7004:
5239:
807:
38:
7268:
778:, 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
3933:
4986:
5372:
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
3687:
3561:
2294:
5355:
4454:
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
4781:
4123:. 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
5487:
4347:
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
1933:
3436:
701:
390:
2185:
3098:
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.
2927:
2858:
3184:
4754:
4264:
3682:
1106:
1838:
4107:
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
3928:{\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}.}
308:
5572:
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
1252:
4179:
1310:
618:
4502:. 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 (
980:
5016:
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
2105:
2055:
4045:
3374:
1979:
5568:
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
5273:
5397:
2409:
4672:
are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval that are square-integrable on this interval, i.e., functions
1640:
1480:
1857:
3288:
3236:
918:
495:
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4315:
3010:
562:
436:
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3049:
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2508:
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6185:
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.
4416:
3963:
6719:
Der
Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry)
6518:"Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)"
4981:{\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}
4445:
1566:
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
4576:
2600:
2446:
1000:
3429:
5543:
4532:
1374:
1340:
3603:
2755:
2720:
2689:
2660:
2628:
2568:
2356:
2325:
2165:
1164:
736:
5382:
5122:
5084:
5254:, 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.
3105:
4679:
4474:
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
5556:
The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the
2123:
similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an
3616:
148:
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
1050:
628:
317:
1781:
5564:. 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
6090:. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true. Thus the two assertions are equivalent.
3556:{\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}}}
1180:
5577:, 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented.
2863:
2794:
6773:
5235:), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.
923:
5258:
It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For
4199:
6717:
3970:
3299:
6259:
Igelnik, B.; Pao, Y.-H. (1995). "Stochastic choice of basis functions in adaptive function approximation and the functional-link net".
6803:
4112:. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through
5242:
Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the
2289:{\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}}
818:. 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
7661:
6862:
257:
6795:
6790:
2003:
of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same
17:
7195:
7253:
4134:
1265:
573:
7810:
1610:
1425:
6655:
6497:
6472:
6444:
6206:
6159:
843:
6703:
6302:; Tyukin, Ivan Y.; Prokhorov, Danil V.; Sofeikov, Konstantin I. (2016). "Approximation with Random Bases: Pro et Contra".
5350:{\displaystyle \left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon }
2060:
2010:
1685:
7845:
7524:
5153:
1938:
2634:
It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of
7304:
6559:
Betrachtungen über einige
Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry)
3094:, respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has
7726:
7243:
7205:
7141:
4644:, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.
3937:
The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here
2373:
6760:
Calcolo
Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva
6304:
3016:, that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to
7577:
7509:
6833:
6647:
6436:
6357:
1718:
is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.
7602:
6983:
6855:
5175:
3576:
3241:
3189:
448:
31:
141:
A vector space can have several bases; however all the bases have the same number of elements, called the
7840:
7088:
6938:
6828:
4274:
2977:
521:
395:
5482:{\displaystyle N\leq {\exp }{\bigl (}{\tfrac {1}{4}}\varepsilon ^{2}n{\bigr )}{\sqrt {-\ln(1-\theta )}}}
4582:
3054:
3019:
2942:
7651:
7471:
6993:
6887:
4097:
775:
6713:
2517:
2484:
2451:
7323:
7233:
6882:
6104:
4387:
7805:
6371:
3941:
7948:
7907:
7825:
7779:
7486:
7225:
7108:
6224:
5165:
4456:
3095:
1677:
1397:
758:
142:
7953:
7877:
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7481:
7451:
7271:
7200:
6978:
6848:
6811:
6525:
6115:
5087:
4423:
4320:
1509:
4331:
In the context of infinite-dimensional vector spaces over the real or complex numbers, the term
7835:
7691:
7035:
6968:
6699:
6668:
6366:
4542:
4499:
2120:
1928:{\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},}
6669:"Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung f(x+y)=f(x)+f(y)"
3102:
Typically, the new basis vectors are given by their coordinates over the old basis, that is,
2585:
2431:
985:
7917:
7872:
7352:
7297:
7050:
7045:
7040:
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6918:
6723:
6149:
5045:
4778:
that is square-integrable on is an "infinite linear combination" of them, in the sense that
3401:
1487:
1482:
Any set of polynomials such that there is exactly one polynomial of each degree (such as the
6633:
6196:
5522:
4510:
1349:
1315:
7892:
7820:
7706:
7572:
7534:
7466:
7060:
7025:
7012:
6903:
6823:
6733:
6608:
6585:
6454:
5187:
4085:
3390:
2733:
2698:
2667:
2638:
2606:
2546:
2334:
2303:
2143:
2124:
1524:
1483:
1142:
714:
709:
119:
2119:
the basis elements by the first natural numbers. Then, the coordinates of a vector form a
8:
7769:
7592:
7582:
7431:
7416:
7372:
7238:
7118:
7093:
6943:
5259:
5101:
5063:
4365:
4089:
1491:
1136:
186:
6118: – Similar to the basis of a vector space, but not necessarily linearly independent
3573:
in the old and the new basis respectively, then the formula for changing coordinates is
2182:-vector space, with addition and scalar multiplication defined component-wise. The map
7902:
7759:
7612:
7426:
7362:
6948:
6688:
6386:
6327:
6309:
6241:
6099:
4361:
4081:
4075:
2297:
819:
156:
127:
84:
7897:
7666:
7641:
7456:
7367:
7347:
7146:
7103:
7030:
6923:
6767:
6692:
6651:
6641:
6629:
6542:
6517:
6493:
6468:
6440:
6425:
6276:
6245:
6202:
6155:
5017:
2412:
1571:
783:
152:
50:
1002:
is any real number. A simple basis of this vector space consists of the two vectors
7912:
7587:
7554:
7539:
7421:
7290:
7151:
7055:
6908:
6742:
6680:
6594:
6567:
6534:
6390:
6376:
6331:
6319:
6299:
6268:
6233:
5037:
5021:
4349:
2359:
2129:
4468:
41:
The same vector can be represented in two different bases (purple and red arrows).
7882:
7830:
7774:
7754:
7656:
7544:
7411:
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7003:
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6109:
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5787:
4381:
4113:
2770:
2575:
1567:
1379:
6349:
5238:
1177:
is a vector space for similarly defined addition and scalar multiplication. Let
696:{\displaystyle \mathbf {v} =a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}}
385:{\displaystyle c_{1}\mathbf {v} _{1}+\cdots +c_{m}\mathbf {v} _{m}=\mathbf {0} }
7922:
7887:
7784:
7617:
7607:
7597:
7519:
7491:
7476:
7461:
7377:
7215:
7136:
6871:
6755:
6615:
6345:
4653:
4641:
4479:
4357:
2571:
1412:
1023:
811:
791:
204:
7867:
6580:
6323:
4578:
of real numbers that have only finitely many non-zero elements, with the norm
7942:
7859:
7764:
7676:
7549:
7248:
7171:
7131:
7098:
7078:
6731:
Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940",
6546:
6513:
5625:
4460:
4353:
4120:
4080:
If one replaces the field occurring in the definition of a vector space by a
3564:
3296:
over the old and the new basis respectively, the change-of-basis formula is
2425:
4376:
of rational numbers, Hamel bases are uncountable, and have specifically the
7927:
7731:
7716:
7681:
7529:
7514:
7181:
7070:
7020:
6913:
6747:
6599:
6538:
6280:
5033:
4487:
4464:
4448:
4093:
1517:
833:
195:
177:
135:
57:
37:
6572:Éléments d'histoire des mathématiques (Elements of history of mathematics)
6557:
4131:(that is an abelian group that has a finite basis), then there is a basis
2922:{\displaystyle B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})}
2853:{\displaystyle B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})}
7815:
7789:
7711:
7400:
7339:
7161:
7126:
7083:
6928:
6664:
6145:
5182:-dimensional ball with respect to Lebesgue measure, it can be shown that
5141:
4495:
4377:
4340:
4071:
3607:
The formula can be proven by considering the decomposition of the vector
1673:
837:
806:
771:
223:
46:
6222:
Kuczma, Marek (1970). "Some remarks about additive functions on cones".
3179:{\displaystyle \mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.}
7696:
7190:
6933:
6684:
6481:
6381:
6237:
5041:
1393:
754:
251:
6272:
4749:{\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .}
2111:, and are different. It is therefore often convenient to work with an
7671:
7622:
6988:
5250:. Boxplots show the second and third quartiles of this data for each
2116:
1652:
5549:. This property of random bases is a manifestation of the so-called
7701:
7686:
7156:
6314:
5137:
4537:
4451:) is the smallest infinite cardinal, the cardinal of the integers.
4259:{\displaystyle a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}}
2695:
may be defined as the isomorphism that maps the canonical basis of
1419:
787:
3677:{\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},}
840:
is a vector space under the operations of component-wise addition
7395:
7357:
6840:
6815:
5796:
has a maximal element. In other words, there exists some element
2726:. In other words, it is equivalent to define an ordered basis of
1101:{\displaystyle \mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.}
767:
itself to check for linear independence in the above definition.
91:. The coefficients of this linear combination are referred to as
5008:. But many square-integrable functions cannot be represented as
7721:
7313:
7166:
5124:
points in general position, in a projective space of dimension
1833:{\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}}
1700:
213:
6642:
Hermann
Grassmann. Translated by Lloyd C. Kannenberg. (2000),
6086:
This proof relies on Zorn's lemma, which is equivalent to the
5504:
random vectors are all pairwise ε-orthogonal with probability
5152:
consists of one point by edge of a polygonal cone. See also a
5012:
linear combinations of these basis functions, which therefore
1742:
elements is a basis if and only if it is linearly independent.
6635:
Die
Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik
6298:
2168:
1167:
303:{\displaystyle \{\mathbf {v} _{1},\dotsc ,\mathbf {v} _{m}\}}
5618:
is nonempty since the empty set is an independent subset of
5580:
5186:
randomly and independently chosen vectors will form a basis
2935:, it is often useful to express the coordinates of a vector
1247:{\displaystyle \mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)}
4119:
A module over the integers is exactly the same thing as an
1753:
elements is a basis if and only if it is a spanning set of
7282:
4506:) normed spaces that have countable Hamel bases. Consider
1490:) is also a basis. (Such a set of polynomials is called a
750:, and by the first property they are uniquely determined.
