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[[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'''.]]
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In [[mathematics]], the '''standard basis''' (also called '''natural basis''' or '''canonical basis''') for a [[Euclidean space]] is the set of [[unit vector]]s pointing in the direction of the axes of a [[Cartesian coordinate system]]. For example, the standard basis for the [[Euclidean plane]] is formed by vectors
:<math>\mathbf{e}_x = (1,0),\quad \mathbf{e}_y = (0,1),</math>
and the standard basis for [[three-dimensional space]] is formed by vectors
:<math>\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 these 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 vectors. Each of these vectors is sometimes referred to as the [[Versor#Definition in geometry and physics|versor]] of the corresponding Cartesian axis.


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. For example, every vector '''v''' in three-dimensional space can be written uniquely as
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:<math>v_x\,\mathbf{e}_x + v_y\,\mathbf{e}_y + v_z\,\mathbf{e}_z,</math>
the [[scalar (mathematics)|scalars]] ''v''<sub>''x''</sub>,&nbsp;''v''<sub>''y''</sub>,&nbsp;''v''<sub>''z''</sub> being the [[scalar component]]s of the vector '''v'''.
 
In <math>n</math>-[[dimension (linear algebra)|dimensional]] Euclidean space, the standard basis consists of ''n'' distinct vectors
:<math>\{ \mathbf{e}_i : 1\leq i\leq n\},</math>
where '''e'''<sub>''i''</sub> denotes the vector with a 1 in the <math>i</math>th [[coordinate]] and 0's elsewhere.
 
== 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,
 
:<math>v_1 = \left( {\sqrt 3 \over 2} , {1 \over 2} \right) \,</math>
:<math>v_2 = \left( {1 \over 2} , {-\sqrt 3 \over 2} \right) \,</math>
 
are orthogonal unit vectors, but the orthonormal basis they form 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 family
 
:<math>{(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 ''i≠j'' and equal to 1 if ''i=j''.
This family is the ''canonical'' basis of the ''R''-module ([[free module]])
 
:<math>R^{(I)}</math>
 
of all families
 
:<math>f=(f_i)</math>
 
from ''I'' into a [[ring (mathematics)|ring]] ''R'', which are zero except for a finite number of indices, if we interpret 1 as 1<sub>''R''</sub>, the unit in ''R''.
 
==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]]
*[[Examples of vector spaces#Generalized coordinate space]]
 
==References==
*{{cite book
| last      = Ryan
| first      = Patrick J.
| title      = Euclidean and non-Euclidean geometry: an analytical approach
| publisher  = Cambridge; New York: Cambridge University Press
| date      = 1986
| pages      =
| isbn      = 0-521-27635-7
}} (page 198)
 
*{{cite book
| last      = Schneider
| first      = Philip J.
| coauthors  = 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)
 
[[Category:Linear algebra]]

Latest revision as of 19:02, 24 December 2014

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