4905:
164:
5169:
1898:
1051:
Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle.
2828:
2757:
2686:
2557:
3013:
2034:
1283:
3095:
1670:
1804:
1491:
3671:
In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a
317:
347:
131:
2223:
1537:
2426:
2369:
726:
3324:
2126:
1416:
2309:
764:
2460:
1815:
2615:
4312:
1040:
of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional
Euclidean vector space, the orthogonal complement of a
1024:
2905:
790:
4410:
4194:
3807:
3760:
3713:
3369:
2465:
914:
815:
608:
2910:
1321:
4464:
4437:
4366:
4339:
4248:
4221:
4150:
4123:
4006:
3979:
3403:
3178:
977:
635:
583:
2860:
4092:
3949:
3926:
3903:
3880:
3857:
3834:
4484:
4268:
4066:
4046:
4026:
3659:
3639:
3619:
3599:
3579:
3559:
3539:
3519:
3499:
3479:
3451:
3431:
3155:
2577:
1697:
1561:
1364:
1344:
1145:
1122:
1102:
835:
675:
655:
556:
526:
506:
486:
462:
442:
422:
277:
82:
3257:
2762:
2691:
2620:
1729:
1593:
1177:
4763:
1910:
In other words, every pair of them (excluding pairing of a function with itself) is orthogonal, and the norm of each is 1. See in particular the
1926:
5154:
4694:
4667:
4637:
1185:
2428:
are orthogonal to each other. The dot product of these vectors is zero. We can then make the generalization to consider the vectors in
3813:
to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes:
3020:
1421:
Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product.
4581:
3221:
1601:
1737:
5144:
1434:
5106:
5042:
387:
286:
4710:
3329:
845:
4884:
4756:
947:
generalizes the case of an inner product. When the bilinear form applied to two vectors results in zero, then they are
4989:
4839:
849:
322:
91:
4894:
4788:
2130:
1499:
2374:
2317:
5193:
5134:
4783:
133:. Depending on the bilinear form, the vector space may contain non-zero self-orthogonal vectors. In the case of
5009:
4530:
2039:
1376:
5126:
4714:
3284:
3267:
956:
2228:
5198:
5172:
4879:
4749:
688:
1893:{\displaystyle \delta _{ij}=\left\{{\begin{matrix}1,&&i=j\\0,&&i\neq j\end{matrix}}\right.}
5203:
4936:
4869:
4859:
4729:
4515:
3454:
3198:
2431:
1911:
1082:
952:
372:
153:
142:
5096:
4951:
4946:
4941:
4874:
4819:
4736:. Leipzig: G.J.Göschensche Verlagshandlung. Volume 1 (Sammlung Schubert XXXV): Die linearen Räume, 1902.
4520:
4500:
3278:
3210:
1037:
841:
529:
731:
4961:
4926:
4913:
4804:
3263:
3217:
2582:
1068:
536:
357:
149:
138:
4273:
5139:
5019:
4994:
4844:
4658:
3206:
3098:
2865:
1045:
1029:), or one of the vectors is zero. Hence orthogonality of vectors is an extension of the concept of
1002:
257:
186:
769:
4849:
4371:
4155:
3271:
1425:
3765:
3718:
3674:
3348:
5047:
5004:
4931:
4824:
4690:
4684:
4663:
4633:
4577:
4537:
4525:
4510:
3665:
1074:
368:
353:
4627:
886:
5052:
4956:
4809:
4505:
4495:
1291:
219:
190:
163:
33:
4442:
4415:
4344:
4317:
4226:
4199:
4128:
4101:
3984:
3957:
3381:
3160:
962:
613:
561:
5111:
4904:
4864:
4854:
3375:
2836:
1904:
1125:
1041:
988:
682:
402:
171:
4071:
2823:{\displaystyle {\begin{bmatrix}0&0&1&0&0&1&0&0\end{bmatrix}}}
2752:{\displaystyle {\begin{bmatrix}0&1&0&0&1&0&0&1\end{bmatrix}}}
2681:{\displaystyle {\begin{bmatrix}1&0&0&1&0&0&1&0\end{bmatrix}}}
932:
to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in
3931:
3908:
3885:
3862:
3839:
3816:
795:
588:
5116:
5101:
5037:
4772:
4542:
4469:
4253:
4051:
4031:
4011:
3644:
3624:
3604:
3584:
3564:
3544:
3524:
3504:
3484:
3464:
3436:
3416:
3181:
3104:
2562:
1682:
1546:
1349:
1329:
1130:
1107:
1087:
992:
921:
876:
820:
660:
640:
541:
511:
491:
471:
447:
427:
407:
361:
262:
134:
67:
38:
3227:
1702:
1566:
1150:
5187:
5149:
5072:
5032:
4999:
4979:
3371:
3342:
3270:. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to
3194:
1078:
1030:
944:
227:
157:
42:
17:
2907:
