Editing Basis (linear algebra) (section)

== 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]] and [[spanning set]]s 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 resolution]]s.

A module over the integers is exactly the same thing as an [[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 [[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 space]]s. 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]] of the continuum, which is the [[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]]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 space|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]]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]], 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]], [[projective space]], [[convex set]], and [[Cone (linear algebra)|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]] is the set of the vertices of its [[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]] in {{math|'''R'''<sup>''n''</sup>}} with a [[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
 | first1 = Alexander N. | last1 = Gorban | author1-link = Aleksandr Gorban
 | first2 = Ivan Y. | last2 = Tyukin | first3 = Danil V. | last3 = Prokhorov | first4 = Konstantin I. | last4 = Sofeikov
 | journal = [[Information Sciences (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 |author-link=Shiri 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.