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73-413: Voudon may refer to: Voudon (algebra) , a 256-dimensional algebra over the real numbers Alternative spelling for voudou practitioner Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Voudon . If an internal link led you here, you may wish to change the link to point directly to

146-406: A 1 e 1 , … , a k e k {\displaystyle a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}} is a basis of G , for some nonzero integers a 1 , … , a k {\displaystyle a_{1},\ldots ,a_{k}} . For details, see Free abelian group § Subgroups . In

219-506: A i , j v i = ∑ i = 1 n ( ∑ j = 1 n a i , j y j ) v i . {\displaystyle \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}.} The change-of-basis formula results then from

292-661: A k , b k . But many 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 . The geometric notions of an affine space , projective space , convex set , and cone have related notions of basis . An affine basis for an n -dimensional affine space

365-450: A ≤ 2 n − 1 {\displaystyle e_{a},1\leq a\leq 2^{n}-1} , is encoded in the properties of the projective space P G ( n − 1 , 2 ) {\displaystyle PG(n-1,2)} if these imaginary units are regarded as points and distinguished triads of them { e a , e b , e c } , 1 ≤

438-407: A < b < c ≤ 2 n − 1 {\displaystyle \{e_{a},e_{b},e_{c}\},1\leq a<b<c\leq 2^{n}-1} and e a ⋅ e b = ± e c {\displaystyle e_{a}\cdot e_{b}=\pm e_{c}} , as lines. This projective space is seen to feature two distinct kinds of lines according as

511-514: A + b = c {\displaystyle a+b=c} or a + b ≠ c {\displaystyle a+b\neq c} . Furthermore, Saniga, Holweck & Pracna (2015) state that: The corresponding point-line incidence structure is found to be a specific binomial configuration C n {\displaystyle {\mathcal {C}}_{n}} ; in particular, C 3 {\displaystyle {\mathcal {C}}_{3}} ( octonions )

584-438: A Hilbert basis (linear programming) . For a probability distribution in R with a probability density function , such as the equidistribution in an n -dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one , which is due to the fact that n linearly dependent vectors x 1 , ..., x n in R should satisfy

657-412: 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

730-464: A linearly independent set L of n elements of V , one may replace n well-chosen elements of S by the elements of L to get a spanning set containing L , having its other elements in S , and having the same number of elements as S . 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

803-448: A basis of R . More generally, if F is a field , the set F n {\displaystyle F^{n}} of n -tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let e i = ( 0 , … , 0 , 1 , 0 , … , 0 ) {\displaystyle \mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)} be

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876-580: A basis of V . By definition of a basis, every v in V may be written, in a unique way, as v = λ 1 b 1 + ⋯ + λ n b n , {\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},} where the coefficients λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are scalars (that is, elements of F ), which are called

949-388: 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 e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} of H and an integer 0 ≤ k ≤ n such that

1022-420: Is n + 1 {\displaystyle n+1} points in general linear position . A projective basis is n + 2 {\displaystyle n+2} points in general position, in a projective space of dimension n . A convex basis of a polytope is the set of the vertices of its convex hull . A cone basis consists of one point by edge of a polygonal cone. See also

1095-622: Is isomorphic to the Pasch (6 2 ,4 3 )-configuration, C 4 {\displaystyle {\mathcal {C}}_{4}} ( sedenions ) is the famous Desargues (10 3 )-configuration, C 5 {\displaystyle {\mathcal {C}}_{5}} (32-nions) coincides with the Cayley–Salmon (15 4 ,20 3 )-configuration found in the well-known Pascal mystic hexagram and C 6 {\displaystyle {\mathcal {C}}_{6}} (64-nions)

1168-405: Is a linear combination of the unit trigintaduonions e 0 {\displaystyle e_{0}} , e 1 {\displaystyle e_{1}} , e 2 {\displaystyle e_{2}} , e 3 {\displaystyle e_{3}} , ..., e 31 {\displaystyle e_{31}} , which form a basis of

1241-572: Is a linear isomorphism from the vector space F n {\displaystyle F^{n}} onto V . In other words, F n {\displaystyle F^{n}} is the coordinate space of V , and the n -tuple φ − 1 ( v ) {\displaystyle \varphi ^{-1}(\mathbf {v} )} is the coordinate vector of v . The inverse image by φ {\displaystyle \varphi } of b i {\displaystyle \mathbf {b} _{i}}

