View source for Basis (linear algebra)

{{short description|Set of vectors used to define coordinates}}
[[File:3d two bases same vector.svg|130px|thumb|The same vector can be represented in two different bases (purple and red arrows).]]
In [[Mathematics|mathematics]], a [[Set (mathematics)|set]] {{mvar|B}} of vectors in a [[Vector space|vector space]] {{math|''V''}} is called a '''basis''' ({{plural form}}: '''bases''') if every element of {{math|''V''}} may be written in a unique way as a finite [[Linear combination|linear combination]] of elements of {{mvar|B}}. The coefficients of this linear combination are referred to as '''components''' or '''coordinates''' of the vector with respect to {{mvar|B}}. The elements of a basis are called '''{{visible anchor|basis vectors}}'''.

Equivalently, a set {{mvar|B}} is a basis if its elements are linearly independent and every element of {{mvar|V}} is a [[Linear combination|linear combination]] of elements of {{mvar|B}}.<ref>{{cite book |last=Halmos |first=Paul Richard |year=1987 |title=Finite-Dimensional Vector Spaces |edition=4th |publisher=Springer |location=New York |url=https://books.google.com/books?id=mdWeEhA17scC&pg=PA10 |page=10 |isbn=978-0-387-90093-3 }}</ref> In other words, a basis is a linearly independent spanning set.

A vector space can have several bases; however all the bases have the same number of elements, called the [[Dimension (vector space)|''dimension'' of the vector space]].

This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

== Definition ==
A '''basis''' {{math|''B''}} of a [[Vector space|vector space]] {{math|''V''}} over a [[Field (mathematics)|field]] {{math|''F''}} (such as the real numbers {{math|'''R'''}} or the [[Complex number|complex number]]s {{math|'''C'''}}) is a linearly independent [[Subset|subset]] of {{math|''V''}} that [[Linear span|span]]s {{math|''V''}}. This means that a subset {{mvar|B}} of {{math|''V''}} is a basis if it satisfies the two following conditions:
;''linear independence''
: for every [[Finite set|finite]] subset <math>\{\mathbf v_1, \dotsc, \mathbf v_m\}</math> of {{mvar|B}}, if <math>c_1 \mathbf v_1 + \cdots + c_m \mathbf v_m = \mathbf 0</math> for some <math>c_1,\dotsc,c_m</math> in {{math|''F''}}, then {{nowrap|<math>c_1 = \cdots = c_m = 0</math>;}}
;''spanning property''
: for every vector {{math|'''v'''}} in {{math|''V''}}, one can choose <math>a_1,\dotsc,a_n</math> in {{math|''F''}} and <math>\mathbf v_1, \dotsc, \mathbf v_n</math> in {{mvar|B}} such that {{nowrap|<math>\mathbf v = a_1 \mathbf v_1 + \cdots + a_n \mathbf v_n</math>.}}

The [[Scalar (mathematics)|scalar]]s <math>a_i</math> are called the coordinates of the vector {{math|'''v'''}} with respect to the basis {{math|''B''}}, and by the first property they are uniquely determined.

A vector space that has a [[Finite set|finite]] basis is called [[Dimension (vector space)|finite-dimensional]]. In this case, the finite subset can be taken as {{math|''B''}} itself to check for linear independence in the above definition.

It is often convenient or even necessary to have an [[Total order|ordering]] on the basis vectors, for example, when discussing [[Orientation (vector space)|orientation]], or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an '''ordered basis''', which is therefore not simply an unstructured [[Set (mathematics)|set]], but a [[Sequence|sequence]], an [[Indexed family|indexed family]], or similar; see {{slink||Ordered bases and coordinates}} below.

== Examples ==
[[File:Basis graph (no label).svg|thumb|400px|This picture illustrates the [[Standard basis|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|ordered pair]]s of [[Real number|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|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|polynomial ring]]s.  If {{mvar|F}} is a field, the collection {{math|''F''[''X'']}} of all [[Polynomial|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|monomial basis]] {{mvar|B}}, consisting of all [[Monomial|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|polynomial sequence]].)  But there are also many bases for {{math|''F''[''X'']}} that are not of this form.

