Basis (linear algebra)

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Basis (linear algebra)
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{{redirect|Basis vector|basis vector in the context of crystals|Crystal structure|a more general concept in physics|Frame of reference}}{{redirects here|Basis (mathematics)||Basis (disambiguation)#Mathematics{{!}}Basis}}(File:3d two bases same vector.svg|130px|thumb|The same vector can be represented in two different bases (purple and red arrows).)In mathematics, a set {{mvar|B}} of elements (vectors) in a vector space {{math|V}} is called a basis, if every element of {{math|V}} may be written in a unique way as a (finite) linear combination of elements of {{mvar|B}}. The coefficients of this linear combination are referred to as components or coordinates on {{mvar|B}} of the vector. The elements of a basis are called {{visible anchor|basis vectors}}.Equivalently {{mvar|B}} is a basis if its elements are linearly independent and every element of {{mvar|V}} is a linear combination of elements of {{mvar|B}}.BOOK, Halmos, Paul Richard, Paul Halmos, 1987, Finite-Dimensional Vector Spaces, 4th, Springer, New York,weblink 10, 978-0-387-90093-3, In more general terms, 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.


A basis {{math|B}} of a vector space {{math|V}} over a field {{math|F}} (such as the real numbers {{math|R}} or the complex numbers {{math|C}}) is a linearly independent subset of {{math|V}} that spans {{mvar|V}}.This means that a subset {{mvar|B}} of {{math|V}} is a basis if it satisfies the two following conditions:
  • the linear independence property:

for every finite subset {{math|{b1, ..., b'n}{{void}}}} of {{mvar|B}} and every {{math|a1, ..., a'n}} in {{math|F}}, if {{math|1=a1b1 + â‹…â‹…â‹… + a'n'b'n = 0}}, then necessarily {{math|1=a1 = â‹…â‹…â‹… = a'n = 0}};
  • the spanning property:

for every (vector) {{math|v}} in {{math|V}}, it is possible to choose {{math|v1, ..., v'n}} in {{math|F}} and {{math|b1, ..., b'n}} in {{mvar|B}} such that {{math|1=v = v1b1 + â‹…â‹…â‹… + v'n'bn}}.
The scalars {{math|vi}} are called the coordinates of the vector {{math|v}} with respect to the basis {{math|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 subset {{math|{b1, ..., bn}{{void}}}} that is considered (twice) in the above definition may be chosen as {{mvar|B}} itself.It is often convenient or even necessary to have an ordering on the basis vectors, e.g. for 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. For example, when dealing with (m, n)-matrices, the {{math|(i, j)th}} element (in the {{mvar|i}}th row and {{mvar|j}}th column) can be referred to the {{math|(mâ‹…(j - 1) + i)}}th element of a basis consisting of the (m, n)-unit-matrices (varying column-indices before row-indices). For emphasizing that an order has been chosen, one speaks of an ordered basis, which is therefore not simply an unstructured set, but e.g. a sequence, or an indexed family, or similar; see Ordered bases and coordinates below.


File:Basis graph (no label).svg|thumb|400px|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 dependentlinearly dependent

(a, b) + (c, d) = (a + c, b+d),
and scalar multiplication
lambda (a,b) = (lambda a, lambda b),
where lambda is any real number. A simple basis of this vector space, called the standard basis consists of the two vectors {{math|1=e1 = (1,0)}} and {{math|1=e2 = (0,1)}}, since, any vector {{math|1=v = (a, b)}} of {{math|R2}} may be uniquely written as
v= ae_1+be_2.
Any other pair of linearly independent vectors of {{math|R2}}, such as {{math|(1, 1)}} and {{math|(−1, 2)}}, forms also a basis of R2.
  • More generally, if {{mvar|F}} is a field, the set F^n of {{mvar|n}}-tuples of elements of {{mvar|F}} is a vector space for similarly defined addition and scalar multiplication. Let

e_i = (0, ldots, 0,1,0,ldots, 0)
be the {{mvar|n}}-tuple with all components equal to 0, except the {{mvar|i}}th, which is 1. Then e_1, ldots, e_n is a basis of F^n, which is called the standard basis of F^n.

B={1, X, X^2, ldots}.
Any set of polynomials such that there is exactly one polynomial of each degree is also a basis. Such a set of polynomials is called a polynomial sequence. Example (among many) of such polynomial sequences are Bernstein basis polynomials, and Chebyshev polynomials.


Many properties of finite bases result from the Steinitz exchange lemma, which states that, given a finite spanning set {{mvar|S}} and a linearly independent subset {{mvar|L}} of {{mvar|n}} elements of {{mvar|S}}, one may replace {{mvar|n}} well chosen elements of {{mvar|S}} by the elements of {{mvar|L}} for getting a spanning set containing {{mvar|L}}, having its other elements in {{mvar|S}}, and having the same number of elements as {{mvar|S}}.Most properties resulting from the 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 {{mvar|V}} is a vector space over a field {{mvar|F}}, then:
  • If {{mvar|L}} is a linearly independent subset of a spanning set {{math|S ⊆ V}}, then there is a basis {{mvar|B}} such that

Lsubseteq Bsubseteq S.
  • {{mvar|V}} has a basis (this is the preceding property with {{mvar|L}} being the empty set, and {{math|1=S = V}}).
  • All bases of {{mvar|V}} have the same cardinality, which is called the dimension of {{mvar|V}}. This is the dimension theorem.
  • A generating set {{mvar|S}} is a basis of {{mvar|V}} if and only if it is minimal, that is, no proper subset of {{mvar|S}} is also a generating set of {{mvar|V}}.
  • A linearly independent set {{mvar|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 {{mvar|V}} is a vector space of dimension {{mvar|n}}, then:
  • A subset of {{mvar|V}} with {{mvar|n}} elements is a basis if and only if it is linearly independent.
  • A subset of {{mvar|V}} with {{mvar|n}} elements is a basis if and only if it is spanning set of {{mvar|V}}.

