Английская Википедия:Basis (linear algebra)

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Файл:3d two bases same vector.svg
The same vector can be represented in two different bases (purple and red arrows).

In mathematics, a set Шаблон:Mvar of vectors in a vector space Шаблон:Math is called a basis (Шаблон:Plural form: bases) if every element of Шаблон:Math may be written in a unique way as a finite linear combination of elements of Шаблон:Mvar. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to Шаблон:Mvar. The elements of a basis are called Шаблон:Visible anchor.

Equivalently, a set Шаблон:Mvar is a basis if its elements are linearly independent and every element of Шаблон:Mvar is a linear combination of elements of Шаблон:Mvar.[1] 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 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 Шаблон:Math of a vector space Шаблон:Math over a field Шаблон:Math (such as the real numbers Шаблон:Math or the complex numbers Шаблон:Math) is a linearly independent subset of Шаблон:Math that spans Шаблон:Math. This means that a subset Шаблон:Mvar of Шаблон:Math is a basis if it satisfies the two following conditions:

linear independence
for every finite subset <math>\{\mathbf v_1, \dotsc, \mathbf v_m\}</math> of Шаблон:Mvar, 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 Шаблон:Math, then Шаблон:Nowrap
spanning property
for every vector Шаблон:Math in Шаблон:Math, one can choose <math>a_1,\dotsc,a_n</math> in Шаблон:Math and <math>\mathbf v_1, \dotsc, \mathbf v_n</math> in Шаблон:Mvar such that Шаблон:Nowrap

The scalars <math>a_i</math> are called the coordinates of the vector Шаблон:Math with respect to the basis Шаблон:Math, 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 Шаблон:Math 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 Шаблон:Slink below.

Examples

Файл:Basis graph (no label).svg
This picture illustrates the standard basis in R2. 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 Шаблон:Math of the ordered pairs of real numbers 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 Шаблон:Math and Шаблон:Math. These vectors form a basis (called the standard basis) because any vector Шаблон:Math of Шаблон:Math 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 Шаблон:Math, such as Шаблон:Math and Шаблон:Math, forms also a basis of Шаблон:Math.

More generally, if Шаблон:Mvar is a field, the set <math>F^n</math> of [[tuple|Шаблон:Mvar-tuples]] of elements of Шаблон:Mvar 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 Шаблон:Mvar-tuple with all components equal to 0, except the Шаблон:Mvarth, 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 rings. If Шаблон:Mvar is a field, the collection Шаблон:Math of all polynomials in one indeterminate Шаблон:Mvar with coefficients in Шаблон:Mvar is an Шаблон:Mvar-vector space. One basis for this space is the monomial basis Шаблон:Mvar, consisting of all monomials: <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 basis polynomials or Chebyshev polynomials) is also a basis. (Such a set of polynomials is called a polynomial sequence.) But there are also many bases for Шаблон:Math 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 Шаблон:Mvar, given a finite spanning set Шаблон:Mvar and a linearly independent set Шаблон:Mvar of Шаблон:Mvar elements of Шаблон:Mvar, one may replace Шаблон:Mvar well-chosen elements of Шаблон:Mvar by the elements of Шаблон:Mvar to get a spanning set containing Шаблон:Mvar, having its other elements in Шаблон:Mvar, and having the same number of elements as Шаблон:Mvar.

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 or a weaker form of it, such as the ultrafilter lemma.

If Шаблон:Mvar is a vector space over a field Шаблон:Mvar, then:

If Шаблон:Mvar is a vector space of dimension Шаблон:Mvar, then:

Coordinates Шаблон:Anchor

Let Шаблон:Mvar be a vector space of finite dimension Шаблон:Mvar over a field Шаблон:Mvar, and <math display="block">B = \{\mathbf b_1, \ldots, \mathbf b_n\}</math> be a basis of Шаблон:Mvar. By definition of a basis, every Шаблон:Math in Шаблон:Mvar 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 Шаблон:Mvar), which are called the coordinates of Шаблон:Math over Шаблон:Mvar. 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 Шаблон:Math, 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, <math>F^n</math> be the set of the [[tuple|Шаблон:Mvar-tuples]] of elements of Шаблон:Mvar. This set is an Шаблон:Mvar-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 Шаблон:Mvar. In other words, <math>F^n</math> is the coordinate space of Шаблон:Mvar, and the Шаблон:Mvar-tuple <math>\varphi^{-1}(\mathbf v)</math> is the coordinate vector of Шаблон:Math.

The inverse image by <math>\varphi</math> of <math>\mathbf b_i</math> is the Шаблон:Mvar-tuple <math>\mathbf e_i</math> all of whose components are 0, except the Шаблон:Mvarth that is 1. The <math>\mathbf e_i</math> form an ordered basis of <math>F^n</math>, which is called its standard basis or canonical basis. The ordered basis Шаблон:Mvar is the image by <math>\varphi</math> of the canonical basis of Шаблон:Nowrap

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

Change of basis

Шаблон:Main Let Шаблон:Math be a vector space of dimension Шаблон:Mvar over a field Шаблон:Math. 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 Шаблон:Math, it is often useful to express the coordinates of a vector Шаблон:Mvar 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 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 Шаблон:Math 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 Шаблон:Math.