are orthogonal with respect to a unit weight function on the interval from −1 to 1:
5082:
4971:
4921:
4814:
4548:
242:
61:
3715:
Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes
2552:{\displaystyle \mathbf {v} _{k}=\sum _{i=0 \atop ai+k<n}^{n/a}\mathbf {e} _{i}}
848:
if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are
5062:
5027:
4984:
4829:
3328:
The
Chebyshev polynomials of the second kind are orthogonal with respect to the
996:
933:
925:
380:
235:
24:
4597:
3008:{\displaystyle \int _{-1}^{1}\left(2t+3\right)\left(45t^{2}+9t-17\right)\,dt=0}
5091:
4834:
3809:. Each of the 6 orthogonal planes shares an axis with 4 of the others, and is
1056:
1055:
In four-dimensional
Euclidean space, the orthogonal complement of a line is a
937:
204:
of a light signal, a vector is orthogonal to itself if it lies on this line).
4889:
920:
may also refer to the magnitude of a vector. In particular, a set is called
394:
vectors by the same conformal linear transformation will keep those vectors
201:
1370:
if their inner product (equivalently, the value of this integral) is zero:
2029:{\displaystyle (1,3,2)^{\text{T}},(3,-1,0)^{\text{T}},(1,3,-5)^{\text{T}}}
5057:
959:. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given
215:
4553:
1278:{\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)g(x)w(x)\,dx.}
5067:
1026:
4466:
may or may not intersect; if they intersect then they intersect at
175:
167:
Orthogonality and rotation of coordinate systems compared between
4741:
390:
preserves angles and distance ratios, meaning that transforming
4745:
3090:{\displaystyle 1,\sin {(nx)},\cos {(nx)}\mid n\in \mathbb {N} }
860:
if each pairing of them is orthogonal. Such a set is called an
1665:{\displaystyle \langle f_{i},f_{j}\rangle _{w}=0\mid i\neq j.}
3281:
of the first kind are orthogonal with respect to the measure
1799:{\displaystyle \langle f_{i},f_{j}\rangle _{w}=\delta _{ij},}
1887:
1486:{\displaystyle \|f\|_{w}={\sqrt {\langle f,f\rangle _{w}}}}
4651:
4649:
312:{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle }
1048:
through the origin perpendicular to it, and vice versa.
924:(orthogonal plus normal) if it is an orthogonal set of
3287:
2771:
2700:
2629:
1840:
1005:
4472:
4445:
4418:
4374:
4347:
4320:
4276:
4256:
4229:
4202:
4158:
4131:
4104:
4074:
4054:
4034:
4014:
3987:
3960:
3934:
3911:
3888:
3865:
3842:
3819:
3768:
3721:
3677:
3647:
3627:
3607:
3587:
3567:
3547:
3527:
3507:
3487:
3467:
3439:
3419:
3384:
3351:
3230:
3163:
3107:
3023:
2913:
2868:
2839:
2765:
2694:
2623:
2585:
2565:
2468:
2434:
2377:
2320:
2231:
2133:
2042:
1929:
1818:
1740:
1705:
1685:
1604:
1569:
1549:
1502:
1437:
1379:
1352:
1332:
1294:
1188:
1153:
1133:
1110:
1090:
965:
889:
856:
A set of vectors in an inner product space is called
823:
798:
772:
734:
691:
663:
643:
616:
591:
564:
544:
514:
494:
474:
450:
430:
410:
325:
289:
265:
94:
70:
5125:
5081:
5018:
4970:
4912:
4797:
4478:
4458:
4431:
4404:
4360:
4333:
4306:
4262:
4242:
4215:
4188:
4144:
4117:
4086:
4060:
4040:
4020:
4000:
3973:
3943:
3920:
3897:
3874:
3851:
3828:
3801:
3754:
3707:
3653:
3633:
3613:
3593:
3573:
3553:
3533:
3513:
3493:
3473:
3445:
3425:
3397:
3363:
3318:
3251:
3172:
3149:
3089:
3007:
2899:
2854:
2822:
2751:
2680:
2609:
2571:
2551:
2454:
2420:
2363:
2303:
2217:
2120:
2028:
1892:
1798:
1723:
1691:
1664:
1587:
1555:
1531:
1485:
1410:
1358:
1338:
1315:
1277:
1171:
1139:
1116:
1096:
1077:, it is common to use the following to define the
1018:
971:
908:
829:
809:
784:
758:
720:
669:
649:
629:
602:
577:
550:
520:
500:
480:
456:
436:
416:
341:
311:
271:
125:
76:
31:is the generalization of the geometric notion of
3762:that we have in 3 dimensions, and also 3 others
1059:and vice versa, and that of a plane is a plane.