== Properties ==

Many properties of finite bases result from the [[Steinitz exchange lemma]], which states that, for any vector space {{mvar|V}}, given a finite spanning set {{mvar|S}} and a linearly independent set {{mvar|L}} of {{mvar|n}} elements of {{mvar|V}}, one may replace {{mvar|n}} well-chosen elements of {{mvar|S}} by the elements of {{mvar|L}} to get a spanning set containing {{mvar|L}}, having its other elements in {{mvar|S}}, and having the same number of elements as {{mvar|S}}.

Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require the [[Axiom of choice|axiom of choice]] or a weaker form of it, such as the ultrafilter lemma.

If {{mvar|V}} is a vector space over a field {{mvar|F}}, then:
* If {{mvar|L}} is a linearly independent subset of a spanning set {{math|''S'' ⊆ ''V''}}, then there is a basis {{mvar|B}} such that <math display="block">L\subseteq B\subseteq S.</math>
* {{mvar|V}} has a basis (this is the preceding property with {{mvar|L}} being the [[Empty set|empty set]], and {{math|1=''S'' = ''V''}}).
* All bases of {{mvar|V}} have the same [[Cardinality|cardinality]], which is called the [[Dimension (vector space)|dimension]] of {{mvar|V}}. This is the [[Dimension theorem for vector spaces|dimension theorem]].
* A generating set {{mvar|S}} is a basis of {{mvar|V}} if and only if it is minimal, that is, no [[Subset|proper subset]] of {{mvar|S}} is also a generating set of {{mvar|V}}.
* A linearly independent set {{mvar|L}} is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.

If {{mvar|V}} is a vector space of dimension {{mvar|n}}, then:
* A subset of {{mvar|V}} with {{mvar|n}} elements is a basis if and only if it is linearly independent.
* A subset of {{mvar|V}} with {{mvar|n}} elements is a basis if and only if it is a spanning set of {{mvar|V}}.

== Coordinates {{anchor|Ordered bases and coordinates}} ==

Let {{mvar|V}} be a vector space of finite dimension {{mvar|n}} over a field {{mvar|F}}, and 
<math display="block">B = \{\mathbf b_1, \ldots, \mathbf b_n\}</math>
be a basis of {{mvar|V}}. By definition of a basis, every {{math|'''v'''}} in {{mvar|V}} may be written, in a unique way, as
<math display="block">\mathbf v = \lambda_1 \mathbf b_1 + \cdots + \lambda_n \mathbf b_n,</math>
where the coefficients <math>\lambda_1, \ldots, \lambda_n</math> are scalars (that is, elements of {{mvar|F}}), which are called the ''coordinates'' of {{math|'''v'''}} over {{mvar|B}}. However, if one talks of the ''set'' of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same ''set'' of coefficients. For example, <math>3 \mathbf b_1 + 2 \mathbf b_2</math> and <math>2 \mathbf b_1 + 3 \mathbf b_2</math> have the same set of coefficients {{math|{2, 3}<nowiki/>}}, and are different. It is therefore often convenient to work with an '''ordered basis'''; this is typically done by [[Index set|indexing]] the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis is also called a '''frame''', a word commonly used, in various contexts, for referring to a sequence of data allowing defining coordinates.

Let, as usual, <math>F^n</math> be the set of the [[Tuple|{{mvar|n}}-tuples]] of elements of {{mvar|F}}. This set is an {{mvar|F}}-vector space, with addition and scalar multiplication defined component-wise. The map 
<math display="block">\varphi: (\lambda_1, \ldots, \lambda_n) \mapsto \lambda_1 \mathbf b_1 + \cdots + \lambda_n \mathbf b_n</math>
is a linear isomorphism from the vector space <math>F^n</math> onto {{mvar|V}}. In other words, <math>F^n</math> is the [[Coordinate space|coordinate space]] of {{mvar|V}}, and the {{mvar|n}}-tuple <math>\varphi^{-1}(\mathbf v)</math> is the [[Coordinate vector|coordinate vector]] of {{math|'''v'''}}.

