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In mathematics, the standard basis (also called natural basis or canonical basis) of a
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 $\mathbb^n$ or $\mathbb^n$) 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 ...
$\mathbb^2$ formed by the pairs of real numbers, the standard basis is formed by the vectors :$\mathbf_x = \left(1,0\right),\quad \mathbf_y = \left(0,1\right).$ 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 ...
$\mathbb^3$ is formed by vectors :$\mathbf_x = \left(1,0,0\right),\quad \mathbf_y = \left(0,1,0\right),\quad \mathbf_z=\left(0,0,1\right).$ 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 ''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 :$v_x\,\mathbf_x + v_y\,\mathbf_y + v_z\,\mathbf_z,$ 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 $\mathbb R^n$, the standard basis consists of ''n'' distinct vectors :$\,$ where e''i'' denotes the vector with a 1 in the th coordinate and 0's elsewhere. Standard bases can be defined for other
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 ...
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 $\mathcal_$, 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 :$\mathbf_ = \begin 1 & 0 \\ 0 & 0 \end,\quad \mathbf_ = \begin 0 & 1 \\ 0 & 0 \end,\quad \mathbf_ = \begin 0 & 0 \\ 1 & 0 \end,\quad \mathbf_ = \begin 0 & 0 \\ 0 & 1 \end.$

# Properties

By definition, the standard basis is a
sequence 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. :$v_1 = \left\left( , \right\right) \,$ :$v_2 = \left\left( , \right\right) \,$ 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 of
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 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 :$_= \left( \left(\delta_ \right)_ \right)_$ where $I$ is any set and $\delta_$ 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 :$f=\left(f_i\right)$ 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 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 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.