In mathematics, a set B of elements (vectors) in a vector space V is called a basis, if every element of V may be written in a unique way as a (finite) linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates on B of the vector. The elements of a basis are called basis vectors.
Equivalently B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.^{[1]} In more general terms, a basis is a linearly independent spanning set.
A vector space can have generally several bases; however all the bases have the same number of elements, called the dimension of the vector space.
A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V. This means that, a subset B of V is a basis if it satisfies the two following conditions:
The scalars v_{i} are called the coordinates of the vector v with respect to the basis B, and by the first property they are uniquely determined.
A vector space that has a finite basis is called finitedimensional. In this case, the subset {b_{1}, ..., b_{n}} that is considered (twice) in the above definition may be chosen as B itself.
It is often convenient to ordering the basis vectors, typically when one consider the coefficients of a vector on 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. Generally, this ordering is implicitly done by numbering the basis elements. For example, when using matrices, the ith row, and ith column refer to the ith element of a basis of some vector space. For emphasizing that an order has been chosen, one speaks of an ordered basis, which is therefore a sequence rather than a set; see Ordered bases and coordinates below.
Many properties of finite bases result from the Steinitz exchange lemma, which states that, given a finite spanning set S and a linearly independent subset L of n elements of S, one may replace n well chosen elements of S by the elements of L for getting a spanning set containing L, having its other elements in S, and having the same number of elements as S.
Most properties resulting from Steinitz exchange lemma remain true when there is no finite spanning set, but their proof in the infinite case requires generally the axiom of choice or a weaker form of it, such as the ultrafilter lemma.
If V is a vector space over a field F, then:
If V is a vector space of dimension n, then:
Let V be a vector space of finite dimension n over a field F, and
be a basis of V. By definition of a basis, for every v in V may be written, in a unique way,
where the coefficients are scalars (that is, elements of F), which are called the coordinates of v over B. However, if one talks of the set of the coefficients, one looses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, and have the same set of coefficients {2, 3}, and are different. It is therefore often convenient to work with an ordered basis; this is typically done by 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, be the set of the ntuples of elements of F. This set is an Fvector space, with addition and scalar multiplication defined componentwise. The map
is a linear isomorphism from the vector space onto V. In other words, is the coordinate space of V, and the ntuple is the coordinate vector of v.
The inverse image by of is the ntuple whose all components are 0, except the ith that is 1. The form an ordered basis of which is called its standard basis or canonical basis. The ordered basis B is the image by of the canonical basis of
It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of and that every linear isomorphism from onto V may be defined as the isomorphism that maps the canonical basis of onto a given ordered basis of V. In other words it is equivalent to define an ordered basis of V, or a linear isomorphism from onto V.
Let V be a vector space of dimension n over a field F. Given two (ordered) bases and of V, it is often useful to express the coordinates of a vector x with respect to in terms of the coordinates with respect to This can be done by the change of base formula, is described below. The subscripts o and n have been chosen because it is customary to refer to as the old basis and to as the new basis. It is useful to describe the old coordinates in terms of the new ones, because in general one has an expression, in which the old coordinates are to be substituted by these terms in the new coordinates, thus yielding an equivalent expression, involving the new coordinates, instead of the old ones.
Typically, the new basis vectors are given by their coordinates over the old basis, that is,
If and are the coordinates of a vector v over the old and the new basis respectively, one has
The formula for changing the coordinates with respect to the other basis results from the uniqueness of the decomposition of a vector over a basis, and is thus
for i = 1, ..., n.
This formula may be concisely written in matrix notation. Let A be the matrix of the and
be the column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is
If one replaces the field occurring in the definition of a vector space by a ring, one gets the definition of a module. For modules, linear independence and spanning sets are defined exactly as for vector spaces, although "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 nonfree modules through free resolutions.
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 group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), there is a basis of H and an integer 0 ≤ k ≤ n such that is a basis of G, for some nonzero integers For details, see Free abelian group § Subgroups.