The inverse image by <math>\varphi</math> of <math>\mathbf b_i</math> is the {{mvar|n}}-tuple <math>\mathbf e_i</math> all of whose components are 0, except the {{mvar|i}}th that is 1. The <math>\mathbf e_i</math> form an ordered basis of <math>F^n</math>, which is called its [[Standard basis|standard basis]] or [[Canonical basis|canonical basis]]. The ordered basis {{mvar|B}} is the image by <math>\varphi</math> of the canonical basis of {{nowrap|<math>F^n</math>.}}

It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of {{nowrap|<math>F^n</math>,}} and that every linear isomorphism from <math>F^n</math> onto {{mvar|V}} may be defined as the isomorphism that maps the canonical basis of <math>F^n</math> onto a given ordered basis of {{mvar|V}}. In other words, it is equivalent to define an ordered basis of {{mvar|V}}, or a linear isomorphism from <math>F^n</math> onto {{mvar|V}}.

== Change of basis ==
{{main|Change of basis}}
Let {{math|''V''}} be a vector space of dimension {{mvar|n}} over a field {{math|''F''}}. Given two (ordered) bases <math>B_\text{old} = (\mathbf v_1, \ldots, \mathbf v_n)</math> and <math>B_\text{new} = (\mathbf w_1, \ldots, \mathbf w_n)</math> of {{math|''V''}}, it is often useful to express the coordinates of a vector {{mvar|x}} with respect to <math>B_\mathrm{old}</math> in terms of the coordinates with respect to <math>B_\mathrm{new}.</math> This can be done by the ''change-of-basis formula'', that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to <math>B_\mathrm{old}</math> and <math>B_\mathrm{new}</math> as the ''old basis'' and the ''new basis'', respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has [[Expression (mathematics)|expressions]] involving the old coordinates, and if one wants to obtain equivalent expressions in terms of the new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates.

Typically, the new basis vectors are given by their coordinates over the old basis, that is, 
<math display="block">\mathbf w_j = \sum_{i=1}^n a_{i,j} \mathbf v_i.</math>
If <math>(x_1, \ldots, x_n)</math> and <math>(y_1, \ldots, y_n)</math> are the coordinates of a vector {{math|'''x'''}} over the old and the new basis respectively, the change-of-basis formula is 
<math display="block">x_i = \sum_{j=1}^n a_{i,j}y_j,</math>
for {{math|1=''i'' = 1, ..., ''n''}}.

This formula may be concisely written in [[Matrix (mathematics)|matrix]] notation. Let {{mvar|A}} be the matrix of the {{nowrap|<math>a_{i,j}</math>,}} and
<math display="block">X= \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} \quad \text{and} \quad Y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}</math>
be the column vectors of the coordinates of {{math|'''v'''}} in the old and the new basis respectively, then the formula for changing coordinates is
<math display="block">X = A Y.</math>

The formula can be proven by considering the decomposition of the vector {{math|'''x'''}} on the two bases: one has 
<math display="block">\mathbf x = \sum_{i=1}^n x_i \mathbf v_i,</math>
and
<math display="block">\mathbf x =\sum_{j=1}^n y_j \mathbf w_j
= \sum_{j=1}^n y_j\sum_{i=1}^n a_{i,j}\mathbf v_i
= \sum_{i=1}^n \biggl(\sum_{j=1}^n a_{i,j}y_j\biggr)\mathbf v_i.</math>

The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here {{nowrap|<math>B_\text{old}</math>;}} that is
<math display="block">x_i = \sum_{j=1}^n a_{i,j} y_j,</math>
for {{math|1=''i'' = 1, ..., ''n''}}.

== Related notions ==
===Free module===
{{main|Free module|Free abelian group}} 
If one replaces the field occurring in the definition of a vector space by a [[Ring (mathematics)|ring]], one gets the definition of a [[Module (mathematics)|module]]. For modules, [[Linear independence|linear independence]] and spanning sets are defined exactly as for vector spaces, although "[[Generating set of a module|generating set]]" is more commonly used than that of "spanning set".

Like for vector spaces, a ''basis'' of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not every module has a basis. A module that has a basis is called a ''free module''. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions.

A module over the integers is exactly the same thing as an [[Abelian group|abelian group]]. Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of a free abelian group is a free abelian group, and, if {{mvar|G}} is a subgroup of a finitely generated free abelian group {{mvar|H}} (that is an abelian group that has a finite basis), then there is a basis <math>\mathbf e_1, \ldots, \mathbf e_n</math> of {{mvar|H}} and an integer {{math|0 ≤ ''k'' ≤ ''n''}} such that <math>a_1 \mathbf e_1, \ldots, a_k \mathbf e_k</math> is a basis of {{mvar|G}}, for some nonzero integers {{nowrap|<math>a_1, \ldots, a_k</math>.}} For details, see {{slink|Free abelian group|Subgroups}}.