875:, particularly in the geometric sense as in the
528:that is orthogonal to a given subspace is its
342:{\displaystyle \mathbf {u} \perp \mathbf {v} }
126:{\displaystyle B(\mathbf {u} ,\mathbf {v} )=0}
4757:
4719:(3rd ed.). New York: Dover. p. 124.
4626:Trefethen, Lloyd N. & Bau, David (1997).
4572:J.A. Wheeler; C. Misner; K.S. Thorne (1973).
2218:{\displaystyle \ (3)(1)+(-1)(3)+(0)(-5)=0\ ,}
1532:{\displaystyle {f_{i}\mid i\in \mathbb {N} }}
8:
2617:, these vectors are orthogonal, for example
2421:{\displaystyle (0,1,0,1,\ldots )^{\text{T}}}
2364:{\displaystyle (1,0,1,0,\ldots )^{\text{T}}}
1768:
1741:
1632:
1605:
1472:
1459:
1445:
1438:
1393:
1380:
1202:
1189:
709:
692:
306:
290:
148:The concept has been used in the context of
1675:The members of such a set of functions are
4764:
4750:
4742:
4686:Orthogonal arrays: theory and applications
3319:{\textstyle {\frac {1}{\sqrt {1-x^{2}}}}.}
2121:{\displaystyle (1)(3)+(3)(-1)+(2)(0)=0\ ,}
1411:{\displaystyle \langle f,g\rangle _{w}=0.}
4471:
4450:
4444:
4423:
4417:
4373:
4352:
4346:
4325:
4319:
4275:
4255:
4234:
4228:
4207:
4201:
4157:
4136:
4130:
4109:
4103:
4073:
4053:
4033:
4013:
3992:
3986:
3965:
3959:
3933:
3910:
3887:
3864:
3841:
3818:
3767:
3720:
3676:
3646:
3626:
3606:
3586:
3566:
3546:
3526:
3506:
3486:
3466:
3438:
3418:
3389:
3383:
3350:
3304:
3288:
3286:
3229:
3162:
3157:, or any other closed interval of length
3106:
3083:
3082:
3059:
3036:
3022:
2992:
2966:
2926:
2918:
2912:
2876:
2867:
2838:
2766:
2764:
2695:
2693:
2624:
2622:
2584:
2564:
2543:
2538:
2527:
2523:
2488:
2475:
2470:
2467:
2446:
2441:
2437:
2436:
2433:
2412:
2376:
2355:
2319:
2230:
2132:
2041:
2020:
1986:
1952:
1928:
1839:
1823:
1817:
1784:
1771:
1761:
1748:
1739:
1704:
1684:
1635:
1625:
1612:
1603:
1568:
1548:
1524:
1523:
1508:
1503:
1501:
1475:
1457:
1448:
1436:
1396:
1378:
1351:
1331:
1293:
1265:
1223:
1218:
1205:
1187:
1152:
1132:
1109:
1089:
1006:
1004:
999:is zero, i.e. they make an angle of 90° (
964:
900:
888:
822:
797:
771:
750:
733:
690:
662:
642:
621:
615:
590:
569:
563:
543:
513:
493:
473:
449:
429:
409:
334:
326:
324:
301:
293:
288:
264:
109:
101:
93:
69:
162:
4564:
3193:Various polynomial sequences named for
2304:{\displaystyle (1)(1)+(3)(3)+(2)(-5)=0}
319:is zero. This relationship is denoted
5155:Comparison of linear algebra libraries
1428:with respect to this inner product as
721:{\displaystyle \langle m',m\rangle =0}
4576:. W.H. Freeman & Co. p. 58.