30:"Basis (mathematics)" redirects here. For other uses, see
6796:
Proof that any subspace basis has same number of elements
6581:"A general outline of the genesis of vector space theory"
4174:{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}}
4127:
is a subgroup of a finitely generated free abelian group
1305:{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}}
613:{\displaystyle \mathbf {v} _{1},\dotsc ,\mathbf {v} _{n}}
241:
is a basis if it satisfies the two following conditions:
6435:, Contemporary Mathematics volume 31, Providence, R.I.:
6120:
Pages displaying short descriptions of redirect targets
6083:, and this proves that every vector space has a basis.
4478:
is an infinite-dimensional normed vector space that is
5420:
4585:
3511:
3451:
5525:
5400:
5276:
5104:
5066:
4784:
4682:
4545:
4513:
4426:
4390:
4277:
4202:
4137:
4100:" is more commonly used than that of "spanning set".
3973:
3944:
3690:
3619:
3579:
3439:
3404:
3302:
3244:
3192:
3108:
3057:
3022:
2980:
2945:
2866:
2797:
2736:
2701:
2670:
2641:
2609:
2588:
2549:
2520:
2487:
2454:
2434:
2376:
2337:
2306:
2188:
2146:
2063:
2013:
1941:
1860:
1784:
1613:
1428:
1352:
1318:
1268:
1183:
1145:
1053:
988:
926:
846:
717:
631:
576:
524:
451:
398:
320:
260:
6350:"Proportional concentration phenomena of the sphere"
4096:
are defined exactly as for vector spaces, although "
975:{\displaystyle \lambda (a,b)=(\lambda a,\lambda b),}
770:
It is often convenient or even necessary to have an
134:. In other words, a basis is a linearly independent
2100:{\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}}
2050:{\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}}
1593:is a linearly independent subset of a spanning set
774:on the basis vectors, for example, when discussing
6294:
6292:
6290:
5953:, that is, it is a linearly independent subset of
5603:be the set of all linearly independent subsets of
5537:
5481:
5349:
5116:
5078:
4980:
4748:
4630:
4570:
4526:
4439:
4410:
4309:
4258:
4173:
4040:{\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},}
4039:
3957:
3927:
3676:
3597:
3555:
3423:
3369:{\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},}
3368:
3282:
3230:
3178:
3078:
3043:
3004:
2966:
2921:
2852:
2749:
2714:
2683:
2654:
2622:
2594:
2562:
2535:
2502:
2469:
2440:
2403:
2350:
2319:
2288:
2159:
2099:
2049:
1973:
1927:
1832:
1634:
1474:
1368:
1334:
1304:
1258:-tuple with all components equal to 0, except the
1246:
1158:
1108:Any other pair of linearly independent vectors of
1100:
994:
974:
912:
795:
761:. In this case, the finite subset can be taken as
730:
695:
612:
556:
489:
430:
384:
302:
5772:is nonempty, and every totally ordered subset of
5381:independent random vectors from a ball (they are
4955:
4821:
3905:
3851:
1974:{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}
1647:has a basis (this is the preceding property with
7940:
6426:"Existence of bases implies the axiom of choice"
6154:(4th ed.). New York: Springer. p. 10.
4786:
4599:
4459:– a large class of vector spaces including e.g.
1508:Many properties of finite bases result from the
1411:-vector space. One basis for this space is the
782:, which is therefore not simply an unstructured
151:Basis vectors find applications in the study of
6287:
6127: – Basis used to express spherical tensors
5560:-dimensional cube as a function of dimension,
5246:-dimensional cube as a function of dimension,
6804:"Linear combinations, span, and basis vectors"
7298:
6856:
5628:by inclusion, which is denoted, as usual, by
5446:
5414:
5224:(zero determinant of the matrix with columns
4774:are linearly independent, and every function
6814:from the original on 2021-11-17 – via
6624:(in French), Chez Firmin Didot, père et fils
5698:, which is a linearly independent subset of
4990:for suitable (real or complex) coefficients
4592:
4586:
2974:in terms of the coordinates with respect to
1827:
1791:
1559:, and having the same number of elements as
1466:
1435:
297:
261:
6112: – Coordinate change in linear algebra
5683:is totally ordered, every finite subset of
2404:{\displaystyle \varphi ^{-1}(\mathbf {v} )}
83:may be written in a unique way as a finite
7305:
7291:
6863:
6849:
6772:: CS1 maint: location missing publisher (
6258:
2510:all of whose components are 0, except the
1699:if and only if it is minimal, that is, no
1512:, which states that, for any vector space
1378:A different flavor of example is given by
1022:. These vectors form a basis (called the
6746:
6628:
6598:
6380:
6370:
6313:
5666:(which are themselves certain subsets of
5581:Proof that every vector space has a basis
4730:
3389:This formula may be concisely written in
738:are called the coordinates of the vector
27:Set of vectors used to define coordinates
7662:Covariance and contravariance of vectors
6698:
6566:
6506:
6344:
5237:
4372:viewed as a vector space over the field
1635:{\displaystyle L\subseteq B\subseteq S.}
1475:{\displaystyle B=\{1,X,X^{2},\ldots \}.}
805:
36:
6787:Instructional videos from Khan Academy
6614:
6552:
5811:satisfying the condition that whenever
5383:independent and identically distributed
2664:and that every linear isomorphism from
14:
7941:
7254:Comparison of linear algebra libraries
6712:
6646:, Kannenberg, L.C., Providence, R.I.:
6578:
6512:
6221:
6144:
6035:), this contradicts the maximality of
5357:(that is, cosine of the angle between
1770:be a vector space of finite dimension
1494:.) But there are also many bases for
7286:
6844:
6754:
6730:
6663:
6460:
6423:
6417:
6403:
6174:
5389:be a small positive number. Then for
5178:, such as the equidistribution in an
3283:{\displaystyle (y_{1},\ldots ,y_{n})}
3231:{\displaystyle (x_{1},\ldots ,x_{n})}
913:{\displaystyle (a,b)+(c,d)=(a+c,b+d)}
490:{\displaystyle c_{1}=\cdots =c_{m}=0}
103:. The elements of a basis are called
6480:
6194:
5882:is a linearly independent subset of
5660:be the union of all the elements of
5591:be any vector space over some field
5515:growth exponentially with dimension
5391:
1854:may be written, in a unique way, as
1570:or a weaker form of it, such as the
796:§ Ordered bases and coordinates
4631:{\textstyle \|x\|=\sup _{n}|x_{n}|}
4321:Free abelian group § Subgroups
4310:{\displaystyle a_{1},\ldots ,a_{k}}
3005:{\displaystyle B_{\mathrm {new} }.}
557:{\displaystyle a_{1},\dotsc ,a_{n}}
431:{\displaystyle c_{1},\dotsc ,c_{m}}
70:
24:
7525:Tensors in curvilinear coordinates
6870:
6791:Introduction to bases of subspaces
6071:is linearly independent and spans
5154:Hilbert basis (linear programming)
5140:is the set of the vertices of its
4796:
4740:
4428:
4397:
4368:. In the case of the real numbers
4060:
3079:{\displaystyle B_{\mathrm {new} }}
3070:
3067:
3064:
3044:{\displaystyle B_{\mathrm {old} }}
3035:
3032:
3029:
2993:
2990:
2987:
2967:{\displaystyle B_{\mathrm {old} }}
2958:
2955:
2952:
2764:
2107:have the same set of coefficients
1981:are scalars (that is, elements of
1844:. By definition of a basis, every
25:
7965:
6781:
5757:, that contains every element of
5020:of these spaces are essential in
1551:to get a spanning set containing
7267:
7266:
7244:Basic Linear Algebra Subprograms
7002:
6621:Théorie analytique de la chaleur
6151:Finite-Dimensional Vector Spaces
5551:measure concentration phenomenon
5190:, which is due to the fact that
4656:, one learns that the functions
4246:
4215:
4161:
4140:
3912:
3815:
3732:
3692:
3661:
3621:
3290:are the coordinates of a vector
3163:
3111:
2906:
2885:
2837:
2816:
2536:{\displaystyle \mathbf {e} _{i}}
2523:
2503:{\displaystyle \mathbf {e} _{i}}
2490:
2470:{\displaystyle \mathbf {b} _{i}}
2457:
2394:
2276:
2245:
2087:
2069:
2037:
2019:
1912:
1881:
1862:
1817:
1796:
1292:
1271:
1186:
1085:
1067:
1055:
683:
652:
633:
600:
579:
378:
364:
333:
287:
266:
7142:Seven-dimensional cross product
6201:. Berlin: Springer. p. 7.
5159:
4498:. This is a consequence of the
4411:{\displaystyle 2^{\aleph _{0}}}
4380:of the continuum, which is the
4084:, one gets the definition of a
3499:
3493:
2781:be a vector space of dimension
2730:, or a linear isomorphism from
1999:. However, if one talks of the
1726:is a vector space of dimension
1581:is a vector space over a field
1555:, having its other elements in
118:is a basis if its elements are
6397:
6338:
6252:
6215:
6188:
6179:
6168:
6138:
5713:is linearly independent. Thus
5474:
5462:
5332:
5326:
5321:
5315:
4949:
4943:
4793:
4716:
4710:
4624:
4609:
4565:
4552:
4065:
3958:{\displaystyle B_{\text{old}}}
3277:
3245:
3225:
3193:
2916:
2880:
2847:
2811:
2722:onto a given ordered basis of
2398:
2390:
2230:
2227:
2195:
2007:of coefficients. For example,
1761:
1241:
1199:
966:
948:
942:
930:
907:
883:
877:
865:
859:
847:
99:of the vector with respect to
13:
1:
7578:Exterior covariant derivative
7510:Tensor (intrinsic definition)
6648:American Mathematical Society
6616:Fourier, Jean Baptiste Joseph
6437:American Mathematical Society
6412:
6358:Israel Journal of Mathematics
5692:is a subset of an element of
5060:-dimensional affine space is
1503:
810:This picture illustrates the
162:
7603:Raising and lowering indices
6984:Eigenvalues and eigenvectors
5947:. This set is an element of
5220:should satisfy the equation
5176:probability density function
5032:The geometric notions of an
4270:, for some nonzero integers
2791:. Given two (ordered) bases
2115:; this is typically done by
1707:is also a generating set of
212:) is a linearly independent
7:
7841:Gluon field strength tensor
7312:
6829:Encyclopedia of Mathematics
6722:(in German), archived from
6574:(in French), Paris: Hermann
6093:
5918:would not be an element of
5903:that is not in the span of
5647:that is totally ordered by
5194:linearly dependent vectors
5027:
4490:), then any Hamel basis of
4440:{\displaystyle \aleph _{0}}
4326:
3613:on the two bases: one has
1714:A linearly independent set
1500:that are not of this form.