Coordinates {{anchor|Ordered bases and coordinates}}

Let {{mvar|V}} be a vector space of finite dimension {{mvar|n}} over a field {{mvar|F}}, and
B={b_1, ldots, b_n}
be a basis of {{mvar|V}}. By definition of a basis, for every {{mvar|v}} in {{mvar|V}} may be written, in a unique way,
v=lambda_1 b_1 + cdots + lambda_n b_n,
where the coefficients lambda_1, ldots, lambda_n are scalars (that is, elements of {{mvar|F}}), which are called the coordinates of {{mvar|v}} over {{mvar|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, 3b_1 +2b_2 and 2b_1 +3b_2 have the same set of coefficients {{math|{2, 3}{{void}}}}, 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, F^n be the set of the {{mvar|n}}-tuples of elements of {{mvar|F}}. This set is an {{mvar|F}}-vector space, with addition and scalar multiplication defined component-wise. The map
varphi: (lambda_1, ldots, lambda_n) mapsto lambda_1 b_1 + cdots + lambda_n b_n
is a linear isomorphism from the vector space F^n onto {{mvar|V}}. In other words, F^n is the coordinate space of {{mvar|V}}, and the {{mvar|n}}-tuple varphi^{-1}(v) is the coordinate vector of {{mvar|v}}.The inverse image by varphi of b_i is the {{mvar|n}}-tuple e_i all of whose components are 0, except the {{mvar|i}}th that is 1. The e_i form an ordered basis of F^n, which is called its standard basis or canonical basis. The ordered basis {{mvar|B}} is the image by varphi of the canonical basis of F^n. It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of F^n, and that every linear isomorphism from F^n onto {{mvar|V}} may be defined as the isomorphism that maps the canonical basis of F^n onto a given ordered basis of {{mvar|V}}. In other words it is equivalent to define an ordered basis of {{mvar|V}}, or a linear isomorphism from F^n onto {{mvar|V}}.

Change of basis

Let {{math|V}} be a vector space of dimension {{mvar|n}} over a field {{math|F}}. Given two (ordered) bases B_mathrm {old}=(v_1, ldots, v_n) and B_mathrm {new}=(w_1, ldots, w_n) of {{math|V}}, it is often useful to express the coordinates of a vector {{mvar|x}} with respect to B_mathrm {old} in terms of the coordinates with respect to B_mathrm {new}. 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 B_mathrm {old} and B_mathrm {new} 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,
w_j=sum_{i=1}^n a_{i,j}v_i.
If (x_1, ldots, x_n) and (y_1, ldots, y_n) are the coordinates of a vector {{mvar|x}} over the old and the new basis respectively, the change-of-basis formula is
x_i = sum_{j=1}^n a_{i,j}y_j,
for {{math|1=i = 1, ..., n}}.This formula may be concisely written in matrix notation. Let {{mvar|A}} be the matrix of the a_{i,j}, and
X= begin{pmatrix}x_1vdotsx_nend{pmatrix}quad and quad Y= begin{pmatrix}y_1vdotsy_nend{pmatrix}
be the column vectors of the coordinates of {{mvar|v}} 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 vector {{mvar|x}} on the two bases: one has
x=sum_{i=1}^n x_i v_i,
x&=sum_{j=1}^n y_j w_j
&=sum_{j=1}^n y_jsum_{i=1}^n a_{i,j}v_i
&=sum_{i=1}^n left(sum_{j=1}^n a_{i,j}y_jright)v_i.
end{align}The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here B_mathrm {old}; that is
x_i = sum_{j=1}^n a_{i,j}y_j,
for {{math|1=i = 1, ..., n}}.

Related notions

Free module

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 group, and, if {{mvar|G}} is a subgroup of a finitely generated free abelian group {{mvar|H}} (that is an abelian group that has a finite basis), there is a basis e_1, ldots, e_n of {{mvar|H}} and an integer {{math|0 ≤ k ≤ n}} such that a_1e_1, ldots, a_ke_k is a basis of {{mvar|G}}, for some nonzero integers a_1, ldots, a_k. For details, see {{slink|Free abelian group|Subgroups}}.


In the context of infinite-dimensional vector spaces over the real or complex numbers, the term {{visible anchor|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 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 2^{aleph_0}, where aleph_0 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 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, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (non-complete) normed spaces which have countable Hamel bases. Consider c_{00}, the space of the sequences x=(x_n) of real numbers which have only finitely many non-zero elements, with the norm |x|=sup_n |x_n|. Its standard basis, consisting of the sequences having only one non-zero 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 square-integrable on this interval, i.e., functions f satisfying
int_0^{2pi} left|f(x)right|^2,dx

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