In the context of infinitedimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel) 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 infinitedimensional vector spaces are endowed with extra structure. The most important alternatives are orthogonal bases on Hilbert spaces, Schauder bases, and 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 of the continuum, which is the cardinal number where 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 spaces – a large class of vector spaces including e.g. Hilbert spaces, Banach spaces, or Fréchet spaces.
The preference of other types of bases for infinitedimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If X is an infinitedimensional normed vector space which 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, finitedimensional spaces have by definition finite bases and there are infinitedimensional (noncomplete) normed spaces which have countable Hamel bases. Consider , the space of the sequences of real numbers which have only finitely many nonzero elements, with the norm Its standard basis, consisting of the sequences having only one nonzero element, which is equal to 1, is a countable Hamel basis.
In the study of Fourier series, one learns that the functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } 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 squareintegrable on this interval, i.e., functions f satisfying
The functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are linearly independent, and every function f that is squareintegrable on [0, 2π] is an "infinite linear combination" of them, in the sense that
for suitable (real or complex) coefficients a_{k}, b_{k}. But many^{[2]} squareintegrable 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.
The geometric notions of an affine space, projective space, convex set, and cone have related notions of basis.^{[3]} An affine basis for an ndimensional affine space is points in general linear position. A projective basis is points in general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis^{[4]} consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).
For a probability distribution in R^{n} with a probability density function, such as the equidistribution in a ndimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that n linearly dependent vectors x_{1}, ..., x_{n} in R^{n} should satisfy the equation det[x_{1}, ..., x_{n}] = 0 (zero determinant of the matrix with columns x_{i}), and the set of zeros of a nontrivial polynomial has zero measure. This observation has led to techniques for approximating random bases.^{[5]}^{[6]}
It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of εorthogonality is used. For spaces with inner product, x is εorthogonal to y if (that is, cosine of the angle between x and y is less than 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 ndimensional ball. Choose N independent random vectors from a ball (they are independent and identically distributed). Let θ be a small positive number. Then for


(Eq. 1) 
N random vectors are all pairwise εorthogonal with probability 1 − θ.^{[6]} This N growth exponentially with dimension n and for sufficiently big n. This property of random bases is a manifestation of the socalled measure concentration phenomenon.^{[7]}
The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the ndimensional cube [−1, 1]^{n} 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 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 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 where constructed numerically for each dimension. Distribution of the length of these chains is presented.
Let V be any vector space over some field F. Let X be the set of all linearly independent subsets of V.
The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by ⊆.
Let Y be a subset of X that is totally ordered by ⊆, and let L_{Y} be the union of all the elements of Y (which are themselves certain subsets of V).
Since (Y, ⊆) is totally ordered, every finite subset of L_{Y} is a subset of an element of Y, which is a linearly independent subset of V, and hence every finite subset of L_{Y} is linearly independent. Thus L_{Y} is linearly independent, so L_{Y} is an element of X. Therefore, L_{Y} is an upper bound for Y in (X, ⊆): it is an element of X, that contains every element Y.
As X is nonempty, and every totally ordered subset of (X, ⊆) has an upper bound in X, Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L_{max} of X satisfying the condition that whenever L_{max} ⊆ L for some element L of X, then L = L_{max}.
It remains to prove that L_{max} is a basis of V. Since L_{max} belongs to X, we already know that L_{max} is a linearly independent subset of V.
If L_{max} would not span V, there would exist some vector w of V that cannot be expressed as a linear combination of elements of L_{max} (with coefficients in the field F). In particular, w cannot be an element of L_{max}. Let L_{w} = L_{max} ∪ {w}. This set is an element of X, that is, it is a linearly independent subset of V (because w is not in the span of L_{max}, and L_{max} is independent). As L_{max} ⊆ L_{w}, and L_{max} ≠ L_{w} (because L_{w} contains the vector w that is not contained in L_{max}), this contradicts the maximality of L_{max}. Thus this shows that L_{max} spans V.
Hence L_{max} is linearly independent and spans V. It is thus a basis of 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. Conversely, it may be proved that if every vector space has a basis, then the axiom of choice is true; thus the two assertions are equivalent.
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