1484:Bernstein basis polynomials
1386:is a field, the collection
1047:may be uniquely written as
801:
231:. This means that a subset
10:
7970:
7652:Cartan formalism (physics)
7472:Penrose graphical notation
6464:Matrices and vector spaces
6461:Brown, William A. (1991),
5891:If there were some vector
5377:-dimensional ball. Choose
4647:
4103:Like for vector spaces, a
4069:
2768:
2602:of the canonical basis of
920:and scalar multiplication
753:A vector space that has a
744:with respect to the basis
29:
7858:
7798:
7747:
7740:
7632:
7563:
7500:
7444:
7391:
7338:
7331:
7324:Glossary of tensor theory
7320:
7262:
7224:
7180:
7117:
7069:
7011:
7000:
6896:
6878:
6808:Essence of linear algebra
6579:Dorier, Jean-Luc (1995),
6324:10.1016/j.ins.2015.09.021
6105:Basis of a linear program
6026:that is not contained in
5843:It remains to prove that
5260:spaces with inner product
5048:have related notions of
4571:{\displaystyle x=(x_{n})}
4457:topological vector spaces
2543:form an ordered basis of
7908:Gregorio Ricci-Curbastro
7780:Riemann curvature tensor
7487:Van der Waerden notation
6714:Möbius, August Ferdinand
6225:Aequationes Mathematicae
6131:
6077:. It is thus a basis of
5166:probability distribution
3012:This can be done by the
2595:{\displaystyle \varphi }
2441:{\displaystyle \varphi }
1985:), which are called the
1603:, then there is a basis
1543:well-chosen elements of
1122:, forms also a basis of
995:{\displaystyle \lambda }
7878:Elwin Bruno Christoffel
7811:Angular momentum tensor
7482:Tetrad (index notation)
7452:Abstract index notation
6705:Lectures on Quaternions
6700:Hamilton, William Rowan
6679:(3), Leipzig: 459–462,
6526:Fundamenta Mathematicae
6467:, New York: M. Dekker,
6424:Blass, Andreas (1984),
6261:IEEE Trans. Neural Netw
6195:Rees, Elmer G. (2005).
6116:Frame of a vector space
6044:. Thus this shows that
5873:, we already know that
5088:general linear position
3424:{\displaystyle a_{i,j}}
3014:change-of-basis formula
1935:where the coefficients
1510:Steinitz exchange lemma
18:Basis of a vector space
7692:Levi-Civita connection
6969:Row and column vectors
6748:10.1006/hmat.1995.1025
6600:10.1006/hmat.1995.1024
6539:10.4064/fm-3-1-133-181
5963:is not in the span of
5780:has an upper bound in
5751:: it is an element of
5737:is an upper bound for
5539:
5538:{\displaystyle N\gg n}
5483:
5351:
5255:
5118:
5080:
4982:
4859:
4750:
4632:
4572:
4528:
4527:{\displaystyle c_{00}}
4500:Baire category theorem
4441:
4412:
4311:
4260:
4175:
4041:
4007:
3959:
3929:
3876:
3848:
3796:
3765:
3719:
3678:
3648:
3599:
3567:of the coordinates of
3557:
3425:
3370:
3336:
3284:
3232:
3180:
3144:
3080:
3045:
3006:
2968:
2923:
2854:
2751:
2716:
2685:
2656:
2624:
2596:
2570:, which is called its
2564:
2537:
2504:
2471:
2442:
2405:
2352:
2321:
2300:from the vector space
2290:
2161:
2101:
2051:
1975:
1929:
1834:
1676:, which is called the
1636:
1476:
1370:
1369:{\displaystyle F^{n}.}
1336:
1335:{\displaystyle F^{n},}
1306:
1248:
1160:
1102:
996:
976:
914:
823:
732:
697:
614:
558:
491:
432:
386:
304:
77:) if every element of
42:
7918:Jan Arnoldus Schouten
7873:Augustin-Louis Cauchy
7353:Differential geometry
6974:Row and column spaces
6919:Scalar multiplication
6708:, Royal Irish Academy
6673:Mathematische Annalen
6507:Historical references
5545:for sufficiently big
5540:
5484:
5352:
5241:
5119:
5081:
4983:
4839:
4751:
4633:
4573:
4529:
4442:
4413:
4312:
4261:
4176:
4042:
3987:
3960:
3930:
3856:
3828:
3776:
3745:
3699:
3679:
3628:
3600:
3598:{\displaystyle X=AY.}
3558:
3426:
3397:be the matrix of the
3371:
3316:
3285:
3233:
3181:
3124:
3081:
3046:
3007:
2969:
2924:
2855:
2752:
2750:{\displaystyle F^{n}}
2717:
2715:{\displaystyle F^{n}}
2686:
2684:{\displaystyle F^{n}}
2657:
2655:{\displaystyle F^{n}}
2625:
2623:{\displaystyle F^{n}}
2597:
2565:
2563:{\displaystyle F^{n}}
2538:
2505:
2472:
2443:
2406:
2353:
2351:{\displaystyle F^{n}}
2322:
2320:{\displaystyle F^{n}}
2291:
2162:
2160:{\displaystyle F^{n}}
2102:
2052:
1976:
1930:
1835:
1637:
1488:Chebyshev polynomials
1477:
1403:with coefficients in
1371:
1337:
1307:
1262:th, which is 1. Then
1249:
1161:
1159:{\displaystyle F^{n}}
1103:
1026:) because any vector
997:
977:
915:
809:
733:
731:{\displaystyle a_{i}}
698:
615:
559:
492:
433:
387:
305:
145:of the vector space.
122:and every element of
40:
7893:Carl Friedrich Gauss
7826:stress–energy tensor
7821:Cauchy stress tensor
7573:Covariant derivative
7535:Antisymmetric tensor
7467:Multi-index notation
7109:Gram–Schmidt process
7061:Gaussian elimination
6734:Historia Mathematica
6586:Historia Mathematica
6488:, Berlin, New York:
6433:Axiomatic set theory
6308:. 364–365: 129–145.
6305:Information Sciences
6300:Gorban, Alexander N.
6146:Halmos, Paul Richard
6020:contains the vector
5981:is independent). As
5523:
5398:
5274:
5188:with probability one
5102:
5064:
4782:
4680:
4583:
4543:
4511:
4424:
4388:
4366:normed linear spaces
4275:
4200:
4135:
3971:
3942:
3688:
3617:
3577:
3437:
3402:
3300:
3242:
3190:
3106:
3055:
3020:
2978:
2943:
2864:
2795:
2734:
2699:
2668:
2639:
2607:
2586:
2578:. The ordered basis
2547:
2518:
2485:
2452:
2432:
2374:
2335:
2304:
2186:
2144:
2061:
2011:
1939:
1858:
1782:
1611:
1525:linearly independent
1426:
1418:, consisting of all
1350:
1342:which is called the
1316:
1266:
1181:
1143:
1051:
986:
924:
844:
715:
629:
574:
522:
449:
396:
318:
258:
120:linearly independent
114:Equivalently, a set
7770:Nonmetricity tensor
7625:(2nd-order tensors)
7593:Hodge star operator
7583:Exterior derivative
7432:Transport phenomena
7417:Continuum mechanics
7373:Multilinear algebra
7239:Numerical stability
7119:Multilinear algebra
7094:Inner product space
6944:Linear independence
6762:(in Italian), Turin
5266:is ε-orthogonal to
5117:{\displaystyle n+2}
5079:{\displaystyle n+1}
4818:
4700:
4090:linear independence
2127:, is also called a
1547:by the elements of
1492:polynomial sequence
1131:More generally, if
246:linear independence
157:frames of reference
7903:Tullio Levi-Civita
7846:Metric tensor (GR)
7760:Levi-Civita symbol
7613:Tensor contraction
7427:General relativity
7363:Euclidean geometry
6949:Linear combination
6810:. August 6, 2016.
6685:10.1007/BF01457624
6630:Grassmann, Hermann
6439:, pp. 31–33,
6418:General references
6382:10.1007/BF02784520
6238:10.1007/BF01844160
6100:Basis of a matroid
5535:
5479:
5429:
5347:
5256:
5114:
5076:
4978:
4801:
4800:
4746:
4683:
4628:
4607:
4568:
4524:
4437:
4408:
4362:Markushevich bases
4307:
4256:
4171:
4076:Free abelian group
4037:
3955:
3925:
3674:
3595:
3553:
3547:
3487:
3421:
3366:
3280:
3228:
3176:
3076:
3041:
3002:
2964:
2919:
2850:
2747:
2712:
2681:
2652:
2620:
2592:
2560:
2533:
2514:th that is 1. The
2500:
2467:
2438:
2401:
2348:
2331:. In other words,
2317:
2298:linear isomorphism
2286:
2167:be the set of the
2157:
2097:
2047:
1971:
1925:
1830:
1632:
1539:, one may replace
1472:
1366:
1332:
1302:
1244:
1156:
1098:
992:
972:
910:
824:
820:linearly dependent
794:, or similar; see
759:finite-dimensional
728:
693:
610:
554:
487:
428:
382:
300:
153:crystal structures
128:linear combination
85:linear combination
43:
7936:
7935:
7898:Hermann Grassmann
7854:
7853:
7806:Moment of inertia
7667:Differential form
7642:Affine connection
7457:Einstein notation
7440:
7439:
7368:Exterior calculus
7348:Coordinate system
7280:
7279:
7147:Geometric algebra
7104:Kronecker product
6939:Linear projection
6924:Vector projection
6657:978-0-8218-2031-5
6568:Bourbaki, Nicolas
6499:978-0-387-96412-6
6474:978-0-8247-8419-5
6446:978-0-8218-5026-8
6273:10.1109/72.471375
6208:978-3-540-12053-7
6198:Notes on Geometry
6161:978-0-387-90093-3
5821:for some element
5722:is an element of
5626:partially ordered
5499:
5498:
5477:
5428:
5018:orthonormal bases
4785:
4598:
4536:the space of the
4319:For details, see
3952:
3497:
2874:
2805:
2413:coordinate vector
2178:. This set is an
1691:A generating set
1686:dimension theorem
1572:ultrafilter lemma
1516:, given a finite
518:, one can choose
506:for every vector
502:spanning property
16:(Redirected from
7961:
7913:Bernhard Riemann
7745:
7744:
7588:Exterior product
7555:Two-point tensor
7540:Symmetric tensor
7422:Electromagnetism
7336:
7335:
7307:
7300:
7293:
7284:
7283:
7270:
7269:
7152:Exterior algebra
7089:Hadamard product
7006:
6994:Linear equations
6865:
6858:
6851:
6842:
6841:
6837:
6819:
6777:
6771:
6763:
6751:
6750:
6727:
6709:
6695:
6660:
6644:Extension Theory
6639:
6625:
6611:
6602:
6575:
6563:
6554:Bolzano, Bernard
6549:
6522:
6502:
6477:
6457:
6430:
6406:
6401:
6395:
6394:
6384:
6374:
6354:
6342:
6336:
6335:
6317:
6296:
6285:
6284:
6267:(6): 1320–1329.