=== Analysis ===
In the context of infinite-dimensional vector spaces over the real or complex numbers, the term '''{{visible anchor|Hamel basis}}''' (named after [[Biography:Georg Hamel|Georg Hamel]]<ref>{{Harvnb|Hamel|1905}}</ref>) or '''algebraic basis''' can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are [[Orthogonal basis|orthogonal bases]] on [[Hilbert space]]s, [[Schauder basis|Schauder bases]], and [[Markushevich basis|Markushevich bases]] on normed linear spaces. In the case of the real numbers '''R''' viewed as a vector space over the field '''Q''' of rational numbers, Hamel bases are uncountable, and have specifically the [[Cardinality|cardinality]] of the continuum, which is the [[Cardinal number|cardinal number]] {{nowrap|<math>2^{\aleph_0}</math>,}} where <math>\aleph_0</math> (aleph-nought) is the smallest infinite cardinal, the cardinal of the integers.

The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for [[Topological vector space|topological vector space]]s – a large class of vector spaces including e.g. [[Hilbert space]]s, [[Banach space]]s, or [[Fréchet space]]s.

The preference of other types of bases for infinite-dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If ''X'' is an infinite-dimensional normed vector space that is complete (i.e. ''X'' is a [[Banach space]]), then any Hamel basis of ''X'' is necessarily uncountable. This is a consequence of the [[Baire category theorem]]. The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (''non-complete'') normed spaces that have countable Hamel bases. Consider {{nowrap|<math>c_{00}</math>,}} the space of the [[Sequence|sequence]]s <math>x=(x_n)</math> of real numbers that have only finitely many non-zero elements, with the norm {{nowrap|<math display="inline">\|x\|=\sup_n |x_n|</math>.}} Its [[Standard basis|standard basis]], consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.

==== Example ====
In the study of [[Fourier series]], one learns that the functions {{math|1={1} ∪ { sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... }<nowiki/>}} are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions ''f'' satisfying
<math display="block">\int_0^{2\pi} \left|f(x)\right|^2\,dx < \infty.</math>

The functions {{math|1={1} ∪ { sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... }<nowiki/>}} are linearly independent, and every function ''f'' that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that
<math display="block">\lim_{n\to\infty} \int_0^{2\pi} \biggl|a_0 + \sum_{k=1}^n \left(a_k\cos\left(kx\right)+b_k\sin\left(kx\right)\right)-f(x)\biggr|^2 dx = 0</math>

for suitable (real or complex) coefficients ''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>. But many<ref>Note that one cannot say "most" because the cardinalities of the two sets (functions that can and cannot be represented with a finite number of basis functions) are the same.</ref> square-integrable functions cannot be represented as ''finite'' linear combinations of these basis functions, which therefore ''do not'' comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in [[Fourier analysis]].

===Geometry===
The geometric notions of an [[Affine space|affine space]], [[Projective space|projective space]], [[Convex set|convex set]], and cone have related notions of {{anchor|affine basis}} ''basis''.<ref>{{cite book |title=Notes on Geometry |first=Elmer G. |last=Rees |location=Berlin |publisher=Springer |year=2005 |url=https://books.google.com/books?id=JkzPRaihGIYC&pg=PA7 |page=7 |isbn=978-3-540-12053-7 }}</ref> An '''affine basis''' for an ''n''-dimensional affine space is <math>n+1</math> points in general linear position. A '''{{visible anchor|projective basis}}''' is <math>n+2</math> points in general position, in a projective space of dimension ''n''. A '''{{visible anchor|convex basis}}''' of a [[Polytope|polytope]] is the set of the vertices of its [[Convex hull|convex hull]]. A '''{{visible anchor|cone basis}}'''<ref>{{cite journal |title=Some remarks about additive functions on cones |first=Marek |last=Kuczma |journal=[[Aequationes Mathematicae]] |year=1970 |volume=4 |issue=3 |pages=303–306 |doi=10.1007/BF01844160 |s2cid=189836213 }}</ref> consists of one point by edge of a polygonal cone. See also a [[Hilbert basis (linear programming)]].