7:
2455:{\displaystyle \mathbb {Z} _{2}^{n}}
2036:are orthogonal to each other, since
1033:vectors to spaces of any dimension.
4662:. Vintage books. pp. 417–419.
3954:More generally, two flat subspaces
3374:are said to be orthogonal if their
3266:are orthogonal with respect to the
3220:are orthogonal with respect to the
3209:are orthogonal with respect to the
2489:
792:are orthogonal if each element of
14:
4683:Hedayat, A.; et al. (1999).
1496:The members of a set of functions
817:is orthogonal to each element of
759:{\displaystyle S'\subseteq M^{*}}
5168:
5167:
5145:Basic Linear Algebra Subprograms
4903:
3180:. This fact is a central one in
2539:
2471:
488:is orthogonal to each vector in
335:
327:
302:
294:
110:
102:
5043:Seven-dimensional cross product
4125:is orthogonal to every line in
3481:is orthogonal to every line in
3097:are orthogonal with respect to
2610:{\displaystyle 1\leq k\leq a-1}
928:. As a result, use of the term
388:conformal linear transformation
141:are used to form an orthogonal
4387:
4381:
4307:{\displaystyle \dim(S)>M+N}
4289:
4283:
4171:
4165:
3951:intersect only at the origin.
3905:intersect only at the origin;
3859:intersect only at the origin;
3796:
3769:
3749:
3722:
3702:
3678:
3330:Wigner semicircle distribution
3246:
3231:
3144:
3129:
3123:
3108:
3069:
3060:
3046:
3037:
2409:
2378:
2352:
2321:
2292:
2283:
2280:
2274:
2268:
2262:
2259:
2253:
2247:
2241:
2238:
2232:
2200:
2191:
2188:
2182:
2176:
2170:
2167:
2158:
2152:
2146:
2143:
2137:
2103:
2097:
2094:
2088:
2082:
2073:
2070:
2064:
2058:
2052:
2049:
2043:
2017:
1995:
1983:
1961:
1949:
1930:
1718:
1706:
1582:
1570:
1304:
1298:
1262:
1256:
1250:
1244:
1238:
1232:
1166:
1154:
1124:with respect to a nonnegative
1019:{\textstyle {\frac {\pi }{2}}}
114:
98:
1:
4368:may or may not intersect. If
3461:if and only if every line in
3197:of the past are sequences of
2900:{\displaystyle 45t^{2}+9t-17}
991:, two vectors are orthogonal
883:-axis is normal to the curve
4885:Eigenvalues and eigenvectors
4547:Pan-orthogonality occurs in
4250:intersect at a single point
3541:intersect at a single point
785:{\displaystyle S\subseteq M}
4405:{\displaystyle \dim(S)=M+N}
4189:{\displaystyle \dim(S)=M+N}
867:In certain cases, the word
508:. The largest subspace of
5220:
4734:Mehrdimensionale Geometrie
3802:{\displaystyle (wx,wy,wz)}
3755:{\displaystyle (xy,xz,yz)}
3581:intersects with a line in
3501:. In that case the planes
2559:for some positive integer
1066:
1044:through the origin is the
444:of an inner product space
15:
5163:
4901:
4779:
4689:. Springer. p. 168.
4611:Bourbaki, "ch. II §2.4",
3708:{\displaystyle (w,x,y,z)}
3405:combinations of entries.
