Standard Basis
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the standard basis (also called natural basis or
canonical basis In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: * In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the ...
) 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 can ...
(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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, the standard basis is formed by the vectors :\mathbf_x = (1,0),\quad \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 (geometry), position of an element (i.e., Point (m ...
\mathbb^3 is formed by vectors :\mathbf_x = (1,0,0),\quad \mathbf_y = (0,1,0),\quad \mathbf_z=(0,0,1). 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 v ...
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_xv_yv_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 coordi ...
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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
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 can ...
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 exa ...
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 expone ...
s and is commonly called monomial basis. 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 calle ...
of orthogonal
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 un ...
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( , \right) \, :v_2 = \left( , \right) \, are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, 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 exa ...
s in ''n'' indeterminates over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, 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 expone ...
s. All of the preceding are special cases of the family :_= ( (\delta_ )_ )_ 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=(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 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 essen ...
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 represent ...
of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
is established by the
Poincaré–Birkhoff–Witt theorem In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poi ...
. 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 r ...
, the standard basis vectors for a given Euclidean space are sometimes referred to as the
versors In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Wil ...
of the axes of the corresponding Cartesian coordinate system.


See also

*
Canonical units A canonical unit is a unit of measurement agreed upon as default in a certain context. In astrodynamics In astrodynamics, canonical units are defined in terms of some important object’s orbit that serves as a reference. In this system, a refere ...
*


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