Inmathematics, asetB of elements of avector spaceV is called abasis (pl.:bases) if every element ofV can be written in a unique way as a finitelinear combination of elements ofB. The coefficients of this linear combination are referred to ascomponents orcoordinates of the vector with respect toB. The elements of a basis are calledbasis vectors.
Equivalently, a setB is a basis if its elements arelinearly independent and every element ofV is alinear combination of elements ofB.[1] In other words, a basis is a linearly independentspanning set.
A vector space can have several bases; however all the bases have the same number of elements, called thedimension 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.
Basis vectors find applications in the study ofcrystal structures andframes of reference.
AbasisB of avector spaceV over afieldF (such as thereal numbersR or thecomplex numbersC) is a linearly independentsubset ofV thatspansV. This means that a subsetB ofV is a basis if it satisfies the two following conditions:
Thescalars are called the coordinates of the vectorv with respect to the basisB, and by the first property they are uniquely determined.
A vector space that has afinite basis is calledfinite-dimensional. In this case, the finite subset can be taken asB itself to check for linear independence in the above definition.
It is often convenient or even necessary to have anordering on the basis vectors, for example, when discussingorientation, 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 anordered basis, which is therefore not simply an unstructuredset, but asequence, anindexed family, or similar; see§ Ordered bases and coordinates below.
.svg)
The setR2 of theordered pairs ofreal numbers is a vector space under the operations of component-wise addition and scalar multiplication where is any real number. A simple basis of this vector space consists of the two vectorse1 = (1, 0) ande2 = (0, 1). These vectors form a basis (called thestandard basis) because any vectorv = (a,b) ofR2 may be uniquely written as Any other pair of linearly independent vectors ofR2, such as(1, 1) and(−1, 2), forms also a basis ofR2.
More generally, ifF is afield, the set ofn-tuples of elements ofF is a vector space for similarly defined addition and scalar multiplication. Let be then-tuple with all components equal to 0, except theith, which is 1. Then is a basis of which is called thestandard basis of
A different flavor of example is given bypolynomial rings. IfF is a field, the collectionF[X] of allpolynomials in oneindeterminateX with coefficients inF is anF-vector space. One basis for this space is themonomial basisB, consisting of allmonomials: Any set of polynomials such that there is exactly one polynomial of each degree (such as theBernstein basis polynomials orChebyshev polynomials) is also a basis. (Such a set of polynomials is called apolynomial sequence.) But there are also many bases forF[X] that are not of this form.
Many properties of finite bases result from theSteinitz exchange lemma, which states that, for any vector spaceV, given a finitespanning setS and alinearly independent setL ofn elements ofV, one may replacen well-chosen elements ofS by the elements ofL to get a spanning set containingL, having its other elements inS, and having the same number of elements asS.
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 theaxiom of choice or a weaker form of it, such as theultrafilter lemma.
IfV is a vector space over a fieldF, then:
IfV is a vector space of dimensionn, then:
LetV be a vector space of finite dimensionn over a fieldF, andbe a basis ofV. By definition of a basis, everyv inV may be written, in a unique way, aswhere the coefficients are scalars (that is, elements ofF), which are called thecoordinates ofv overB. However, if one talks of theset of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the sameset of coefficients. For example, and have the same set of coefficients{2, 3}, and are different. It is therefore often convenient to work with anordered basis; this is typically done byindexing the basis elements by the first natural numbers. Then, the coordinates of a vector form asequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with anorigin, is also called acoordinate frame or simply aframe (for example, aCartesian frame or anaffine frame).
Let, as usual, be the set of then-tuples of elements ofF. This set is anF-vector space, with addition and scalar multiplication defined component-wise. The mapis alinear isomorphism from the vector space ontoV. In other words, is thecoordinate space ofV, and then-tuple is thecoordinate vector ofv.
Theinverse image by of is then-tuple all of whose components are 0, except theith that is 1. The form an ordered basis of, which is called itsstandard basis orcanonical basis. The ordered basisB 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 ontoV may be defined as the isomorphism that maps the canonical basis of onto a given ordered basis ofV. In other words, it is equivalent to define an ordered basis ofV, or a linear isomorphism from ontoV.
LetV be a vector space of dimensionn over a fieldF. Given two (ordered) bases and ofV, it is often useful to express the coordinates of a vectorx with respect to in terms of the coordinates with respect to This can be done by thechange-of-basis formula, that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to and as theold basis and thenew basis, respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one hasexpressions 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,If and are the coordinates of a vectorx over the old and the new basis respectively, the change-of-basis formula isfori = 1, ...,n.
This formula may be concisely written inmatrix notation. LetA be the matrix of the, andbe thecolumn vectors of the coordinates ofv in the old and the new basis respectively, then the formula for changing coordinates is
The formula can be proven by considering the decomposition of the vectorx on the two bases: one hasand
The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here; that isfori = 1, ...,n.
If one replaces the field occurring in the definition of a vector space by aring, one gets the definition of amodule. For modules,linear independence andspanning sets are defined exactly as for vector spaces, although "generating set" is more commonly used than that of "spanning set".