3364:{\displaystyle n\times n}
916:at the origin. However,
379:and normalized (they are
360:whose column vectors are
4629:Numerical linear algebra
3268:exponential distribution
957:hyperbolic orthogonality
3561:, so that if a line in
983:Euclidean vector spaces
909:{\displaystyle y=x^{2}}
375:whose vectors are both
283:if their inner product
4870:Row and column vectors
4516:Orthogonal polynomials
4480:
4460:
4433:
4406:
4362:
4335:
4308:
4264:
4244:
4217:
4190:
4146:
4119:
4094:dimensions are called
4088:
4062:
4042:
4022:
4002:
3975:
3945:
3922:
3899:
3876:
3853:
3830:
3803:
3756:
3709:
3655:
3635:
3615:
3595:
3575:
3555:
3535:
3515:
3495:
3475:
3455:four-dimensional space
3447:
3427:
3399:
3365:
3320:
3253:
3199:orthogonal polynomials
3189:Orthogonal polynomials
3174:
3151:
3091:
3009:
2901:
2856:
2824:
2753:
2682:
2611:
2573:
2553:
2536:
2456:
2422:
2365:
2305:
2219:
2122:
2030:
1912:orthogonal polynomials
1894:
1800:
1725:
1693:
1666:
1589:
1557:
1533:
1487:
1412:
1360:
1340:
1326:We say that functions
1317:
1316:{\displaystyle w(x)=1}
1279:
1173:
1141:
1118:
1098:
1020:
973:
953:pseudo-Euclidean plane
943:A vector space with a
910:
831:
811:
786:
760:
722:
671:
651:
631:
604:
579:
552:
522:
502:
482:
458:
438:
418:
343:
313:
273:
205:
154:orthogonal polynomials
127:
78:
4875:Row and column spaces
4820:Scalar multiplication
4521:Orthogonal trajectory
4501:Orthogonal complement
4481:
4461:
4459:{\displaystyle S_{2}}
4434:
4432:{\displaystyle S_{1}}
4407:
4363:
4361:{\displaystyle S_{2}}
4336:
4334:{\displaystyle S_{1}}
4309:
4265:
4245:
4243:{\displaystyle S_{2}}
4218:
4216:{\displaystyle S_{1}}
4191:
4147:
4145:{\displaystyle S_{2}}
4120:
4118:{\displaystyle S_{1}}
4096:completely orthogonal
4089:
4063:
4048:of a Euclidean space
4043:
4023:
4003:
4001:{\displaystyle S_{2}}
3976:
3974:{\displaystyle S_{1}}
3946:
3923:
3900:
3877:
3854:
3831:
3811:completely orthogonal
3804:
3757:
3710:
3656:
3636:
3616:
3596:
3576:
3556:
3536:
3516:
3496:
3476:
3459:completely orthogonal
3448:
3428:
3409:Completely orthogonal
3400:
3398:{\displaystyle n^{2}}
3366:
3321:
3279:Chebyshev polynomials
3254:
3213:with zero mean value.
3211:Gaussian distribution
3175:
3173:{\displaystyle 2\pi }
3152:
3092:
3010:
2902:
2857:
2825:
2754:
2683:
2612:
2574:
2554:
2484:
2457:
2423:
2366:
2306:
2220:
2123:
2031:
1895:
1801:
1726:
1694:
1667:
1590:
1558:
1534:
1488:
1413:
1361:
1341:
1318:
1280:
1174:
1142:
1119:
1099:
1038:orthogonal complement
1021:
974:
972:{\displaystyle \phi }
911:
842:term rewriting system
832:
812:
787:
761:
723:
672:
652:
632:
630:{\displaystyle M^{*}}
605:
580:
578:{\displaystyle M^{*}}
553:
530:orthogonal complement
523:
503:
483:
459:
439:
419:
344:
314:
274:
166:
128:
79:
5010:Gram–Schmidt process
4962:Gaussian elimination
4632:. SIAM. p. 13.
4531:Gram–Schmidt process
4470:
4443:
4416:
4372:
4345:
4318:
4274:
4254:
4227:
4200:
4156:
4129:
4102:
4072:
4052:
4032:
4012:
3985:
3958:
3932:
3909:
3886:
3863:
3840:
3817:
3766:
3719:
3675:
3645:
3625:
3605:
3601:, they intersect at
3585:
3565:
3545:
3525:
3505:
3485:
3465:
3437:
3417:
3382:
3378:yields all possible
3349:
3285:
3264:Laguerre polynomials
3228:
3222:uniform distribution
3218:Legendre polynomials
3161:
3105:
3021:
2911:
2866:
2855:{\displaystyle 2t+3}
2837:
2763:
2692:
2621:
2583:
2563:
2466:
2432:
2375:
2318:
2229:
2131:
2040:
1927:
1816:
1738:
1703:
1683:
1602:
1567:
1547:
1500:
1435:
1377:
1350:
1330:
1292:
1186:
1151:
1131:
1108:
1088:
1069:Orthogonal functions
1063:Orthogonal functions
1003:
963:
887:
821:
796:
770:
732:
689:
661:
641:
614:
589:
562:
542:
512:
492:
472:
466:orthogonal subspaces
448:
428:
408:
323:
287:
263:
196:(red lines labelled
150:orthogonal functions
139:orthogonal functions
92:
68:
16:For other uses, see
5140:Numerical stability
5020:Multilinear algebra
4995:Inner product space
4845:Linear independence
4659:The Road to Reality
4656:R. Penrose (2007).