6256:
6250:
6249:
6219:
6213:
6212:
6192:
6186:
6183:
6177:
6172:
6166:
6165:
6142:
6121:
6082:
6076:
6070:
6058:
6052:
6043:
6034:
6025:
6019:
6010:
5995:
5980:
5971:
5958:
5952:
5946:
5926:
5917:
5911:
5902:
5896:
5887:
5881:
5872:
5866:
5857:
5851:
5839:
5830:
5824:
5820:
5810:
5804:
5795:
5785:
5779:
5771:
5762:
5756:
5750:
5742:
5736:
5727:
5721:
5712:
5703:
5697:
5691:
5682:
5671:
5665:
5659:
5650:
5646:
5640:
5631:
5623:
5617:
5608:
5602:
5596:
5590:
5571:
5567:
5548:
5544:
5542:
5541:
5536:
5518:
5514:
5510:
5503:
5488:
5486:
5485:
5480:
5478:
5452:
5450:
5449:
5440:
5439:
5430:
5421:
5418:
5417:
5411:
5392:
5368:
5364:
5360:
5356:
5354:
5353:
5348:
5340:
5336:
5335:
5324:
5308:
5303:
5299:
5295:
5234:
5223:
5219:
5213:
5202:
5193:
5185:
5173:
5150:
5149:
5134:
5133:
5123:
5121:
5120:
5115:
5096:
5095:
5094:projective basis
5085:
5083:
5082:
5077:
5038:projective space
5022:Fourier analysis
4987:
4985:
4984:
4979:
4965:
4964:
4959:
4958:
4936:
4932:
4931:
4927:
4909:
4908:
4896:
4892:
4874:
4873:
4858:
4853:
4835:
4834:
4825:
4824:
4817:
4809:
4799:
4773:
4772:= 1, 2, 3, ... }
4755:
4753:
4752:
4747:
4729:
4728:
4723:
4719:
4699:
4691:
4671:
4670:= 1, 2, 3, ... }
4652:In the study of
4639:
4637:
4635:
4634:
4629:
4627:
4622:
4621:
4612:
4606:
4577:
4575:
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4531:
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4438:
4436:
4435:
4419:
4417:
4415:
4414:
4409:
4407:
4406:
4405:
4404:
4350:orthogonal bases
4337:
4336:
4318:
4316:
4314:
4313:
4308:
4306:
4305:
4287:
4286:
4269:
4265:
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4257:
4255:
4254:
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4243:
4242:
4224:
4223:
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4195:
4184:
4180:
4178:
4177:
4172:
4170:
4169:
4164:
4149:
4148:
4143:
4130:
4126:
4114:free resolutions
4056:
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3522:
3498:
3495:
3492:
3491:
3484:
3483:
3463:
3462:
3432:
3430:
3428:
3427:
3422:
3420:
3419:
3396:
3385:
3375:
3373:
3372:
3367:
3362:
3361:
3352:
3351:
3335:
3330:
3312:
3311:
3295:
3289:
3287:
3286:
3281:
3276:
3275:
3257:
3256:
3237:
3235:
3234:
3229:
3224:
3223:
3205:
3204:
3185:
3183:
3182:
3177:
3172:
3171:
3166:
3160:
3159:
3143:
3138:
3120:
3119:
3114:
3085:
3083:
3082:
3077:
3075:
3074:
3073:
3050:
3048:
3047:
3042:
3040:
3039:
3038:
3011:
3009:
3008:
3003:
2998:
2997:
2996:
2973:
2971:
2970:
2965:
2963:
2962:
2961:
2939:with respect to
2938:
2934:
2928:
2926:
2925:
2920:
2915:
2914:
2909:
2894:
2893:
2888:
2876:
2875:
2872:
2859:
2857:
2856:
2851:
2846:
2845:
2840:
2825:
2824:
2819:
2807:
2806:
2803:
2790:
2784:
2780:
2760:
2756:
2754:
2753:
2748:
2746:
2745:
2729:
2725:
2721:
2719:
2718:
2713:
2711:
2710:
2694:
2690:
2688:
2687:
2682:
2680:
2679:
2663:
2661:
2659:
2658:
2653:
2651:
2650:
2631:
2629:
2627:
2626:
2621:
2619:
2618:
2601:
2599:
2598:
2593:
2582:is the image by
2581:
2569:
2567:
2566:
2561:
2559:
2558:
2542:
2540:
2539:
2534:
2532:
2531:
2526:
2513:
2509:
2507:
2506:
2501:
2499:
2498:
2493:
2480:
2476:
2474:
2473:
2468:
2466:
2465:
2460:
2447:
2445:
2444:
2439:
2420:
2410:
2408:
2407:
2402:
2397:
2389:
2388:
2369:
2365:
2360:coordinate space
2357:
2355:
2354:
2349:
2347:
2346:
2330:
2326:
2324:
2323:
2318:
2316:
2315:
2295:
2293:
2292:
2287:
2285:
2284:
2279:
2273:
2272:
2254:
2253:
2248:
2242:
2241:
2226:
2225:
2207:
2206:
2181:
2177:
2171:
2166:
2164:
2163:
2158:
2156:
2155:
2130:coordinate frame
2110:
2106:
2104:
2103:
2098:
2096:
2095:
2090:
2078:
2077:
2072:
2056:
2054:
2053:
2048:
2046:
2045:
2040:
2028:
2027:
2022:
1998:
1994:
1984:
1980:
1978:
1977:
1972:
1970:
1969:
1951:
1950:
1934:
1932:
1931:
1926:
1921:
1920:
1915:
1909:
1908:
1890:
1889:
1884:
1878:
1877:
1865:
1853:
1849:
1843:
1839:
1837:
1836:
1831:
1826:
1825:
1820:
1805:
1804:
1799:
1777:
1773:
1769:
1756:
1752:
1748:
1741:
1737:
1729:
1725:
1717:
1710:
1706:
1698:
1694:
1683:
1671:
1664:
1650:
1646:
1641:
1639:
1638:
1633:
1606:
1602:
1592:
1584:
1580:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1522:
1515:
1499:
1481:
1479:
1478:
1473:
1459:
1458:
1417:
1410:
1406:
1402:
1391:
1385:
1380:polynomial rings
1375:
1373:
1372:
1367:
1362:
1361:
1341:
1339:
1338:
1333:
1328:
1327:
1311:
1309:
1308:
1303:
1301:
1300:
1295:
1280:
1279:
1274:
1261:
1257:
1253:
1251:
1250:
1245:
1195:
1194:
1189:
1176:
1170:
1165:
1163:
1162:
1157:
1155:
1154:
1134:
1127:
1121:
1117:
1113:
1107:
1105:
1104:
1099:
1094:
1093:
1088:
1076:
1075:
1070:
1058:
1046:
1040:
1021:
1011:
1001:
999:
998:
993:
981:
979:
978:
973:
919:
917:
916:
911:
831:
766:
757:basis is called
749:
743:
737:
735:
734:
729:
727:
726:
704:
702:
700:
699:
694:
692:
691:
686:
680:
679:
661:
660:
655:
649:
648:
636:
623:
619:
617:
616:
611:
609:
608:
603:
588:
587:
582:
569:
563:
561:
560:
555:
553:
552:
534:
533:
517:
511:
498:
496:
494:
493:
488:
480:
479:
461:
460:
443:
437:
435:
434:
429:
427:
426:
408:
407:
391:
389:
388:
383:
381:
373:
372:
367:
361:
360:
342:
341:
336:
330:
329:
313:
309:
307:
306:
301:
296:
295:
290:
275:
274:
269:
240:
234:
230:
221:
211:
202:
193:
184:
175:
133:
125:
117:
109:
108:
102:
90:
82:
72:
64:
56:of vectors in a
55:
21:
7969:
7968:
7964:
7963:
7962:
7960:
7959:
7958:
7949:Axiom of choice
7939:
7938:
7937:
7932:
7883:Albert Einstein
7850:
7831:Einstein tensor
7794:
7775:Ricci curvature
7755:Kronecker delta
7741:Notable tensors
7736:
7657:Connection form
7634:
7628:
7559:
7545:Tensor operator
7502:
7496:
7436:
7412:Computer vision
7405:
7387:
7383:Tensor calculus
7327:
7316:
7311:
7281:
7276:
7258:
7220:
7176:
7113:
7065:
7007:
6998:
6964:Change of basis
6954:Multilinear map
6892:
6874:
6869:
6822:
6802:
6784:
6765:
6764:
6756:Peano, Giuseppe
6658:
6520:
6509:
6500:
6490:Springer-Verlag
6475:
6447:
6428:
6420:
6415:
6410:
6409:
6402:
6398:
6372:10.1.1.417.2375
6352:
6346:Artstein, Shiri
6343:
6339:
6297:
6288:
6257:
6253:
6220:
6216:
6209:
6193:
6189:
6184:
6180:
6173:
6169:
6162:
6143:
6139:
6134:
6125:Spherical basis
6119:
6110:Change of basis
6096:
6088:axiom of choice
6078:
6072:
6069:
6063:
6054:
6051:
6045:
6042:
6036:
6033:
6027:
6021:
6018:
6012:
6009:
6003:
5997:
5994:
5988:
5982:
5979:
5973:
5970:
5964:
5954:
5948:
5940:
5934:
5928:
5925:
5919:
5913:
5910:
5904:
5898:
5892:
5883:
5880:
5874:
5868:
5865:
5859:
5853:
5850:
5844:
5838:
5832:
5826:
5822:
5818:
5812:
5806:
5803:
5797:
5791:
5781:
5773:
5767:
5758:
5752:
5744:
5738:
5735:
5729:
5723:
5720:
5714:
5711:
5705:
5699:
5693:
5690:
5684:
5676:
5667:
5661:
5658:
5652:
5648:
5642:
5641:be a subset of
5636:
5629:
5619:
5613:
5604:
5598:
5592:
5586:
5583:
5569:
5565:
5546:
5524:
5521:
5520:
5516:
5512:
5505:
5501:
5451:
5445:
5444:
5435:
5431:
5419:
5413:
5412:
5407:
5399:
5396:
5395:
5366:
5362:
5358:
5325:
5314:
5313:
5309:
5304:
5285:
5281:
5277:
5275:
5272:
5271:
5233:
5225:
5221:
5215:
5212:
5204:
5201:
5195:
5191:
5183:
5169:
5162:
5147:
5146:
5131:
5130:
5103:
5100:
5099:
5093:
5092:
5065:
5062:
5061:
5030:
5007:
4998:
4960:
4954:
4953:
4952:
4920:
4916:
4904:
4900:
4885:
4881:
4869:
4865:
4864:
4860:
4854:
4843:
4830:
4826:
4820:
4819:
4810:
4805:
4789:
4783:
4780:
4779:
4759:
4724:
4706:
4702:
4701:
4692:
4687:
4681:
4678:
4677:
4657:
4650:
4623:
4617:
4613:
4608:
4602:
4584:
4581:
4580:
4579:
4559:
4555:
4544:
4541:
4540:
4518:
4514:
4512:
4509:
4508:
4507:
4494:is necessarily
4431:
4427:
4425:
4422:
4421:
4400:
4396:
4395:
4391:
4389:
4386:
4385:
4384:
4382:cardinal number
4345:algebraic basis
4334:
4333:
4329:
4301:
4297:
4282:
4278:
4276:
4273:
4272:
4271:
4267:
4250:
4245:
4244:
4238:
4234:
4219:
4214:
4213:
4207:
4203:
4201:
4198:
4197:
4186:
4185:and an integer
4182:
4165:
4160:
4159:
4144:
4139:
4138:
4136:
4133:
4132:
4128:
4124:
4088:. For modules,
4078:
4070:Main articles:
4068:
4063:
4061:Related notions
4048:
4028:
4024:
4012:
4008:
4002:
3991:
3978:
3974:
3972:
3969:
3968:
3949:
3945:
3943:
3940:
3939:
3938:
3916:
3911:
3910:
3904:
3903:
3897:
3893:
3881:
3877:
3871:
3860:
3850:
3849:
3843:
3832:
3819:
3814:
3813:
3801:
3797:
3791:
3780:
3770:
3766:
3760:
3749:
3736:
3731:
3730:
3724:
3720:
3714:
3703:
3691:
3689:
3686:
3685:
3665:
3660:
3659:
3653:
3649:
3643:
3632:
3620:
3618:
3615:
3614:
3608:
3578:
3575:
3574:
3568:
3546:
3545:
3539:
3535:
3532:
3531:
3525:
3524:
3518:
3514:
3507:
3506:
3494:
3486:
3485:
3479:
3475:
3472:
3471:
3465:
3464:
3458:
3454:
3447:
3446:
3438:
3435:
3434:
3409:
3405:
3403:
3400:
3399:
3398:
3394:
3377:
3357:
3353:
3341:
3337:
3331:
3320:
3307:
3303:
3301:
3298:
3297:
3291:
3271:
3267:
3252:
3248:
3243:
3240:
3239:
3219:
3215:
3200:
3196:
3191:
3188:
3187:
3167:
3162:
3161:
3149:
3145:
3139:
3128:
3115:
3110:
3109:
3107:
3104:
3103:
3063:
3062:
3058:
3056:
3053:
3052:
3028:
3027:
3023:
3021:
3018:
3017:
2986:
2985:
2981:
2979:
2976:
2975:
2951:
2950:
2946:
2944:
2941:
2940:
2936:
2930:
2910:
2905:
2904:
2889:
2884:
2883:
2871:
2867:
2865:
2862:
2861:
2841:
2836:
2835:
2820:
2815:
2814:
2802:
2798:
2796:
2793:
2792:
2786:
2782:
2776:
2773:
2771:Change of basis
2767:
2765:Change of basis
2758:
2741:
2737:
2735:
2732:
2731:
2727:
2723:
2706:
2702:
2700:
2697:
2696:
2692:
2675:
2671:
2669:
2666:
2665:
2646:
2642:
2640:
2637:
2636:
2635:
2614:
2610:
2608:
2605:
2604:
2603:
2587:
2584:
2583:
2579:
2576:canonical basis
2554:
2550:
2548:
2545:
2544:
2527:
2522:
2521:
2519:
2516:
2515:
2511:
2494:
2489:
2488:
2486:
2483:
2482:
2478:
2461:
2456:
2455:
2453:
2450:
2449:
2433:
2430:
2429:
2416:
2393:
2381:
2377:
2375:
2372:
2371:
2367:
2363:
2342:
2338:
2336:
2333:
2332:
2328:
2311:
2307:
2305:
2302:
2301:
2280:
2275:
2274:
2268:
2264:
2249:
2244:
2243:
2237:
2233:
2221:
2217:
2202:
2198:
2187:
2184:
2183:
2179:
2175:
2174:of elements of
2169:
2151:
2147:
2145:
2142:
2141:
2140:Let, as usual,
2108:
2091:
2086:
2085:
2073:
2068:
2067:
2062:
2059:
2058:
2041:
2036:
2035:
2023:
2018:
2017:
2012:
2009:
2008:
1996:
1990:
1982:
1965:
1961:
1946:
1942:
1940:
1937:
1936:
1916:
1911:
1910:
1904:
1900:
1885:
1880:
1879:
1873:
1869:
1861:
1859:
1856:
1855:
1851:
1845:
1841:
1821:
1816:
1815:
1800:
1795:
1794:
1783:
1780:
1779:
1775:
1771:
1767:
1764:
1754:
1750:
1746:
1739:
1735:
1727:
1723:
1715:
1708:
1704:
1696:
1692:
1681:
1669:
1656:
1648:
1644:
1612:
1609:
1608:
1604:
1594:
1590:
1582:
1578:
1568:axiom of choice
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1520:
1513:
1506:
1495:
1454:
1450:
1427:
1424:
1423:
1415:
1408:
1404:
1400:
1387:
1383:
1357:
1353:
1351:
1348:
1347:
1323:
1319:
1317:
1314:
1313:
1296:
1291:
1290:
1275:
1270:
1269:
1267:
1264:
1263:
1259:
1255:
1190:
1185:
1184:
1182:
1179:
1178:
1174:
1173:of elements of
1168:
1150:
1146:
1144:
1141:
1140:
1132:
1123:
1119:
1115:
1109:
1089:
1084:
1083:
1071:
1066:
1065:
1054:
1052:
1049:
1048:
1042:
1027:
1019:
1013:
1009:
1003:
987:
984:
983:
925:
922:
921:
845:
842:
841:
827:
804:
762:
745:
739:
722:
718:
716:
713:
712:
687:
682:
681:
675:
671:
656:
651:
650:
644:
640:
632:
630:
627:
626:
625:
621:
604:
599:
598:
583:
578:
577:
575:
572:
571:
565:
548:
544:
529:
525:
523:
520:
519:
513:
507:
475:
471:
456:
452:
450:
447:
446:
445:
439:
422:
418:
403:
399:
397:
394:
393:
377:
368:
363:
362:
356:
352:
337:
332:
331:
325:
321:
319:
316:
315:
311:
291:
286:
285:
270:
265:
264:
259:
256:
255:
236:
232:
226:
217:
207:
205:complex numbers
198:
189:
180:
171:
165:
131:
130:of elements of
123:
115:
106:
105:
100:
88:
87:of elements of
78:
60:
53:
35:
28:
23:
22:
15:
12:
11:
5:
7967:
7957:
7956:
7954:Linear algebra
7951:
7934:
7933:
7931:
7930:
7925:
7923:Woldemar Voigt
7920:
7915:
7910:
7905:
7900:
7895:
7890:
7888:Leonhard Euler
7885:
7880:
7875:
7870:
7864:
7862:
7860:Mathematicians
7856:
7855:
7852:
7851:
7849:
7848:
7843:
7838:
7833:
7828:
7823:
7818:
7813:
7808:
7802:
7800:
7796:
7795:
7793:
7792:
7787:
7785:Torsion tensor
7782:
7777:
7772:
7767:
7762:
7757:
7751:
7749:
7742:
7738:
7737:
7735:
7734:
7729:
7724:
7719:
7714:
7709:
7704:
7699:
7694:
7689:
7684:
7679:
7674:
7669:
7664:
7659:
7654:
7649:
7644:
7638:
7636:
7630:
7629:
7627:
7626:
7620:
7618:Tensor product
7615:
7610:
7608:Symmetrization
7605:
7600:
7598:Lie derivative
7595:
7590:
7585:
7580:
7575:
7569:
7567:
7561:
7560:
7558:
7557:
7552:
7547:
7542:
7537:
7532:
7527:
7522:
7520:Tensor density
7517:
7512:
7506:
7504:
7498:
7497:
7495:
7494:
7492:Voigt notation
7489:
7484:
7479:
7477:Ricci calculus
7474:
7469:
7464:
7462:Index notation
7459:
7454:
7448:
7446:
7442:
7441:
7438:
7437:
7435:
7434:
7429:
7424:
7419:
7414:
7408:
7406:
7404:
7403:
7398:
7392:
7389:
7388:
7386:
7385:
7380:
7378:Tensor algebra
7375:
7370:
7365:
7360:
7358:Dyadic algebra
7355:
7350:
7344:
7342:
7333:
7329:
7328:
7321:
7318:
7317:
7310:
7309:
7302:
7295:
7287:
7278:
7277:
7275:
7274:
7263:
7260:
7259:
7257:
7256:
7251:
7246:
7241:
7236:
7234:Floating-point
7230:
7228:
7222:
7221:
7219:
7218:
7216:Tensor product
7213:
7208:
7203:
7201:Function space
7198:
7193:
7187:
7185:
7178:
7177:
7175:
7174:
7169:
7164:
7159:
7154:
7149:
7144:
7139:
7137:Triple product
7134:
7129:
7123:
7121:
7115:
7114:
7112:
7111:
7106:
7101:
7096:
7091:
7086:
7081:
7075:
7073:
7067:
7066:
7064:
7063:
7058:
7053:
7051:Transformation
7048:
7043:
7041:Multiplication
7038:
7033:
7028:
7023:
7017:
7015:
7009:
7008:
7001:
6999:
6997:
6996:
6991:
6986:
6981:
6976:
6971:
6966:
6961:
6956:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6916:
6911:
6906:
6900:
6898:
6897:Basic concepts
6894:
6893:
6891:
6890:
6885:
6879:
6876:
6875:
6872:Linear algebra
6868:
6867:
6860:
6853:
6845:
6839:
6838:
6820:
6800:
6799:
6798:
6793:
6783:
6782:External links
6780:
6779:
6778:
6752:
6741:(3): 262–303,
6728:
6710:
6696:
6661:
6656:
6626:
6612:
6593:(3): 227–261,
6576:
6564:
6550:
6514:Banach, Stefan
6508:
6505:
6504:
6503:
6498:
6486:Linear algebra
6478:
6473:
6458:
6445:
6419:
6416:
6414:
6411:
6408:
6407:
6396:
6365:(1): 337–358.