===Random basis===
For a [[Probability distribution|probability distribution]] in {{math|'''R'''<sup>''n''</sup>}} with a [[Physics:Probability density function|probability density function]], such as the equidistribution in an ''n''-dimensional ball with respect to Lebesgue measure, it can be shown that {{mvar|n}} randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that {{mvar|n}} linearly dependent vectors {{math|'''x'''<sub>1</sub>}}, ..., {{math|'''x'''<sub>''n''</sub>}} in {{math|'''R'''<sup>''n''</sup>}} should satisfy the equation {{math|1=det['''x'''<sub>1</sub> ⋯ '''x'''<sub>''n''</sub>] = 0}} (zero determinant of the matrix with columns {{math|'''x'''<sub>''i''</sub>}}), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.<ref>{{cite journal |first1=B. |last1=Igelnik |first2=Y.-H. |last2=Pao |title=Stochastic choice of basis functions in adaptive function approximation and the functional-link net |journal=IEEE Trans. Neural Netw. |volume=6 |issue=6 |year=1995 |pages=1320–1329 |doi=10.1109/72.471375 |pmid=18263425 }}</ref><ref name = "GorbanTyukin2016">{{cite journal
 | first2 = Ivan Y. | last2 = Tyukin | first3 = Danil V. | last3 = Prokhorov | first4 = Konstantin I. | last4 = Sofeikov
 | journal = Information Sciences
 | title = Approximation with Random Bases: Pro et Contra
 | pages = 129–145
 | doi = 10.1016/j.ins.2015.09.021 | volume = 364-365 | year = 2016 | arxiv = 1506.04631 | s2cid = 2239376 }}</ref>

[[File:Random almost orthogonal sets.png|thumb|270px|Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the ''n''-dimensional cube {{math|[−1, 1]<sup>''n''</sup>}} as a function of dimension, ''n''. Boxplots show the second and third quartiles of this data for each ''n'', red bars correspond to the medians, and blue stars indicate means. Red curve shows theoretical bound given by Eq. (1) and green curve shows a refined estimate.<ref name = "GorbanTyukin2016"/>]]

It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For [[Inner product space|spaces with inner product]], ''x'' is ε-orthogonal to ''y'' if <math>\left|\left\langle x,y \right\rangle\right| / \left(\left\|x\right\|\left\|y\right\|\right) < \varepsilon</math> (that is, cosine of the angle between {{mvar|x}} and {{mvar|y}} is less than {{mvar|ε}}).

In high dimensions, two independent random vectors are with high probability almost orthogonal, and the number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in ''n''-dimensional ball. Choose ''N'' independent random vectors from a ball (they are [[Independent and identically distributed random variables|independent and identically distributed]]). Let ''θ'' be a small positive number. Then for
{{NumBlk||<math display="block">N\leq {\exp}\bigl(\tfrac14\varepsilon^2n\bigr)\sqrt{-\ln(1-\theta)}</math>|Eq. 1}}

{{mvar|N}} random vectors are all pairwise ε-orthogonal with probability {{math|1 − ''θ''}}.<ref name = "GorbanTyukin2016"/> This {{mvar|N}} growth exponentially with dimension {{mvar|n}} and <math>N\gg n</math> for sufficiently big {{mvar|n}}. This property of random bases is a manifestation of the so-called {{em|measure concentration phenomenon}}.<ref>{{cite journal |first=Shiri |last=Artstein |title=Proportional concentration phenomena of the sphere |journal=[[Israel Journal of Mathematics]] |volume=132 |year=2002 |issue=1 |pages=337–358 |doi=10.1007/BF02784520 |doi-access=free|url=http://www.tau.ac.il/~shiri/israelj/ISRAJ.pdf |citeseerx=10.1.1.417.2375|s2cid=8095719}}</ref>

The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the ''n''-dimensional cube {{math|[−1, 1]<sup>''n''</sup>}} as a function of dimension, ''n''. A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If the angle between the vectors was within {{math|π/2 ± 0.037π/2}} then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within {{math|π/2 ± 0.037π/2}} then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of the chain) is recorded. For each ''n'', 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented.