1314-413: Is a basis of V . Since L max belongs to X , we already know that L max is a linearly independent subset of V . If there were some vector w of V that is not in the span of L max , then w would not be an element of L max either. Let L w = L max ∪ { w } . This set is an element of X , that is, it is a linearly independent subset of V (because w

1387-399: Is a linearly independent subset of V that spans V . This means that a subset B of V is a basis if it satisfies the two following conditions: The scalars a i {\displaystyle a_{i}} 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

1460-400: Is a manifestation of the so-called measure concentration phenomenon . 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] 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

1533-734: Is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame ). Let, as usual, F n {\displaystyle F^{n}} 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 φ : ( λ 1 , … , λ n ) ↦ λ 1 b 1 + ⋯ + λ n b n {\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}}

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1606-521: Is an element of X . Therefore, L Y 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 L max of X satisfying the condition that whenever L max ⊆ L for some element L of X , then L = L max . It remains to prove that L max

1679-529: Is any real number. A simple basis of this vector space consists of the two vectors e 1 = (1, 0) and e 2 = (0, 1) . These vectors form a basis (called the standard basis ) because any vector v = ( a , b ) of R may be uniquely written as v = a e 1 + b e 2 . {\displaystyle \mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.} Any other pair of linearly independent vectors of R , such as (1, 1) and (−1, 2) , forms also

1752-469: Is customary to refer to B o l d {\displaystyle B_{\mathrm {old} }} and B n e w {\displaystyle 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

1825-421: Is denoted, as usual, by ⊆ . Let Y be a subset of X that is totally ordered by ⊆ , and let L Y 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 L Y is a subset of an element of Y , which is a linearly independent subset of V , and hence L Y is linearly independent. Thus L Y

1898-484: Is distributive over addition. As with the sedenions, the trigintaduonions contain zero divisors and are thus not a division algebra . Whereas octonion unit multiplication patterns can be geometrically represented by PG(2,2) (also known as the Fano plane ) and sedenion unit multiplication by PG(3,2) , trigintaduonion unit multiplication can be geometrically represented by PG(4,2). This can be also extended to PG(5,2) for

1971-732: 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 ∫ 0 2 π | f ( x ) | 2 d x < ∞ . {\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .} The functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are linearly independent, and every function f that

2044-525: 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 F [ X ] that are not of this form. Many properties of finite bases result from the Steinitz exchange lemma , which states that, for any vector space V , given a finite spanning set S and

2117-444: 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 : B = { 1 , X , X 2 , … } . {\displaystyle B=\{1,X,X^{2},\ldots \}.} Any set of polynomials such that there

2190-602: Is identical with a particular (21 5 ,35 3 )-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration . The configuration of 2 n {\displaystyle 2^{n}} -nions can thus be generalized as: ( n + 1 2 ) n − 1 , ( n + 1 3 ) 3 {\displaystyle {\binom {n+1}{2}}_{n-1},{\binom {n+1}{3}}_{3}} The multiplication of

2263-777: 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 c 00 {\displaystyle c_{00}} ,

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2336-453: Is neither commutative nor associative . However, being products of a Cayley–Dickson construction, trigintaduonions have the property of power associativity , which can be stated as that, for any element x {\displaystyle x} of T {\displaystyle \mathbb {T} } , the power x n {\displaystyle x^{n}} is well defined. They are also flexible , and multiplication

2409-442: Is not in the span of L max , and L max is independent). As L max ⊆ L w , and L max ≠ L w (because L w contains the vector w that is not contained in L max ), this contradicts the maximality of L max . Thus this shows that L max spans V . Hence L max is linearly independent and spans V . It is thus a basis of V , and this proves that every vector space has

2482-439: Is often useful to express the coordinates of a vector x with respect to B o l d {\displaystyle B_{\mathrm {old} }} in terms of the coordinates with respect to B n e w . {\displaystyle 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

2555-743: Is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that lim n → ∞ ∫ 0 2 π | a 0 + ∑ k = 1 n ( a k cos ⁡ ( k x ) + b k sin ⁡ ( k x ) ) − f ( x ) | 2 d x = 0 {\displaystyle \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} for suitable (real or complex) coefficients

