Basis (linear algebra)

File:3d two bases same vector.svg

The same vector can be represented in two different bases (purple and red arrows).

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

Equivalently, a set Template:Mvar is a basis if its elements are linearly independent and every element of Template:Mvar is a linear combination of elements of Template:Mvar.<ref>Template:Cite book</ref> 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.

Basis vectors find applications in the study of crystal structures and frames of reference.

Definition

A basis Template:Math of a vector space Template:Math over a field Template:Math (such as the real numbers Template:Math or the complex numbers Template:Math) is a linearly independent subset of Template:Math that spans Template:Math. This means that a subset Template:Mvar of Template: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 Template: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 Template:Math, then Template:Nowrap

spanning property

for every vector Template:Math in Template:Math, one can choose <math>a_1,\dotsc,a_n</math> in Template:Math and <math>\mathbf v_1, \dotsc, \mathbf v_n</math> in Template:Mvar such that Template:Nowrap

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

Examples

File: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 Template: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 Template:Math and Template:Math. These vectors form a basis (called the standard basis) because any vector Template:Math of Template: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 Template:Math, such as Template:Math and Template:Math, forms also a basis of Template:Math.

More generally, if Template:Mvar is a field, the set <math>F^n</math> of [[tuple|Template:Mvar-tuples]] of elements of Template: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 Template:Mvar-tuple with all components equal to 0, except the Template: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 Template:Mvar is a field, the collection Template:Math of all polynomials in one indeterminate Template:Mvar with coefficients in Template:Mvar is an Template:Mvar-vector space. One basis for this space is the monomial basis Template: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 Template: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 Template:Mvar, given a finite spanning set Template:Mvar and a linearly independent set Template:Mvar of Template:Mvar elements of Template:Mvar, one may replace Template:Mvar well-chosen elements of Template:Mvar by the elements of Template:Mvar to get a spanning set containing Template:Mvar, having its other elements in Template:Mvar, and having the same number of elements as Template: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 Template:Mvar is a vector space over a field Template:Mvar, then:

• If Template:Mvar is a linearly independent subset of a spanning set Template:Math, then there is a basis Template:Mvar such that <math display="block">L\subseteq B\subseteq S.</math>
• Template:Mvar has a basis (this is the preceding property with Template:Mvar being the empty set, and Template:Math).
• All bases of Template:Mvar have the same cardinality, which is called the dimension of Template:Mvar. This is the dimension theorem.
• A generating set Template:Mvar is a basis of Template:Mvar if and only if it is minimal, that is, no proper subset of Template:Mvar is also a generating set of Template:Mvar.
• A linearly independent set Template:Mvar is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.

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

• A subset of Template:Mvar with Template:Mvar elements is a basis if and only if it is linearly independent.
• A subset of Template:Mvar with Template:Mvar elements is a basis if and only if it is a spanning set of Template:Mvar.

Coordinates Template:Anchor

Let Template:Mvar be a vector space of finite dimension Template:Mvar over a field Template:Mvar, and <math display="block">B = \{\mathbf b_1, \ldots, \mathbf b_n\}</math> be a basis of Template:Mvar. By definition of a basis, every Template:Math in Template: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 Template:Mvar), which are called the coordinates of Template:Math over Template: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 Template: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, especially when used in conjunction with an origin, is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame).

Let, as usual, <math>F^n</math> be the set of the [[tuple|Template:Mvar-tuples]] of elements of Template:Mvar. This set is an Template: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 Template:Mvar. In other words, <math>F^n</math> is the coordinate space of Template:Mvar, and the Template:Mvar-tuple <math>\varphi^{-1}(\mathbf v)</math> is the coordinate vector of Template:Math.

The inverse image by <math>\varphi</math> of <math>\mathbf b_i</math> is the Template:Mvar-tuple <math>\mathbf e_i</math> all of whose components are 0, except the Template: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 Template:Mvar is the image by <math>\varphi</math> of the canonical basis of Template:Nowrap

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

Change of basis

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Let Template:Math be a vector space of dimension Template:Mvar over a field Template: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 Template:Math, it is often useful to express the coordinates of a vector Template: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 Template: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 Template:Math.

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

Related notions

Free module

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Template:Mvar is a subgroup of a finitely generated free abelian group Template: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 Template:Mvar and an integer Template:Math such that <math>a_1 \mathbf e_1, \ldots, a_k \mathbf e_k</math> is a basis of Template:Mvar, for some nonzero integers Template:Nowrap For details, see Template:Slink.

Analysis

In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Template:Visible anchor (named after Georg Hamel<ref>Template:Harvnb</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 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 Template: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 Template: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 Template: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 Template: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 Template: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 a_k, b_k. 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 have related notions of Template:Anchor basis.<ref>Template:Cite book</ref> An affine basis for an n-dimensional affine space is <math>n+1</math> points in general linear position. A Template:Visible anchor is <math>n+2</math> points in general position, in a projective space of dimension n. A Template:Visible anchor of a polytope is the set of the vertices of its convex hull. A Template:Visible anchor<ref>Template:Cite journal</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 Template: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 Template:Mvar randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that Template:Mvar linearly dependent vectors Template:Math, ..., Template:Math in Template:Math should satisfy the equation Template:Math (zero determinant of the matrix with columns Template:Math), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.<ref>Template:Cite journal</ref><ref name = "GorbanTyukin2016">Template:Cite journal</ref>

File: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 Template: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.<ref name = "GorbanTyukin2016"/>

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 Template:Mvar and Template:Mvar is less than Template: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 Template:NumBlk

Template:Mvar random vectors are all pairwise ε-orthogonal with probability Template:Math.<ref name = "GorbanTyukin2016"/> This Template:Mvar growth exponentially with dimension Template:Mvar and <math>N\gg n</math> for sufficiently big Template:Mvar. This property of random bases is a manifestation of the so-called Template:Em.<ref>Template:Cite journal</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 Template: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 Template: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 Template: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 Template:Math be any vector space over some field Template:Math. Let Template:Math be the set of all linearly independent subsets of Template:Math.

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

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

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

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

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

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

Hence Template:Math is linearly independent and spans Template:Math. It is thus a basis of Template: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.<ref>Template:Harvnb</ref> Thus the two assertions are equivalent.

See also

• Basis of a matroid
• Basis of a linear program
• Coordinate system
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References

General references

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Historical references

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External links

• Instructional videos from Khan Academy
  • Introduction to bases of subspaces
  • Proof that any subspace basis has same number of elements
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