6337:
6286:
6251:
6232:(3): 303–306.
6214:
6207:
6187:
6178:
6167:
6160:
6136:
6135:
6133:
6130:
6129:
6128:
6122:
6113:
6107:
6102:
6095:
6092:
6065:
6047:
6038:
6029:
6014:
6005:
5999:
5990:
5984:
5975:
5966:
5936:
5930:
5921:
5906:
5876:
5861:
5852:is a basis of
5846:
5834:
5814:
5799:
5731:
5716:
5707:
5686:
5654:
5582:
5579:
5570:π/2 ± 0.037π/2
5566:π/2 ± 0.037π/2
5552:
5534:
5531:
5528:
5497:
5496:
5491:
5489:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5448:
5443:
5438:
5434:
5427:
5424:
5416:
5410:
5406:
5403:
5346:
5343:
5339:
5334:
5331:
5328:
5323:
5320:
5317:
5312:
5307:
5302:
5298:
5294:
5291:
5288:
5284:
5280:
5229:
5208:
5199:
5161:
5158:
5113:
5110:
5107:
5075:
5072:
5069:
5029:
5026:
5003:
4994:
4977:
4974:
4971:
4968:
4963:
4957:
4951:
4948:
4945:
4942:
4939:
4935:
4930:
4926:
4923:
4919:
4915:
4912:
4907:
4903:
4899:
4895:
4891:
4888:
4884:
4880:
4877:
4872:
4868:
4863:
4857:
4852:
4849:
4846:
4842:
4838:
4833:
4829:
4823:
4816:
4813:
4808:
4804:
4798:
4795:
4792:
4788:
4758:The functions
4745:
4742:
4739:
4736:
4733:
4727:
4722:
4718:
4715:
4712:
4709:
4705:
4698:
4695:
4690:
4686:
4654:Fourier series
4649:
4646:
4642:standard basis
4626:
4620:
4616:
4611:
4605:
4601:
4597:
4594:
4591:
4588:
4567:
4562:
4558:
4554:
4551:
4548:
4521:
4517:
4469:Fréchet spaces
4461:Hilbert spaces
4434:
4430:
4403:
4399:
4394:
4358:Schauder bases
4354:Hilbert spaces
4328:
4325:
4304:
4300:
4296:
4293:
4290:
4285:
4281:
4266:is a basis of
4253:
4248:
4241:
4237:
4233:
4230:
4227:
4222:
4217:
4210:
4206:
4168:
4163:
4158:
4155:
4152:
4147:
4142:
4098:generating set
4067:
4064:
4062:
4059:
4036:
4031:
4027:
4021:
4018:
4015:
4011:
4005:
4000:
3997:
3994:
3990:
3986:
3981:
3977:
3948:
3924:
3919:
3914:
3907:
3900:
3896:
3890:
3887:
3884:
3880:
3874:
3869:
3866:
3863:
3859:
3853:
3846:
3841:
3838:
3835:
3831:
3827:
3822:
3817:
3810:
3807:
3804:
3800:
3794:
3789:
3786:
3783:
3779:
3773:
3769:
3763:
3758:
3755:
3752:
3748:
3744:
3739:
3734:
3727:
3723:
3717:
3712:
3709:
3706:
3702:
3698:
3694:
3673:
3668:
3663:
3656:
3652:
3646:
3641:
3638:
3635:
3631:
3627:
3623:
3594:
3591:
3588:
3585:
3582:
3565:column vectors
3550:
3542:
3538:
3534:
3533:
3530:
3527:
3526:
3521:
3517:
3513:
3512:
3510:
3505:
3502:
3490:
3482:
3478:
3474:
3473:
3470:
3467:
3466:
3461:
3457:
3453:
3452:
3450:
3445:
3442:
3418:
3415:
3412:
3408:
3393:notation. Let
3365:
3360:
3356:
3350:
3347:
3344:
3340:
3334:
3329:
3326:
3323:
3319:
3315:
3310:
3306:
3279:
3274:
3270:
3266:
3263:
3260:
3255:
3251:
3247:
3227:
3222:
3218:
3214:
3211:
3208:
3203:
3199:
3195:
3175:
3170:
3165:
3158:
3155:
3152:
3148:
3142:
3137:
3134:
3131:
3127:
3123:
3118:
3113:
3072:
3069:
3066:
3061:
3037:
3034:
3031:
3026:
3001:
2995:
2992:
2989:
2984:
2960:
2957:
2954:
2949:
2918:
2913:
2908:
2903:
2900:
2897:
2892:
2887:
2882:
2879:
2870:
2849:
2844:
2839:
2834:
2831:
2828:
2823:
2818:
2813:
2810:
2801:
2769:Main article:
2766:
2763:
2744:
2740:
2709:
2705:
2678:
2674:
2649:
2645:
2617:
2613:
2591:
2572:standard basis
2557:
2553:
2530:
2525:
2497:
2492:
2464:
2459:
2437:
2400:
2396:
2392:
2387:
2384:
2380:
2345:
2341:
2314:
2310:
2283:
2278:
2271:
2267:
2263:
2260:
2257:
2252:
2247:
2240:
2236:
2232:
2229:
2224:
2220:
2216:
2213:
2210:
2205:
2201:
2197:
2194:
2191:
2154:
2150:
2094:
2089:
2084:
2081:
2076:
2071:
2066:
2044:
2039:
2034:
2031:
2026:
2021:
2016:
1968:
1964:
1960:
1957:
1954:
1949:
1945:
1924:
1919:
1914:
1907:
1903:
1899:
1896:
1893:
1888:
1883:
1876:
1872:
1868:
1864:
1840:be a basis of
1829:
1824:
1819:
1814:
1811:
1808:
1803:
1798:
1793:
1790:
1787:
1763:
1760:
1759:
1758:
1743:
1720:
1719:
1712:
1695:is a basis of
1689:
1684:. This is the
1672:have the same
1666:
1642:
1631:
1628:
1625:
1622:
1619:
1616:
1505:
1502:
1471:
1468:
1465:
1462:
1457:
1453:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1413:monomial basis
1365:
1360:
1356:
1344:standard basis
1331:
1326:
1322:
1312:is a basis of
1299:
1294:
1289:
1286:
1283:
1278:
1273:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1193:
1188:
1153:
1149:
1097:
1092:
1087:
1082:
1079:
1074:
1069:
1064:
1061:
1057:
1024:standard basis
1017:
1007:
991:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
812:standard basis
803:
800:
792:indexed family
725:
721:
706:
705:
690:
685:
678:
674:
670:
667:
664:
659:
654:
647:
643:
639:
635:
607:
602:
597:
594:
591:
586:
581:
551:
547:
543:
540:
537:
532:
528:
504:
499:
486:
483:
478:
474:
470:
467:
464:
459:
455:
425:
421:
417:
414:
411:
406:
402:
380:
376:
371:
366:
359:
355:
351:
348:
345:
340:
335:
328:
324:
299:
294:
289:
284:
281:
278:
273:
268:
263:
248:
164:
161:
26:
9:
6:
4:
3:
2:
7966:
7955:
7952:
7950:
7947:
7946:
7944:
7929:
7926:
7924:
7921:
7919:
7916:
7914:
7911:
7909:
7906:
7904:
7901:
7899:
7896:
7894:
7891:
7889:
7886:
7884:
7881:
7879:
7876:
7874:
7871:
7869:
7866:
7865:
7863:
7861:
7857:
7847:
7844:
7842:
7839:
7837:
7834:
7832:
7829:
7827:
7824:
7822:
7819:
7817:
7814:
7812:
7809:
7807:
7804:
7803:
7801:
7797:
7791:
7788:
7786:
7783:
7781:
7778:
7776:
7773:
7771:
7768:
7766:
7765:Metric tensor
7763:
7761:
7758:
7756:
7753:
7752:
7750:
7746:
7743:
7739:
7733:
7730:
7728:
7725:
7723:
7720:
7718:
7715:
7713:
7710:
7708:
7705:
7703:
7700:
7698:
7695:
7693:
7690:
7688:
7685:
7683:
7680:
7678:
7677:Exterior form
7675:
7673:
7670:
7668:
7665:
7663:
7660:
7658:
7655:
7653:
7650:
7648:
7645:
7643:
7640:
7639:
7637:
7631:
7624:
7621:
7619:
7616:
7614:
7611:
7609:
7606:
7604:
7601:
7599:
7596:
7594:
7591:
7589:
7586:
7584:
7581:
7579:
7576:
7574:
7571:
7570:
7568:
7566:
7562:
7556:
7553:
7551:
7550:Tensor bundle
7548:
7546:
7543:
7541:
7538:
7536:
7533:
7531:
7528:
7526:
7523:
7521:
7518:
7516:
7513:
7511:
7508:
7507:
7505:
7499:
7493:
7490:
7488:
7485:
7483:
7480:
7478:
7475:
7473:
7470:
7468:
7465:
7463:
7460:
7458:
7455:
7453:
7450:
7449:
7447:
7443:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7409:
7407:
7402:
7399:
7397:
7394:
7393:
7390:
7384:
7381:
7379:
7376:
7374:
7371:
7369:
7366:
7364:
7361:
7359:
7356:
7354:
7351:
7349:
7346:
7345:
7343:
7341:
7337:
7334:
7330:
7326:
7325:
7319:
7315:
7308:
7303:
7301:
7296:
7294:
7289:
7288:
7285:
7273:
7265:
7264:
7261:
7255:
7252:
7250:
7249:Sparse matrix
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7231:
7229:
7227:
7223:
7217:
7214:
7212:
7209:
7207:
7204:
7202:
7199:
7197:
7194:
7192:
7189:
7188:
7186:
7184:constructions
7183:
7179:
7173:
7172:Outermorphism
7170:
7168:
7165:
7163:
7160:
7158:
7155:
7153:
7150:
7148:
7145:
7143:
7140:
7138:
7135:
7133:
7132:Cross product
7130:
7128:
7125:
7124:
7122:
7120:
7116:
7110:
7107:
7105:
7102:
7100:
7099:Outer product
7097:
7095:
7092:
7090:
7087:
7085:
7082:
7080:
7079:Orthogonality
7077:
7076:
7074:
7072:
7068:
7062:
7059:
7057:
7056:Cramer's rule