Like for vector spaces, abasis 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 afree module. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules throughfree resolutions.
A module over the integers is exactly the same thing as anabelian 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, ifG is a subgroup of a finitely generated free abelian groupH (that is an abelian group that has a finite basis), then there is a basis ofH and an integer0 ≤k ≤n such that is a basis ofG, for some nonzero integers. For details, seeFree abelian group § Subgroups.
In the context of infinite-dimensional vector spaces over the real or complex numbers, the termHamel basis (named afterGeorg Hamel[2]) oralgebraic 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 areorthogonal bases onHilbert spaces,Schauder bases, andMarkushevich bases onnormed linear spaces. In the case of the real numbersR viewed as a vector space over the fieldQ of rational numbers, Hamel bases are uncountable, and have specifically thecardinality of the continuum, which is thecardinal number, where (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 fortopological vector spaces – a large class of vector spaces including e.g.Hilbert spaces,Banach spaces, orFréchet spaces.
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: IfX is an infinite-dimensional normed vector space that iscomplete (i.e.X is aBanach space), then any Hamel basis ofX is necessarilyuncountable. This is a consequence of theBaire 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, the space of thesequences of real numbers that have only finitely many non-zero elements, with the norm. Itsstandard basis, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.
In the study ofFourier 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 square-integrable on this interval, i.e., functionsf satisfying
The functions{1} ∪ { sin(nx), cos(nx) :n = 1, 2, 3, ... } are linearly independent, and every functionf that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that
for suitable (real or complex) coefficientsak,bk. But many[3] square-integrable functions cannot be represented asfinite linear combinations of these basis functions, which thereforedo 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, whereasorthonormal bases of these spaces are essential inFourier analysis.
The geometric notions of anaffine space,projective space,convex set, andcone have related notions ofbasis.[4] Anaffine basis for ann-dimensional affine space is points ingeneral linear position. Aprojective basis is points in general position, in a projective space of dimensionn. Aconvex basis of apolytope is the set of the vertices of itsconvex hull. Acone basis[5] consists of one point by edge of a polygonal cone. See also aHilbert basis (linear programming).
For aprobability distribution inRn with aprobability density function, such as the equidistribution in ann-dimensional ball with respect to Lebesgue measure, it can be shown thatn randomly and independently chosen vectors will form a basiswith probability one, which is due to the fact thatn linearly dependent vectorsx1, ...,xn inRn should satisfy the equationdet[x1 ⋯xn] = 0 (zero determinant of the matrix with columnsxi), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.[6][7]
It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. Forspaces with inner product,x is ε-orthogonal toy if (that is, cosine of the angle betweenx andy is less thanε).
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 inn-dimensional ball. ChooseN independent random vectors from a ball (they areindependent and identically distributed). Letθ be a small positive number. Then for
| Eq. 1 |
N random vectors are all pairwise ε-orthogonal with probability1 −θ.[7] ThisN growth exponentially with dimensionn and for sufficiently bign. This property of random bases is a manifestation of the so-calledmeasure concentration phenomenon.[8]
The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from then-dimensional 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 withinπ/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 withinπ/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 eachn, 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented.
LetV be any vector space over some fieldF. LetX be the set of all linearly independent subsets ofV.
The setX is nonempty since the empty set is an independent subset ofV, and it ispartially ordered by inclusion, which is denoted, as usual, by⊆.
LetY be a subset ofX that is totally ordered by⊆, and letLY be the union of all the elements ofY (which are themselves certain subsets ofV).
Since(Y, ⊆) is totally ordered, every finite subset ofLY is a subset of an element ofY, which is a linearly independent subset ofV, and henceLY is linearly independent. ThusLY is an element ofX. Therefore,LY is an upper bound forY in(X, ⊆): it is an element ofX, that contains every element ofY.
AsX is nonempty, and every totally ordered subset of(X, ⊆) has an upper bound inX,Zorn's lemma asserts thatX has a maximal element. In other words, there exists some elementLmax ofX satisfying the condition that wheneverLmax ⊆ L for some elementL ofX, thenL = Lmax.
It remains to prove thatLmax is a basis ofV. SinceLmax belongs toX, we already know thatLmax is a linearly independent subset ofV.
If there were some vectorw ofV that is not in the span ofLmax, thenw would not be an element ofLmax either. LetLw = Lmax ∪ {w}. This set is an element ofX, that is, it is a linearly independent subset ofV (becausew is not in the span ofLmax, andLmax is independent). AsLmax ⊆ Lw, andLmax ≠ Lw (becauseLw contains the vectorw that is not contained inLmax), this contradicts the maximality ofLmax. Thus this shows thatLmax spansV.
HenceLmax is linearly independent and spansV. It is thus a basis ofV, and this proves that every vector space has a basis.
This proof relies on Zorn's lemma, which is equivalent to theaxiom of choice. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true.[9] Thus the two assertions are equivalent.
{{citation}}: CS1 maint: location missing publisher (link)