==Proof that every vector space has a basis==

Let {{math|'''V'''}} be any vector space over some field {{math|'''F'''}}. Let {{math|'''X'''}} be the set of all linearly independent subsets of {{math|'''V'''}}.

The set {{math|'''X'''}} is nonempty since the empty set is an independent subset of {{math|'''V'''}}, and it is partially ordered by inclusion, which is denoted, as usual, by {{math|⊆}}.

Let {{math|'''Y'''}} be a subset of {{math|'''X'''}} that is totally ordered by {{math|⊆}}, and let {{math|L<sub>'''Y'''</sub>}} be the union of all the elements of {{math|'''Y'''}} (which are themselves certain subsets of {{math|'''V'''}}).

Since {{math|('''Y''', ⊆)}} is totally ordered, every finite subset of {{math|L<sub>'''Y'''</sub>}} is a subset of an element of {{math|'''Y'''}}, which is a linearly independent subset of {{math|'''V'''}}, and hence {{math|L<sub>'''Y'''</sub>}} is linearly independent. Thus {{math|L<sub>'''Y'''</sub>}} is an element of {{math|'''X'''}}. Therefore, {{math|L<sub>'''Y'''</sub>}} is an upper bound for {{math|'''Y'''}} in {{math|('''X''', ⊆)}}: it is an element of {{math|'''X'''}}, that contains every element of {{math|'''Y'''}}.

As {{math|'''X'''}} is nonempty, and every totally ordered subset of {{math|('''X''', ⊆)}} has an upper bound in {{math|'''X'''}}, [[Zorn's lemma]] asserts that {{math|'''X'''}} has a maximal element. In other words, there exists some element {{math|L<sub>'''max'''</sub>}} of {{math|'''X'''}} satisfying the condition that whenever {{math|L<sub>'''max'''</sub> ⊆ L}} for some element {{math|L}} of {{math|'''X'''}}, then {{math|1=L = L<sub>'''max'''</sub>}}.

It remains to prove that {{math|L<sub>'''max'''</sub>}} is a basis of {{math|'''V'''}}. Since {{math|L<sub>'''max'''</sub>}} belongs to {{math|'''X'''}}, we already know that {{math|L<sub>'''max'''</sub>}} is a linearly independent subset of {{math|'''V'''}}.

If there were some vector {{math|'''w'''}} of {{math|'''V'''}} that is not in the span of {{math|L<sub>'''max'''</sub>}}, then {{math|'''w'''}} would not be an element of {{math|L<sub>'''max'''</sub>}} either. Let {{math|1=L<sub>'''w'''</sub> = L<sub>'''max'''</sub> ∪ {'''w'''}<nowiki/>}}. This set is an element of {{math|'''X'''}}, that is, it is a linearly independent subset of {{math|'''V'''}} (because '''w''' is not in the span of {{math|L<sub>'''max'''</sub>}}, and {{math|L<sub>'''max'''</sub>}} is independent). As {{math|L<sub>'''max'''</sub> ⊆ L<sub>'''w'''</sub>}}, and {{math|L<sub>'''max'''</sub> ≠ L<sub>'''w'''</sub>}} (because {{math|L<sub>'''w'''</sub>}} contains the vector {{math|'''w'''}} that is not contained in {{math|L<sub>'''max'''</sub>}}), this contradicts the maximality of {{math|L<sub>'''max'''</sub>}}. Thus this shows that {{math|L<sub>'''max'''</sub>}} spans {{math|'''V'''}}.

Hence {{math|L<sub>'''max'''</sub>}} is linearly independent and spans {{math|'''V'''}}. It is thus a basis of {{math|'''V'''}}, and this proves that every vector space has a basis.

This proof relies on Zorn's lemma, which is equivalent to the [[Axiom of choice|axiom of choice]]. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true.<ref>{{Harvnb|Blass|1984}}</ref> Thus the two assertions are equivalent.

