Basis (linear algebra)
In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) 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 of the vector with respect to B. The elements of a basis are called basis vectors.
Equivalently, a set B is a basis if its elements are
A vector space can have several bases; however all the bases have the same number of elements, called the 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 B of a
V. This means that a subset B of V is a basis if it satisfies the two following conditions:- linear independence
- for every finite subset of B, if for some in F, then ;
- spanning property
- for every vector v in V, one can choose in F and in B such that .
The scalars 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 finite-dimensional. In this case, the finite subset can be taken as B itself to check for linear independence in the above definition.
It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing 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, but a sequence, an indexed family, or similar; see § Ordered bases and coordinates below.
Examples
The set R2 of the ordered pairs of real numbers is a vector space under the operations of component-wise addition
More generally, if F is a field, the set of n-tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let
A different flavor of example is given by polynomial rings. If F is a field, the collection F[X] of all polynomials in one indeterminate X with coefficients in F is an F-vector space. One basis for this space is the monomial basis B, consisting of all monomials:
Properties
Many properties of finite bases result from the
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
If V is a vector space over a field F, then:
- If L is a linearly independent subset of a spanning set S ⊆ V, then there is a basis B such that
- V has a basis (this is the preceding property with L being the empty set, and S = V).
- All bases of V have the same cardinality, which is called the dimension of V. This is the dimension theorem.
- A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.
- A linearly independent set 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 V is a vector space of dimension n, then:
- A subset of V with n elements is a basis if and only if it is linearly independent.
- A subset of V with n elements is a basis if and only if it is a spanning set of V.
Coordinates
Let V be a vector space of finite dimension n over a field F, and
Let, as usual, be the set of the n-tuples of elements of F. This set is an F-vector space, with addition and scalar multiplication defined component-wise. The map
The
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.
Change of basis
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-basis formula, that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to and 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 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,
This formula may be concisely written in matrix notation. Let A be the matrix of the , and
The formula can be proven by considering the decomposition of the vector x on the two bases: one has
The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here ; that is
Related notions
Free module
If one replaces the field occurring in the definition of a vector space by a
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
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 G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), then 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.
Analysis
In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after
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 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
Example
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 square-integrable 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 square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that
for suitable (real or complex) coefficients ak, bk. But many
Geometry
The geometric notions of an
Random basis
For a
It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, x is ε-orthogonal 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 n-dimensional 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 ε-orthogonal with probability 1 − θ.[7] This N growth exponentially with dimension n and for sufficiently big n. This property of random bases is a manifestation of the so-called measure concentration phenomenon.[8]
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 [−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 were constructed numerically for each dimension. Distribution of the length of these chains is presented.
Proof that every vector space has a basis
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
Let Y be a subset of X that is totally ordered by ⊆, and let LY 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 LY is a subset of an element of Y, which is a linearly independent subset of V, and hence LY is linearly independent. Thus LY is an element of X. Therefore, LY is an upper bound for Y in (X, ⊆): it is an element of X, that contains every element of 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 Lmax of X satisfying the condition that whenever Lmax ⊆ L for some element L of X, then L = Lmax.
It remains to prove that Lmax is a basis of V. Since Lmax belongs to X, we already know that Lmax is a linearly independent subset of V.
If there were some vector w of V that is not in the span of Lmax, then w would not be an element of Lmax either. Let Lw = Lmax ∪ {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 Lmax, and Lmax is independent). As Lmax ⊆ Lw, and Lmax ≠ Lw (because Lw contains the vector w that is not contained in Lmax), this contradicts the maximality of Lmax. Thus this shows that Lmax spans V.
Hence Lmax 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 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.
See also
- Basis of a matroid
- Basis of a linear program
- Change of basis – Coordinate change in linear algebra
- Frame of a vector space– Similar to the basis of a vector space, but not necessarily linearly independent
- Spherical basis – Basis used to express spherical tensors
Notes
- ISBN 978-0-387-90093-3.
- ^ Hamel 1905
- ^ 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.
- ISBN 978-3-540-12053-7.
- S2CID 189836213.
- PMID 18263425.
- ^ S2CID 2239376.
- S2CID 8095719.
- ^ Blass 1984
References
General references
- Blass, Andreas (1984), "Existence of bases implies the axiom of choice" (PDF), Axiomatic set theory, Contemporary Mathematics volume 31, Providence, R.I.: MR 0763890
- Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5
- ISBN 978-0-387-96412-6
Historical references
- ISSN 0016-2736
- Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in German)
- Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (in French), Paris: Hermann
- Dorier, Jean-Luc (1995), "A general outline of the genesis of vector space theory", MR 1347828
- Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (in French), Chez Firmin Didot, père et fils
- ISBN 978-0-8218-2031-5
- S2CID 120063569
- Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy
- Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (in German), archived from the original on 2009-04-12
- Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940",
- Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin
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External links
- Instructional videos from Khan Academy
- "Linear combinations, span, and basis vectors". Essence of linear algebra. August 6, 2016. Archived from the original on 2021-11-17 – via YouTube.
- "Basis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]