2628-513: 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

2701-404: Is the n -tuple e i {\displaystyle \mathbf {e} _{i}} all of whose components are 0, except the i th that is 1. The e i {\displaystyle \mathbf {e} _{i}} form an ordered basis of F n {\displaystyle F^{n}} , which is called its standard basis or canonical basis . The ordered basis B

2774-481: Is the image by φ {\displaystyle \varphi } of the canonical basis of F n {\displaystyle 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 {\displaystyle F^{n}} , and that every linear isomorphism from F n {\displaystyle F^{n}} onto V may be defined as

2847-533: 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

2920-408: Is the trigintaduonion multiplication table for e j , 16 ≤ j ≤ 31 {\displaystyle e_{j},16\leq j\leq 31} . There are 155 distinguished triples (or triads) of imaginary trigintaduonion units in the trigintaduonion multiplication table, which are listed below. In comparison, the octonions have 7 such triples, the sedenions have 35, while

2993-516: Is ε-orthogonal to y if | ⟨ x , y ⟩ | / ( ‖ x ‖ ‖ y ‖ ) < ε {\displaystyle \left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon } (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

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3066-431: The axiom of choice or a weaker form of it, such as the ultrafilter lemma . If V is a vector space over a field F , then: If V is a vector space of dimension n , then: Let V be a vector space of finite dimension n over a field F , and B = { b 1 , … , b n } {\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}} be

3139-725: The column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is X = A Y . {\displaystyle X=AY.} The formula can be proven by considering the decomposition of the vector x on the two bases: one has x = ∑ i = 1 n x i v i , {\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},} and x = ∑ j = 1 n y j w j = ∑ j = 1 n y j ∑ i = 1 n

3212-874: The complex numbers C {\displaystyle \mathbb {C} } : An algebra of dimension 32 over the real numbers R {\displaystyle \mathbb {R} } : R , C , H , O , S {\displaystyle \mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O} ,\mathbb {S} } are all subsets of T {\displaystyle \mathbb {T} } . This relation can be expressed as: R ⊂ C ⊂ H ⊂ O ⊂ S ⊂ T ⊂ ⋯ {\displaystyle \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \mathbb {O} \subset \mathbb {S} \subset \mathbb {T} \subset \cdots } Like octonions and sedenions , multiplication of trigintaduonions

3285-513: The coordinates of v over B . 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, 3 b 1 + 2 b 2 {\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}} and 2 b 1 + 3 b 2 {\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}} have

3358-419: 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 . A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C )

3431-407: The multiplication table for the sedenions . The top left quadrant of the table, for e i , 0 ≤ i ≤ 7 {\displaystyle e_{i},0\leq i\leq 7} and e j , 0 ≤ j ≤ 7 {\displaystyle e_{j},0\leq j\leq 7} , corresponds to the multiplication table for the octonions . Below

3504-452: The n -tuple with all components equal to 0, except the i th, which is 1. Then e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} is a basis of F n , {\displaystyle F^{n},} which is called the standard basis of F n . {\displaystyle F^{n}.} A different flavor of example

3577-529: The ordered pairs of real numbers is a vector space under the operations of component-wise addition ( a , b ) + ( c , d ) = ( a + c , b + d ) {\displaystyle (a,b)+(c,d)=(a+c,b+d)} and scalar multiplication λ ( a , b ) = ( λ a , λ b ) , {\displaystyle \lambda (a,b)=(\lambda a,\lambda b),} where λ {\displaystyle \lambda }

3650-485: The vector space of trigintaduonions. Every trigintaduonion can be represented in the form with real coefficients x i . The trigintaduonions can be obtained by applying the Cayley–Dickson construction to the sedenions , which can be mathematically expressed as T = C D ( S , 1 ) {\displaystyle \mathbb {T} ={\mathcal {CD}}(\mathbb {S} ,1)} . Applying

3723-416: The 64-nions, as explained in the abstract of Saniga, Holweck & Pracna (2015) : Given a 2 n {\displaystyle 2^{n}} -dimensional Cayley–Dickson algebra , where 3 ≤ n ≤ 6 {\displaystyle 3\leq n\leq 6} , we first observe that the multiplication table of its imaginary units e a , 1 ≤