==See also==
*[[Basis of a matroid]]
*Basis of a linear program
*{{Annotated link|Change of basis}}
*{{Annotated link|Frame of a vector space}}
*{{Annotated link|Spherical basis}}

==Notes==
{{Reflist}}

==References==

===General references===
* {{Citation | last1=Blass | first1=Andreas | title=Axiomatic set theory | publisher=American Mathematical Society | location=Providence, R.I. | series=Contemporary Mathematics volume 31 | mr=763890 | year=1984 | chapter=Existence of bases implies the axiom of choice | pages=31–33|isbn=978-0-8218-5026-8 | chapter-url=http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf}}
* {{Citation | last1=Brown | first1=William A. | title=Matrices and vector spaces | publisher=M. Dekker | location=New York | isbn=978-0-8247-8419-5 | year=1991|url=https://books.google.com/books?id=pFQYKlnW5Z0C}}
* {{Citation | last1=Lang | first1=Serge |  title=Linear algebra | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96412-6 | year=1987}}

===Historical references===
* {{Citation | last1=Banach | first1=Stefan |  title=Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations) | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3120.pdf | year=1922 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=3| pages=133–181 |language=fr| doi=10.4064/fm-3-1-133-181 }}
* {{Citation | last1=Bolzano | first1=Bernard |  title=Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) | url=http://dml.cz/handle/10338.dmlcz/400338 | year=1804|language=de}}
* {{Citation | last1=Bourbaki | first1=Nicolas |  title=Éléments d'histoire des mathématiques (Elements of history of mathematics) | publisher=Hermann | location=Paris | year=1969|language=fr}}
* {{Citation | last1=Dorier | first1=Jean-Luc | title=A general outline of the genesis of vector space theory | mr=1347828 | year=1995 | journal=Historia Mathematica | volume=22 | issue=3 | pages=227–261 | doi=10.1006/hmat.1995.1024| url=http://archive-ouverte.unige.ch/unige:16642 | doi-access=free }}
* {{Citation | last1=Fourier | first1=Jean Baptiste Joseph |  title=Théorie analytique de la chaleur | url=https://books.google.com/books?id=TDQJAAAAIAAJ | publisher=Chez Firmin Didot, père et fils | year=1822|language=fr}}
* {{Citation | last1=Grassmann | first1=Hermann |  title=Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik | url=https://books.google.com/books?id=bKgAAAAAMAAJ&pg=PA1| year=1844|language=de}}, reprint: {{Citation | others=Kannenberg, L.C. | title=Extension Theory | publisher=American Mathematical Society | location=Providence, R.I. | isbn=978-0-8218-2031-5 | year=2000 | author=Hermann Grassmann. Translated by Lloyd C. Kannenberg.}}
* {{Citation | last=Hamel | first=Georg |  title=Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung f(x+y)=f(x)+f(y) | journal=Mathematische Annalen | location=Leipzig | volume=60 | pages=459–462 | url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002260395 | year=1905 | issue=3 | doi=10.1007/BF01457624 | s2cid=120063569 | language=de}}
* {{Citation | last1=Hamilton | first1=William Rowan |  title=Lectures on Quaternions | url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 | publisher=Royal Irish Academy | year=1853}}
* {{Citation |last1=Möbius |first1=August Ferdinand |title=Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) |url=http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0 |year=1827 |language=de |url-status=dead |archive-url=https://web.archive.org/web/20090412013616/http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0 |archive-date=2009-04-12 }}
* {{Citation | last1=Moore | first1=Gregory H. | title=The axiomatization of linear algebra: 1875–1940 | year=1995 | journal=Historia Mathematica | volume=22 | issue=3 | pages=262–303 | doi=10.1006/hmat.1995.1025| doi-access=free }}
* {{Citation | last1=Peano | first1=Giuseppe |  title=Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva | year=1888 | location=Turin|language=it}}

==External links==
* Instructional videos from Khan Academy
**[https://web.archive.org/web/20120426050335/http://khanexercises.appspot.com/video?v=zntNi3-ybfQ Introduction to bases of subspaces]
**[https://web.archive.org/web/20120426050418/http://khanexercises.appspot.com/video?v=Zn2K8UIT8r4 Proof that any subspace basis has same number of elements]
* {{springer|title=Basis|id=p/b015350}}

{{linear algebra}}
{{tensors}}

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[[Category:Articles containing proofs]]
[[Category:Axiom of choice]]
[[Category:Linear algebra]]

{{Sourceattribution|Basis (linear algebra)}}