This formula may be concisely written in matrix notation. Let Шаблон:Mvar be the matrix of the Шаблон:Nowrap 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 Шаблон:Math 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 Шаблон:Math 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 Шаблон:Nowrap that is <math display="block">x_i = \sum_{j=1}^n a_{i,j} y_j,</math> for Шаблон:Math.

Related notions

Free module

Шаблон:Main 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 non-free 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 free abelian group, and, if Шаблон:Mvar is a subgroup of a finitely generated free abelian group Шаблон:Mvar (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 Шаблон:Mvar and an integer Шаблон:Math such that <math>a_1 \mathbf e_1, \ldots, a_k \mathbf e_k</math> is a basis of Шаблон:Mvar, for some nonzero integers Шаблон:Nowrap For details, see Шаблон:Slink.

Analysis

In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Шаблон:Visible anchor (named after Georg Hamel[2]) 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 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 Шаблон:Nowrap 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 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 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 Шаблон:Nowrap the space of the sequences <math>x=(x_n)</math> of real numbers that have only finitely many non-zero elements, with the norm Шаблон:Nowrap 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 Шаблон:Math 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 Шаблон:Math 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 ak, bk. But many[3] 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 have related notions of Шаблон:Anchor basis.[4] An affine basis for an n-dimensional affine space is <math>n+1</math> points in general linear position. A Шаблон:Visible anchor is <math>n+2</math> points in general position, in a projective space of dimension n. A Шаблон:Visible anchor of a polytope is the set of the vertices of its convex hull. A Шаблон:Visible anchor[5] consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).

Random basis

For a probability distribution in Шаблон:Math with a probability density function, such as the equidistribution in an n-dimensional ball with respect to Lebesgue measure, it can be shown that Шаблон:Mvar randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that Шаблон:Mvar linearly dependent vectors Шаблон:Math, ..., Шаблон:Math in Шаблон:Math should satisfy the equation Шаблон:Math (zero determinant of the matrix with columns Шаблон:Math), 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]

Файл:Random almost orthogonal sets.png
Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube Шаблон:Math 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.[7]

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 <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 Шаблон:Mvar and Шаблон:Mvar is less than Шаблон:Mvar).

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 Шаблон:NumBlk

Шаблон:Mvar random vectors are all pairwise ε-orthogonal with probability Шаблон:Math.[7] This Шаблон:Mvar growth exponentially with dimension Шаблон:Mvar and <math>N\gg n</math> for sufficiently big Шаблон:Mvar. This property of random bases is a manifestation of the so-called Шаблон:Em.[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 Шаблон:Math 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 Шаблон:Math 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 Шаблон:Math 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 Шаблон:Math be any vector space over some field Шаблон:Math. Let Шаблон:Math be the set of all linearly independent subsets of Шаблон:Math.

The set Шаблон:Math is nonempty since the empty set is an independent subset of Шаблон:Math, and it is partially ordered by inclusion, which is denoted, as usual, by Шаблон:Math.

Let Шаблон:Math be a subset of Шаблон:Math that is totally ordered by Шаблон:Math, and let Шаблон:Math be the union of all the elements of Шаблон:Math (which are themselves certain subsets of Шаблон:Math).

Since Шаблон:Math is totally ordered, every finite subset of Шаблон:Math is a subset of an element of Шаблон:Math, which is a linearly independent subset of Шаблон:Math, and hence Шаблон:Math is linearly independent. Thus Шаблон:Math is an element of Шаблон:Math. Therefore, Шаблон:Math is an upper bound for Шаблон:Math in Шаблон:Math: it is an element of Шаблон:Math, that contains every element of Шаблон:Math.

As Шаблон:Math is nonempty, and every totally ordered subset of Шаблон:Math has an upper bound in Шаблон:Math, Zorn's lemma asserts that Шаблон:Math has a maximal element. In other words, there exists some element Шаблон:Math of Шаблон:Math satisfying the condition that whenever Шаблон:Math for some element Шаблон:Math of Шаблон:Math, then Шаблон:Math.

It remains to prove that Шаблон:Math is a basis of Шаблон:Math. Since Шаблон:Math belongs to Шаблон:Math, we already know that Шаблон:Math is a linearly independent subset of Шаблон:Math.

If there were some vector Шаблон:Math of Шаблон:Math that is not in the span of Шаблон:Math, then Шаблон:Math would not be an element of Шаблон:Math either. Let Шаблон:Math. This set is an element of Шаблон:Math, that is, it is a linearly independent subset of Шаблон:Math (because w is not in the span of Шаблон:Math, and Шаблон:Math is independent). As Шаблон:Math, and Шаблон:Math (because Шаблон:Math contains the vector Шаблон:Math that is not contained in Шаблон:Math), this contradicts the maximality of Шаблон:Math. Thus this shows that Шаблон:Math spans Шаблон:Math.

Hence Шаблон:Math is linearly independent and spans Шаблон:Math. It is thus a basis of Шаблон:Math, 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

Notes

Шаблон:Reflist

References

General references

Historical references

External links

Шаблон:Linear algebra Шаблон:Tensors

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  1. ↑ Шаблон:Cite book
  2. ↑ Шаблон:Harvnb
  3. ↑ 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.
  4. ↑ Шаблон:Cite book
  5. ↑ Шаблон:Cite journal
  6. ↑ Шаблон:Cite journal
  7. ↑ 7,0 7,1 7,2 Шаблон:Cite journal
  8. ↑ Шаблон:Cite journal
  9. ↑ Шаблон:Harvnb
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