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3796-603: The Cayley–Dickson construction to the trigintaduonions yields a 64-dimensional algebra called the 64-ions , 64-nions , sexagintaquatronions , or sexagintaquattuornions , sometimes also known as the chingons . As a result, the trigintaduonions can also be defined as the following. An algebra of dimension 4 over the octonions O {\displaystyle \mathbb {O} } : An algebra of dimension 8 over quaternions H {\displaystyle \mathbb {H} } : An algebra of dimension 16 over

3869-506: The algebras immediately following the sedenions are the pathions (i.e. trigintaduonions), chingons, routons, and voudons. They are summarized as follows. 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

3942-435: 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

4015-448: 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. The set R of

4088-411: 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 ℵ 0 {\displaystyle 2^{\aleph _{0}}} , where ℵ 0 {\displaystyle \aleph _{0}} ( aleph-nought )

4161-435: 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. 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 partially ordered by inclusion, which

4234-515: The context of infinite-dimensional vector spaces over the real or complex numbers, the term 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

4307-400: The coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . This formula may be concisely written in matrix notation. Let A be

4380-424: 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

4453-416: The equation det[ x 1 ⋯ x n ] = 0 (zero determinant of the matrix with columns x i ), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases. 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

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4526-427: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Voudon&oldid=1250510008 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Voudon (algebra) The word trigintaduonion is derived from Latin triginta 'thirty' + duo 'two' +

4599-826: The isomorphism that maps the canonical basis of F n {\displaystyle F^{n}} 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 F n {\displaystyle F^{n}} onto V . Let V be a vector space of dimension n over a field F . Given two (ordered) bases B old = ( v 1 , … , v n ) {\displaystyle B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})} and B new = ( w 1 , … , w n ) {\displaystyle B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})} of V , it

4672-495: The matrix of the a i , j {\displaystyle a_{i,j}} , and X = [ x 1 ⋮ x n ] and Y = [ y 1 ⋮ y n ] {\displaystyle X={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}\quad {\text{and}}\quad Y={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}} be

4745-713: 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 = ∑ i = 1 n a i , j v i . {\displaystyle \mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.} If ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} and ( y 1 , … , y n ) {\displaystyle (y_{1},\ldots ,y_{n})} are

4818-621: 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 N random vectors are all pairwise ε-orthogonal with probability 1 − θ . This N growth exponentially with dimension n and N ≫ n {\displaystyle N\gg n} for sufficiently big n . This property of random bases

4891-425: The same set of coefficients {2, 3} , 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 ,

4964-473: The sexagintaquatronions have 651 (See OEIS A171477). The first computational algorithm for the multiplication of trigintaduonions was developed by Cariow & Cariowa (2014) . The trigintaduonions have applications in particle physics , quantum physics , and other branches of modern physics. More recently, the trigintaduonions and other hypercomplex numbers have also been used in neural network research and cryptography . Robert de Marrais's terms for

5037-436: The space of the sequences x = ( x n ) {\displaystyle x=(x_{n})} of real numbers that have only finitely many non-zero elements, with the norm ‖ x ‖ = sup n | x n | {\textstyle \|x\|=\sup _{n}|x_{n}|} . Its standard basis , consisting of the sequences having only one non-zero element, which

5110-622: The suffix - nion , which is used for hypercomplex number systems. Although trigintaduonion is typically the more widely used term, Robert P. C. de Marrais instead uses the term pathion in reference to the 32 paths of wisdom from the Kabbalistic ( Jewish mystical ) text Sefer Yetzirah , since pathion is shorter and easier to remember and pronounce. It is represented by blackboard bold P {\displaystyle \mathbb {P} } . Other alternative names include 32-ion , 32-nion , 2 -ion , and 2 -nion . Every trigintaduonion

5183-408: The uniqueness of the decomposition of a vector over a basis, here B old {\displaystyle B_{\text{old}}} ; that is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . If one replaces the field occurring in

5256-480: The unit trigintaduonions is illustrated in the two tables below. Combined, they form a single 32×32 table with 1024 cells. Below is the trigintaduonion multiplication table for e j , 0 ≤ j ≤ 15 {\displaystyle e_{j},0\leq j\leq 15} . The top half of this table, for e i , 0 ≤ i ≤ 15 {\displaystyle e_{i},0\leq i\leq 15} , corresponds to

5329-409: 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 linearly independent and every element of V is a linear combination of elements of B . 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

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