Standard basis
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
\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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 \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 for standard-basis vectors, including , , , and . These vectors are sometimes written with a
hat A hat is a Headgear, 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 incorpor ...
to emphasize their status as
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of these. p. 39-40, §2.29 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 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s, whose definition involves coefficients, such as polynomials and matrices. 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 monomials 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 cal ...
of
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
unit vectors. In other words, it is an ordered and orthonormal 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 In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, 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 polynomials in ''n'' indeterminates over a field, namely the monomials. All of the preceding are special cases of the
indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, wher ...
_= ( (\delta_ )_ )_ where I is any set and \delta_ is the Kronecker delta, 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 ''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 which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of representation theory called ''standard monomial theory''. The idea of standard basis in the universal enveloping algebra 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 ident ...
is established by the Poincaré–Birkhoff–Witt theorem. Gröbner bases are also sometimes called standard bases. In
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, 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 *


Citations


References

* * (page 47) * (page 198) * (page 112) {{refend Linear algebra