Editing Basis (linear algebra) (section)

== Examples ==
[[File:Basis graph (no label).svg|thumb|400px|This picture illustrates the [[standard basis]] in '''R'''<sup>2</sup>. The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is [[linearly dependent]] upon them.]]
The set {{math|'''R'''<sup>2</sup>}} of the [[ordered pair]]s of [[real number]]s is a vector space under the operations of component-wise addition
<math display="block">(a, b) + (c, d) = (a + c, b+d)</math> 
and scalar multiplication
<math display="block">\lambda (a,b) = (\lambda a, \lambda b),</math> 
where <math>\lambda</math> is any real number. A simple basis of this vector space consists of the two vectors {{math|1='''e'''<sub>1</sub> = (1, 0)}} and {{math|1='''e'''<sub>2</sub> = (0, 1)}}.  These vectors form a basis (called the [[standard basis]]) because any vector {{math|1='''v''' = (''a'', ''b'')}} of {{math|'''R'''<sup>2</sup>}} may be uniquely written as <math display="block">\mathbf v = a \mathbf e_1 + b \mathbf e_2.</math> Any other pair of linearly independent vectors of {{math|'''R'''<sup>2</sup>}}, such as {{math|(1, 1)}} and {{math|(−1, 2)}}, forms also a basis of {{math|'''R'''<sup>2</sup>}}.

More generally, if {{mvar|F}} is a [[field (mathematics)|field]], the set <math>F^n</math> of [[tuple|{{mvar|n}}-tuples]] of elements of {{mvar|F}} is a vector space for similarly defined addition and scalar multiplication. Let <math display="block">\mathbf e_i = (0, \ldots, 0,1,0,\ldots, 0)</math> be the {{mvar|n}}-tuple with all components equal to 0, except the {{mvar|i}}th, which is 1. Then <math>\mathbf e_1, \ldots, \mathbf e_n</math> is a basis of <math>F^n,</math> which is called the ''standard basis'' of <math>F^n.</math>

A different flavor of example is given by [[polynomial ring]]s.  If {{mvar|F}} is a field, the collection {{math|''F''[''X'']}} of all [[polynomial]]s in one [[indeterminate (variable)|indeterminate]] {{mvar|X}} with coefficients in {{mvar|F}} is an {{mvar|F}}-vector space.  One basis for this space is the [[monomial basis]] {{mvar|B}}, consisting of all [[monomial]]s: <math display="block">B=\{1, X, X^2, \ldots\}.</math> Any set of polynomials such that there is exactly one polynomial of each degree (such as the [[Bernstein polynomial|Bernstein basis polynomial]]s or [[Chebyshev polynomials]]) is also a basis. (Such a set of polynomials is called a [[polynomial sequence]].)  But there are also many bases for {{math|''F''[''X'']}} that are not of this form.