Editing Standard basis (section)

{{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>