including , , , and . These vectors are sometimes written with a hat
A hat is a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorporate mecha ...

to emphasize their status as unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vect ...

s (standard unit vectors).
These vectors are a basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
* Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...

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
:$v\_x\backslash ,\backslash mathbf\_x\; +\; v\_y\backslash ,\backslash mathbf\_y\; +\; v\_z\backslash ,\backslash mathbf\_z,$
the scalars $v\_x$, $v\_y$, $v\_z$ being the scalar components of the vector v.
In the -dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

Euclidean space $\backslash mathbb\; R^n$, the standard basis consists of ''n'' distinct vectors
:$\backslash ,$
where evector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...

s, whose definition involves coefficients, such as polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...

s and matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expo ...

s and is commonly called monomial basis
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely writ ...

. For matrices $\backslash mathcal\_$, the standard basis consists of the ''m''×''n''-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
:$\backslash mathbf\_\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \backslash end,\backslash quad\; \backslash mathbf\_\; =\; \backslash begin\; 0\; \&\; 1\; \backslash \backslash \; 0\; \&\; 0\; \backslash end,\backslash quad\; \backslash mathbf\_\; =\; \backslash begin\; 0\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \backslash end,\backslash quad\; \backslash mathbf\_\; =\; \backslash begin\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end.$
Properties

By definition, the standard basis is asequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...

of orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

unit vectors
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vec ...

. In other words, it is an ordered and orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...

basis.
However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e.
:$v\_1\; =\; \backslash left(\; ,\; \backslash right)\; \backslash ,$
:$v\_2\; =\; \backslash left(\; ,\; \backslash right)\; \backslash ,$
are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

, so the basis with these vectors does not meet the definition of standard basis.
Generalizations

There is a ''standard'' basis also for the ring ofpolynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...

s in ''n'' indeterminates over a field, namely the monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expo ...

s.
All of the preceding are special cases of the family
:$\_=\; (\; (\backslash delta\_\; )\_\; )\_$
where $I$ is any set and $\backslash delta\_$ is the Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...

, equal to zero whenever and equal to 1 if .
This family is the ''canonical'' basis of the ''R''-module ( free module)
:$R^$
of all families
:$f=(f\_i)$
from ''I'' into a ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...

''R'', which are zero except for a finite number of indices, if we interpret 1 as 1Other usages

The existence of other 'standard' bases has become a topic of interest inalgebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, beginning with work of Hodge from 1943 on Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all - dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...

s. It is now a part of 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 studies modules over these abstract algebraic structures. In esse ...

called ''standard monomial theory''. The idea of standard basis in the universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representa ...

of a Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

is established by the Poincaré–Birkhoff–Witt theorem.
Gröbner bases are also sometimes called standard bases.
In physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...

, 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

* Canonical units *References

* (page 198) *{{cite book , last = Schneider , first = Philip J. , author2=Eberly, David H. , title = Geometric tools for computer graphics , publisher = Amsterdam; Boston: Morgan Kaufmann Publishers , date = 2003 , pages = , isbn = 1-55860-594-0 (page 112) Linear algebra