View source for Standard basis

{{Short description|Vectors whose components are all 0 except one that is 1}}
{{no footnotes|date=July 2016}}
{{broader|Canonical basis}}
{{distinguish|text=another name for a [[Gröbner basis]]}}
[[File:3D Vector.svg|right|thumb|300px|Every vector '''a''' in three dimensions is a [[linear combination]] of the standard basis vectors '''i''', '''j''' and '''k'''.]]
In [[mathematics]], the '''standard basis''' (also called '''natural basis''' or '''[[canonical basis]]''') of a [[coordinate vector space]] (such as <math>\mathbb{R}^n</math> or <math>\mathbb{C}^n</math>) is the set of vectors, each of whose components are all zero, except one that equals 1.{{sfn|Roman|2008|p=47|loc=ch. 1}} For example, in the case of the [[Euclidean plane]] <math>\mathbb{R}^2</math> formed by the pairs {{math|(''x'', ''y'')}} of [[real number]]s, the standard basis is formed by the vectors
<math display="block">\mathbf{e}_x = (1,0),\quad \mathbf{e}_y = (0,1).</math>
Similarly, the standard basis for the [[three-dimensional space]] <math>\mathbb{R}^3</math> is formed by vectors
<math display="block">\mathbf{e}_x = (1,0,0),\quad \mathbf{e}_y = (0,1,0),\quad \mathbf{e}_z=(0,0,1).</math>

Here the vector '''e'''<sub>''x''</sub> points in the ''x'' direction, the vector '''e'''<sub>''y''</sub> points in the ''y'' direction, and the vector '''e'''<sub>''z''</sub> points in the ''z'' direction.  There are several common [[mathematical notation|notations]] for standard-basis vectors, including {'''e'''<sub>''x''</sub>,&nbsp;'''e'''<sub>''y''</sub>,&nbsp;'''e'''<sub>''z''</sub>}, {'''e'''<sub>1</sub>,&nbsp;'''e'''<sub>2</sub>,&nbsp;'''e'''<sub>3</sub>}, {'''i''',&nbsp;'''j''',&nbsp;'''k'''}, and {'''x''',&nbsp;'''y''',&nbsp;'''z'''}. These vectors are sometimes written with a [[circumflex|hat]] to emphasize their status as [[unit vector]]s ('''standard unit vectors''').

These vectors are a [[basis (linear algebra)|basis]] in the sense that any other vector can be expressed uniquely as a [[linear combination]] of these.<ref>{{Harvp|Axler|2015}} p. 39-40, §2.29</ref> For example, every vector '''v''' in three-dimensional space can be written uniquely as
<math display="block">v_x\,\mathbf{e}_x + v_y\,\mathbf{e}_y + v_z\,\mathbf{e}_z,</math>
the [[scalar (mathematics)|scalars]] <math>v_x</math>,&nbsp;<math>v_y</math>,&nbsp;<math>v_z</math> being the [[scalar component]]s of the vector '''v'''.

In the {{mvar|n}}-[[dimension (linear algebra)|dimensional]] Euclidean space <math>\mathbb R^n</math>, the standard basis consists of ''n'' distinct vectors
<math display="block">\{ \mathbf{e}_i : 1\leq i\leq n\},</math>
where '''e'''<sub>''i''</sub> denotes the vector with a 1 in the {{mvar|i}}th [[coordinate]] and 0's elsewhere.

Standard bases can be defined for other [[vector space]]s, whose definition involves [[Coefficient|coefficients]], such as [[polynomial]]s and [[matrix (mathematics)|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 [[monomial]]s and is commonly called [[monomial basis]]. For matrices <math>\mathcal{M}_{m \times n}</math>, 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
<math display="block">\mathbf{e}_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\quad
       \mathbf{e}_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\quad
       \mathbf{e}_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},\quad
       \mathbf{e}_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.</math>

== Properties ==
By definition, the standard basis is a [[sequence]] of [[orthogonal]] [[unit vectors]]. In other words, it is an [[ordered basis|ordered]] and [[orthonormal basis|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. ,
<math display="block">v_1 = \left( {\sqrt 3 \over 2} ,  {1 \over 2} \right) \,</math>
<math display="block">v_2 = \left( {1 \over 2} , {-\sqrt 3 \over 2} \right) \,</math>
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]]s in ''n'' indeterminates over a [[field (mathematics)|field]], namely the [[monomial]]s.

All of the preceding are special cases of the [[indexed family]]
<math display="block">{(e_i)}_{i\in I}= ( (\delta_{ij} )_{j \in I} )_{i \in I}</math> 
where <math>I</math> is any set and <math>\delta_{ij}</math> is the [[Kronecker delta]], equal to zero whenever {{nowrap|''i'' ≠ ''j''}} and equal to 1 if {{nowrap|1=''i'' = ''j''}}.
This family is the ''canonical'' basis of the ''R''-module ([[free module]])
<math display="block">R^{(I)}</math>
of all families
<math display="block">f=(f_i)</math> 
from ''I'' into a [[ring (mathematics)|ring]] ''R'', which are [[finite support|zero except for a finite number of indices]], if we interpret 1 as 1<sub>''R''</sub>, the [[Unit (ring theory)|unit]] in ''R''.{{sfn|Roman|2008|p=131|loc=ch. 5}}

==Other usages==
The existence of other 'standard' bases has become a topic of interest in [[algebraic geometry]], beginning with work of [[W. V. D. Hodge|Hodge]] from 1943 on [[Grassmannian]]s. 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]] is established by the [[Poincaré–Birkhoff–Witt theorem]].

[[Gröbner basis|Gröbner bases]] are also sometimes called standard bases.

In [[physics]], the standard basis vectors for a given Euclidean space are sometimes referred to as the [[Versor (physics)|versors]] of the axes of the corresponding Cartesian coordinate system.

==See also==
*[[Canonical units]]
*{{section link|Examples of vector spaces|Generalized coordinate space}}

==Citations==
{{Reflist}}

==References==
{{refbegin}}
* {{ cite book
 |last=Axler
 |first=Sheldon
 |title=Linear Algebra Done Right
 |volume=
 |pages=
 |publication-date=2015
 |series=[[Undergraduate Texts in Mathematics]]
 |orig-date=18 December 2014 
 |edition=3rd
 |publisher=[[Springer Publishing]]
 |isbn=978-3-319-11079-0|author-link=Sheldon Axler}}
*{{cite book
 | last       = Roman
 | first      = Stephen
 | title      = Advanced Linear Algebra
 | edition    = Third
 | series     =[[Graduate Texts in Mathematics]] 
 | publisher  = Springer
 | date       = 2008
 | pages      = 
 | isbn       = 978-0-387-72828-5
 | author-link = Steven Roman
}} (page 47)
*{{cite book
 | last       = Ryan
 | first      = Patrick J.
 | title      = Euclidean and non-Euclidean geometry: an analytical approach
 | publisher  = Cambridge; New York: Cambridge University Press
 | date       = 2000
 | pages      = 
 | isbn       = 0-521-27635-7
}} (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)
{{refend}}

[[Category:Linear algebra]]