7054:
7052:
7049:
7047:
7044:
7042:
7039:
7037:
7034:
7032:
7029:
7027:
7026:Decomposition
7024:
7022:
7019:
7018:
7016:
7014:
7010:
7005:
6995:
6992:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6972:
6970:
6967:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6901:
6899:
6895:
6889:
6886:
6884:
6881:
6880:
6877:
6873:
6866:
6861:
6859:
6854:
6852:
6847:
6846:
6843:
6835:
6831:
6830:
6825:
6821:
6817:
6813:
6809:
6805:
6801:
6797:
6794:
6792:
6789:
6788:
6786:
6785:
6775:
6769:
6761:
6757:
6753:
6749:
6744:
6740:
6736:
6735:
6729:
6726:on 2009-04-12
6725:
6721:
6720:
6715:
6711:
6707:
6706:
6701:
6697:
6694:
6690:
6686:
6682:
6678:
6675:(in German),
6674:
6670:
6666:
6662:
6659:
6653:
6649:
6645:
6637:
6636:
6631:
6627:
6623:
6622:
6617:
6613:
6610:
6606:
6601:
6596:
6592:
6588:
6587:
6582:
6577:
6573:
6569:
6565:
6561:
6560:
6555:
6551:
6548:
6544:
6540:
6536:
6532:
6529:(in French),
6528:
6527:
6519:
6515:
6511:
6510:
6501:
6495:
6491:
6487:
6483:
6479:
6476:
6470:
6466:
6465:
6459:
6456:
6452:
6448:
6442:
6438:
6434:
6427:
6422:
6421:
6405:
6400:
6392:
6388:
6383:
6378:
6373:
6368:
6364:
6360:
6359:
6351:
6347:
6341:
6333:
6329:
6325:
6321:
6316:
6311:
6307:
6306:
6301:
6295:
6293:
6291:
6282:
6278:
6274:
6270:
6266:
6262:
6255:
6247:
6243:
6239:
6235:
6231:
6227:
6226:
6218:
6210:
6204:
6200:
6199:
6191:
6182:
6176:
6171:
6163:
6157:
6153:
6152:
6147:
6141:
6137:
6126:
6123:
6117:
6114:
6111:
6108:
6106:
6103:
6101:
6098:
6097:
6091:
6089:
6084:
6081:
6075:
6068:
6060:
6057:
6050:
6041:
6032:
6024:
6017:
6008:
6002:
5993:
5987:
5978:
5969:
5962:
5957:
5951:
5944:
5939:
5933:
5924:
5916:
5909:
5901:
5895:
5889:
5886:
5879:
5871:
5864:
5856:
5849:
5841:
5837:
5829:
5817:
5809:
5802:
5794:
5790:asserts that
5789:
5784:
5777:
5770:
5764:
5761:
5755:
5748:
5741:
5734:
5728:. Therefore,
5726:
5719:
5710:
5702:
5696:
5689:
5680:
5673:
5670:
5664:
5657:
5645:
5639:
5633:
5627:
5622:
5616:
5610:
5607:
5601:
5595:
5589:
5578:
5576:
5563:
5559:
5554:
5550:
5532:
5529:
5526:
5509:
5495:
5492:
5490:
5471:
5468:
5465:
5459:
5456:
5453:
5441:
5436:
5432:
5425:
5422:
5408:
5404:
5401:
5394:
5393:
5390:
5388:
5384:
5380:
5376:
5370:
5365:is less than
5344:
5341:
5337:
5329:
5318:
5310:
5305:
5300:
5296:
5292:
5289:
5286:
5282:
5278:
5269:
5265:
5261:
5253:
5249:
5245:
5240:
5236:
5232:
5228:
5218:
5211:
5207:
5198:
5189:
5181:
5177:
5172:
5167:
5157:
5155:
5151:
5143:
5139:
5135:
5127:
5111:
5108:
5105:
5097:
5089:
5073:
5070:
5067:
5059:
5055:
5051:
5047:
5043:
5039:
5035:
5025:
5023:
5019:
5015:
5011:
5006:
5002:
4997:
4993:
4988:
4975:
4972:
4969:
4966:
4961:
4946:
4940:
4937:
4933:
4928:
4924:
4921:
4917:
4913:
4910:
4905:
4901:
4897:
4893:
4889:
4886:
4882:
4878:
4875:
4870:
4866:
4861:
4855:
4850:
4847:
4844:
4840:
4836:
4831:
4827:
4814:
4811:
4806:
4802:
4790:
4777:
4771:
4767:
4763:
4756:
4743:
4737:
4734:
4731:
4725:
4720:
4713:
4707:
4703:
4696:
4693:
4688:
4684:
4675:
4669:
4665:
4661:
4655:
4645:
4643:
4618:
4614:
4603:
4595:
4589:
4560:
4556:
4549:
4546:
4539:
4519:
4515:
4505:
4501:
4497:
4493:
4489:
4485:
4481:
4477:
4472:
4470:
4466:
4465:Banach spaces
4462:
4458:
4452:
4450:
4432:
4401:
4392:
4383:
4379:
4375:
4371:
4367:
4363:
4359:
4355:
4351:
4346:
4342:
4339:(named after
4338:
4324:
4322:
4302:
4298:
4294:
4291:
4288:
4283:
4279:
4251:
4239:
4235:
4231:
4228:
4225:
4220:
4208:
4204:
4194:
4190:
4166:
4156:
4153:
4150:
4145:
4122:
4121:abelian group
4117:
4115:
4111:
4106:
4101:
4099:
4095:
4094:spanning sets
4091:
4087:
4083:
4077:
4073:
4058:
4055:
4051:
4034:
4029:
4025:
4019:
4016:
4013:
4009:
4003:
3998:
3995:
3992:
3988:
3984:
3979:
3975:
3946:
3935:
3922:
3917:
3898:
3894:
3888:
3885:
3882:
3878:
3872:
3867:
3864:
3861:
3857:
3844:
3839:
3836:
3833:
3829:
3825:
3820:
3808:
3805:
3802:
3798:
3792:
3787:
3784:
3781:
3777:
3771:
3767:
3761:
3756:
3753:
3750:
3746:
3742:
3737:
3725:
3721:
3715:
3710:
3707:
3704:
3700:
3696:
3671:
3666:
3654:
3650:
3644:
3639:
3636:
3633:
3629:
3625:
3611:
3605:
3592:
3589:
3586:
3583:
3580:
3571:
3566:
3548:
3540:
3536:
3528:
3519:
3515:
3508:
3503:
3500:
3488:
3480:
3476:
3468:
3459:
3455:
3448:
3443:
3440:
3416:
3413:
3410:
3406:
3392:
3387:
3384:
3380:
3363:
3358:
3354:
3348:
3345:
3342:
3338:
3332:
3327:
3324:
3321:
3317:
3313:
3308:
3304:
3294:
3272:
3268:
3264:
3261:
3258:
3253:
3249:
3220:
3216:
3212:
3209:
3206:
3201:
3197:
3173:
3168:
3156:
3153:
3150:
3146:
3140:
3135:
3132:
3129:
3125:
3121:
3116:
3100:
3097:
3093:
3089:
3059:
3024:
3015:
2999:
2982:
2947:
2933:
2911:
2901:
2898:
2895:
2890:
2877:
2868:
2842:
2832:
2829:
2826:
2821:
2808:
2799:
2789:
2785:over a field
2779:
2772:
2762:
2742:
2738:
2707:
2703:
2676:
2672:
2647:
2643:
2632:
2615:
2611:
2589:
2577:
2573:
2555:
2551:
2528:
2495:
2462:
2435:
2427:
2426:inverse image
2422:
2419:
2414:
2385:
2382:
2378:
2361:
2343:
2339:
2312:
2308:
2299:
2281:
2269:
2265:
2261:
2258:
2255:
2250:
2238:
2234:
2222:
2218:
2214:
2211:
2208:
2203:
2199:
2192:
2189:
2173:
2152:
2148:
2138:
2136:
2132:
2131:
2126:
2122:
2118:
2114:
2113:ordered basis
2092:
2082:
2079:
2074:
2064:
2042:
2032:
2029:
2024:
2014:
2006:
2002:
1993:
1988:
1966:
1962:
1958:
1955:
1952:
1947:
1943:
1922:
1917:
1905:
1901:
1897:
1894:
1891:
1886:
1874:
1870:
1866:
1848:
1822:
1812:
1809:
1806:
1801:
1788:
1785:
1774:over a field
1744:
1733:
1732:
1731:
1713:
1702:
1701:proper subset
1690:
1687:
1679:
1675:
1668:All bases of
1667:
1663:
1659:
1654:
1643:
1629:
1626:
1623:
1620:
1617:
1614:
1601:
1597:
1588:
1587:
1586:
1575:
1573:
1569:
1564:
1526:
1519:
1511:
1501:
1498:
1493:
1489:
1485:
1469:
1463:
1460:
1455:
1451:
1447:
1444:
1441:
1438:
1432:
1429:
1421:
1414:
1399:
1398:indeterminate
1395:
1390:
1381:
1376:
1363:
1358:
1354:
1345:
1329:
1324:
1320:
1297:
1287:
1284:
1281:
1276:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1196:
1191:
1172:
1151:
1147:
1138:
1129:
1126:
1112:
1095:
1090:
1080:
1077:
1072:
1062:
1059:
1045:
1038:
1034:
1030:
1025:
1016:
1006:
989:
969:
963:
960:
957:
954:
951:
945:
939:
936:
933:
927:
904:
901:
898:
895:
892:
889:
886:
880:
874:
871:
868:
862:
856:
853:
850:
839:
835:
834:ordered pairs
830:
821:
817:
813:
808:
799:
797:
793:
789:
785:
781:
780:ordered basis
777:
773:
768:
765:
760:
756:
751:
748:
742:
723:
719:
711:
688:
676:
672:
668:
665:
662:
657:
645:
641:
637:
605:
595:
592:
589:
584:
568:
549:
545:
541:
538:
535:
530:
526:
516:
510:
505:
503:
500:
484:
481:
476:
472:
468:
465:
462:
457:
453:
442:
423:
419:
415:
412:
409:
404:
400:
374:
369:
357:
353:
349:
346:
343:
338:
326:
322:
292:
282:
279:
276:
271:
253:
249:
247:
244:
243:
242:
239:
229:
225:
220:
215:
210:
206:
201:
197:
194:(such as the
192:
188:
183:
179:
174:
170:
160:
158:
154:
149:
146:
144:
139:
137:
129:
121:
112:
110:
107:basis vectors
98:
94:
86:
81:
76:
68:
63:
59:
52:
48:
39:
33:
19:
7928:Hermann Weyl
7732:Vector space
7717:Pseudotensor
7682:Fiber bundle
7646:
7635:abstractions
7530:Mixed tensor
7515:Tensor field
7322:
7182:Vector space
6958:
6914:Vector space
6827:
6807:
6759:
6738:
6732:
6724:the original
6718:
6704:
6676:
6672:
6665:Hamel, Georg
6643:
6634:
6620:
6590:
6584:
6571:
6558:
6530:
6524:
6485:
6463:
6432:
6399:
6362:
6356:
6340:
6303:
6264:
6260:
6254:
6229:
6223:
6217:
6197:
6190:
6181:
6170:
6150:
6140:
6085:
6079:
6073:
6066:
6061:
6055:
6048:
6039:
6030:
6022:
6015:
6006:
6000:
5991:
5985:
5976:
5967:
5960:
5955:
5949:
5942:
5937:
5931:
5927:either. Let
5922:
5914:
5907:
5899:
5893:
5890:
5884:
5877:
5869:
5862:
5854:
5847:
5842:
5835:
5827:
5815:
5807:
5800:
5792:
5788:Zorn's lemma
5782:
5775:
5768:
5765:
5759:
5753:
5746:
5739:
5732:
5724:
5717:
5708:
5704:, and hence
5700:
5694:
5687:
5678:
5674:
5668:
5662:
5655:
5643:
5637:
5634:
5624:, and it is
5620:
5614:
5611:
5605:
5599:
5593:
5587:
5584:
5574:
5561:
5557:
5555:
5507:
5500:
5493:
5386:
5378:
5374:
5371:
5267:
5263:
5257:
5251:
5247:
5243:
5230:
5226:
5216:
5209:
5205:
5196:
5179:
5170:
5163:
5160:Random basis
5145:
5132:convex basis
5129:
5125:
5091:
5057:
5054:affine basis
5053:
5049:
5034:affine space
5031:
5013:
5009:
5004:
5000:
4995:
4991:
4989:
4775:
4769:
4765:
4761:
4760:{1} ∪ { sin(
4757:
4673:
4667:
4663:
4659:
4658:{1} ∪ { sin(
4651:
4504:non-complete
4503:
4491:
4488:Banach space
4483:
4475:
4473:
4453:
4449:aleph-nought
4373:
4369:
4344:
4332:
4330:
4192:
4188:
4118:
4109:
4104:
4102:
4079:
4053:
4049:
3936:
3609:
3606:
3569:
3388:
3382:
3378:
3292:
3101:
3091:
3087:
3013:
2931:
2787:
2777:
2774:
2633:
2423:
2417:
2139:
2134:
2133:or simply a
2128:
2112:
2004:
2000:
1991:
1986:
1846:
1765:
1762:Coordinates
1745:A subset of
1734:A subset of
1721:
1661:
1657:
1599:
1595:
1576:
1565:
1535:elements of
1518:spanning set
1507:
1496:
1388:
1377:
1343:
1130:
1124:
1110:
1043:
1036:
1032:
1028:
1014:
1004:
838:real numbers
828:
825:
815:
779:
769:
763:
752:
746:
740:
707:
566:
514:
508:
501:
440:
245:
237:
227:
218:
208:
199:
196:real numbers
190:
181:
178:vector space
172:
168:
166:
150:
147:
140:
136:spanning set
113:
104:
96:
92:
79:
74:
66:
65:is called a
61:
58:vector space
44:
7868:Élie Cartan
7816:Spin tensor
7790:Weyl tensor
7748:Mathematics
7712:Multivector
7503:definitions
7401:Engineering
7340:Mathematics
7162:Multivector
7127:Determinant
7084:Dot product
6929:Linear span
6640:, reprint:
6638:(in German)
6562:(in German)
6533:: 133–181,
6482:Lang, Serge
5867:belongs to
5142:convex hull
4676:satisfying
4496:uncountable
4378:cardinality
4341:Georg Hamel
4335:Hamel basis
4110:free module
4072:Free module
4066:Free module
3096:expressions
1987:coordinates
1674:cardinality
1394:polynomials
776:orientation
97:coordinates
47:mathematics
7943:Categories
7697:Linear map
7565:Operations
7196:Direct sum
7031:Invertible
6934:Linear map
6413:References
6404:Blass 1984
6315:1506.04631
6175:Hamel 1905
5651:, and let
5148:cone basis
5086:points in
5042:convex set
4196:such that
4052:= 1, ...,
3381:= 1, ...,
2366:, and the
1651:being the
1607:such that
1504:Properties
1139:, the set
1114:, such as
822:upon them.
624:such that
250:for every
163:Definition
93:components
7836:EM tensor
7672:Dimension
7623:Transpose
7226:Numerical
6989:Transpose
6834:EMS Press
6693:120063569
6547:0016-2736
6367:CiteSeerX
6246:189836213
6011:(because
5959:(because
5530:≫
5472:θ
5469:−
5460:
5454:−
5433:ε
5405:≤
5345:ε
4938:−
4914:
4879:
4841:∑
4815:π
4803:∫
4797:∞
4794:→
4768:) :
4741:∞
4697:π
4685:∫
4666:) :
4593:‖
4587:‖
4538:sequences
4429:ℵ
4398:ℵ
4292:…
4229:…
4154:…
3989:∑
3858:∑
3830:∑
3778:∑
3747:∑
3701:∑
3630:∑
3529:⋮
3469:⋮
3318:∑
3262:…
3210:…
3126:∑
3092:new basis
3088:old basis
2899:…
2830:…
2590:φ
2436:φ
2383:−
2379:φ
2266:λ
2259:⋯
2235:λ
2231:↦
2219:λ
2212:…
2200:λ
2190:φ
1963:λ
1956:…
1944:λ
1902:λ
1895:⋯
1871:λ
1810:…
1678:dimension
1653:empty set
1624:⊆
1618:⊆
1464:…
1420:monomials
1285:…
1233:…
1209:…
990:λ
961:λ
952:λ
928:λ
666:⋯
593:…
539:…
466:⋯
413:…
392:for some
347:⋯
280:…
143:dimension
7702:Manifold
7687:Geodesic
7445:Notation
7272:Category
7211:Subspace
7206:Quotient
7157:Bivector
7071:Bilinear
7013:Matrices
6888:Glossary
6812:Archived
6768:citation
6758:(1888),
6716:(1827),
6702:(1853),
6667:(1905),
6632:(1844),
6618:(1822),
6570:(1969),
6556:(1804),
6516:(1922),
6484:(1987),
6348:(2002).
6281:18263425
6148:(1987).
6094:See also
5858:. Since
5612:The set
5333:‖
5327:‖
5322:‖
5316:‖
5297:⟩
5283:⟨
5138:polytope
5028:Geometry
4480:complete
4327:Analysis
3967:that is
3090:and the
2121:sequence
2117:indexing
1730:, then:
1585:, then:
1020:= (0, 1)
1010:= (1, 0)
826:The set
802:Examples
788:sequence
786:, but a
772:ordering
7799:Physics
7633:Related
7396:Physics
7314:Tensors
6883:Outline
6836:, 2001
6824:"Basis"
6816:YouTube
6609:1347828
6455:0763890
6391:8095719
6332:2239376
5912:, then
5831:, then
5511:. This
5494:(Eq. 1)
5385:). Let
5222:det = 0
5203:, ...,
5174:with a
5056:for an
4764:), cos(
4662:), cos(
4648:Example
3563:be the
3086:as the
2481:-tuple
2477:is the
2411:is the
2370:-tuple
2358:is the
2172:-tuples
1778:, and
1396:in one
1392:of all
1254:be the
1171:-tuples
1120:(−1, 2)
832:of the
798:below.
710:scalars
444:, then
254:subset
203:or the
185:over a
7727:Vector
7722:Spinor
7707:Matrix
7501:Tensor
7167:Tensor
6979:Kernel
6909:Vector
6904:Scalar
6691:
6654:
6607:
6545:
6496:
6471:
6453:
6443:
6389:
6369:
6330:
6279:
6244:
6205:
6158:
6062:Hence
6053:spans
5996:, and
5972:, and
5675:Since
5597:. Let
5164:For a
5044:, and
5014:do not
5010:finite
4482:(i.e.
4420:where
4360:, and
4086:module
3391:matrix
2125:origin
2109:{2, 3}
1655:, and
1523:and a
1407:is an
1382:. If
1116:(1, 1)
982:where
755:finite
252:finite
214:subset
7647:Basis
7332:Scope
7036:Minor
7021:Block
6959:Basis
6689:S2CID
6521:(PDF)
6429:(PDF)
6387:S2CID
6353:(PDF)
6328:S2CID
6310:arXiv
6242:S2CID
6132:Notes
5833:L = L
5136:of a
5052:. An
5050:basis
4486:is a
4467:, or
4343:) or
4105:basis
2757:onto
2691:onto
2327:onto
2296:is a
2135:frame
1995:over
1749:with
1738:with
1137:field
1135:is a
790:, an
314:, if
224:spans
222:that
187:field
176:of a
169:basis
126:is a
75:bases
67:basis
32:Basis
7191:Dual
7046:Rank
6774:link
6652:ISBN
6543:ISSN
6494:ISBN
6469:ISBN
6441:ISBN
6277:PMID
6203:ISBN
6156:ISBN
5778:, ⊆)
5749:, ⊆)
5681:, ⊆)
5635:Let
5585:Let
5519:and
5506:1 −
5361:and
5342:<
5144:. A
5128:. A
5090:. A
5046:cone
4738:<
4640:Its
4187:0 ≤
4092:and
4082:ring
4074:and
4047:for
3684:and
3433:and
3376:for
3238:and
3051:and
2860:and
2775:Let
2424:The
2057:and
1766:Let
1527:set
1118:and
1012:and
708:The
570:and
155:and
49:, a
6743:doi
6681:doi
6595:doi
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