4598:"Wolfram MathWorld"
4087:{\displaystyle M+N}
3272:gamma distributions
3207:Hermite polynomials
3099:Riemann integration
2931:
2451:
1228:
879:. For example, the
877:normal to a surface
858:pairwise orthogonal
258:inner product space
187:Minkowski spacetime
64:with bilinear form
4850:Linear combination
4476:
4456:
4429:
4402:
4358:
4331:
4304:
4260:
4240:
4213:
4186:
4142:
4115:
4084:
4058:
4038:
4018:
3998:
3971:
3944:{\displaystyle wx}
3941:
3921:{\displaystyle yz}
3918:
3898:{\displaystyle wy}
3895:
3875:{\displaystyle xz}
3872:
3852:{\displaystyle wz}
3849:
3829:{\displaystyle xy}
3826:
3799:
3752:
3705:
3661:are perpendicular
3651:
3631:
3611:
3591:
3571:
3551:
3531:
3511:
3491:
3471:
3443:
3423:
3395:
3361:
3316:
3249:
3170:
3147:
3087:
3005:
2914:
2897:
2852:
2820:
2814:
2749:
2743:
2678:
2672:
2607:
2569:
2549:
2452:
2435:
2418:
2361:
2301:
2215:
2118:
2026:
1890:
1885:
1796:
1721:
1689:
1662:
1585:
1553:
1529:
1483:
1408:
1356:
1336:
1313:
1275:
1214:
1169:
1137:
1114:
1094:
1016:
969:
906:
827:
810:{\displaystyle S'}
807:
782:
756:
718:
667:
647:
627:
603:{\displaystyle m'}
600:
575:
548:
518:
498:
478:
468:if each vector in
454:
434:
414:
339:
309:
269:
206:
123:
74:
5181:
5180:
5048:Geometric algebra
5005:Kronecker product
4840:Linear projection
4825:Vector projection
4716:Regular Polytopes
4696:978-0-387-98766-8
4669:978-0-679-77631-4
4639:978-0-89871-361-9
4538:Orthonormal basis
4526:Orthogonalization
4511:Orthogonal matrix
4479:{\displaystyle O}
4263:{\displaystyle O}
4098:if every line in
4061:{\displaystyle S}
4041:{\displaystyle N}
4021:{\displaystyle M}
3666:Clifford parallel
3654:{\displaystyle B}
3634:{\displaystyle A}
3614:{\displaystyle O}
3594:{\displaystyle B}
3574:{\displaystyle A}
3554:{\displaystyle O}
3534:{\displaystyle B}
3514:{\displaystyle A}
3494:{\displaystyle B}
3474:{\displaystyle A}
3446:{\displaystyle B}
3426:{\displaystyle A}
3311:
3310:
3201:. In particular:
3150:{\displaystyle ,}
3101:on the intervals
2572:{\displaystyle a}
2521:
2415:
2358:
2211:
2136:
2114:
2023:
1989:
1955:
1692:{\displaystyle w}
1556:{\displaystyle w}
1481:
1359:{\displaystyle g}
1339:{\displaystyle f}
1288:In simple cases,
1147:over an interval
1140:{\displaystyle w}
1117:{\displaystyle g}
1097:{\displaystyle f}
1075:integral calculus
1014:
830:{\displaystyle S}
670:{\displaystyle M}
650:{\displaystyle m}
551:{\displaystyle M}
521:{\displaystyle V}
501:{\displaystyle B}
481:{\displaystyle A}
457:{\displaystyle V}
437:{\displaystyle B}
417:{\displaystyle A}
369:orthonormal basis
354:orthogonal matrix
272:{\displaystyle V}
220:Euclidean vectors
174:through circular
77:{\displaystyle B}
5211:
5194:Abstract algebra
5171:
5170:
5053:Exterior algebra
4990:Hadamard product
4907:
4895:Linear equations
4766:
4759:
4752:
4743:
4737:
4727:
4721:
4720:
4707:
4701:
4700:
4680:
4674:
4673:
4653:
4644:
4643:
4623:
4617:
4616:
4608:
4602:
4601:
4594:
4588:
4587:
4569:
4506:Orthogonal group
4496:Imaginary number
4485:
4483:
4482:
4477:
4465:
4463:
4462:
4457:
4455:
4454:
4438:
4436:
4435:
4430:
4428:
4427:
4411:
4409:
4408:
4403:
4367:
4365:
4364:
4359:
4357:
4356:
4340:
4338:
4337:
4332:
4330:
4329:
4313:
4311:
4310:
4305:
4269:
4267:
4266:
4261:
4249:
4247:
4246:
4241:
4239:
4238:
4222:
4220:
4219:
4214:
4212:
4211:
4195:
4193:
4192:
4187:
4151:
4149:
4148:
4143:
4141:
4140:
4124:
4122:
4121:
4116:
4114:
4113:
4093:
4091:
4090:
4085:
4067:
4065:
4064:
4059:
4047:
4045:
4044:
4039:
4027:
4025:
4024:
4019:
4007:
4005:
4004:
3999:
3997:
3996:
3980:
3978:
3977:
3972:
3970:
3969:
3950:
3948:
3947:
3942:
3927:
3925:
3924:
3919:
3904:
3902:
3901:
3896:
3881:
3879:
3878:
3873:
3858:
3856:
3855:
3850:
3835:
3833:
3832:
3827:
3808:
3806:
3805:
3800:
3761:
3759:
3758:
3753:
3714:
3712:
3711:
3706:
3660:
3658:
3657:
3652:
3640:
3638:
3637:
3632:
3620:
3618:
3617:
3612:
3600:
3598:
3597:
3592:
3580:
3578:
3577:
3572:
3560:
3558:
3557:
3552:
3540:
3538:
3537:
3532:
3520:
3518:
3517:
3512:
3500:
3498:
3497:
3492:
3480:
3478:
3477:
3472:
3452:
3450:
3449:
3444:
3432:
3430:
3429:
3424:
3413:Two flat planes
3404:
3402:
3401:
3396:
3394:
3393:
3370:
3368:
3367:
3362:
3325:
3323:
3322:
3317:
3312:
3309:
3308:
3293:
3289:
3258:
3256:
3255:
3252:{\displaystyle }
3250:
3224:on the interval
3179:
3177:
3176:
3171:
3156:
3154:
3153:
3148:
3096:
3094:
3093:
3088:
3086:
3072:
3049:
3014:
3012:
3011:
3006:
2991:
2987:
2971:
2970:
2953:
2949:
2930:
2925:
2906:
2904:
2903:
2898:
2881:
2880:
2861:
2859:
2858:
2853:
2829:
2827:
2826:
2821:
2819:
2818:
2758:
2756:
2755:
2750:
2748:
2747:
2687:
2685:
2684:
2679:
2677:
2676:
2616:
2614:
2613:
2608:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2548:
2547:
2542:
2535:
2531:
2522:
2520:
2500:
2480:
2479:
2474:
2461:
2459:
2458:
2453:
2450:
2445:
2440:
2427:
2425:
2424:
2419:
2417:
2416:
2413:
2370:
2368:
2367:
2362:
2360:
2359:
2356:
2310:
2308:
2307:
2302:
2224:
2222:
2221:
2216:
2209:
2134:
2127:
2125:
2124:
2119:
2112:
2035:
2033:
2032:
2027:
2025:
2024:
2021:
1991:
1990:
1987:
1957:
1956:
1953:
1899:
1897:
1896:
1891:
1889:
1886:
1872:
1850:
1831:
1830:
1805:
1803:
1802:
1797:
1792:
1791:
1776:
1775:
1766:
1765:
1753:
1752:
1730:
1728:
1727:
1724:{\displaystyle }
1722:
1699:on the interval
1698:
1696:
1695:
1690:
1679:with respect to
1671:
1669:
1668:
1663:
1640:
1639:
1630:
1629:
1617:
1616:
1594:
1592:
1591:
1588:{\displaystyle }
1586:
1563:on the interval
1562:
1560:
1559:
1554:
1543:with respect to
1538:
1536:
1535:
1530:
1528:
1527:
1513:
1512:
1492:
1490:
1489:
1484:
1482:
1480:
1479:
1458:
1453:
1452:
1417:
1415:
1414:
1409:
1401:
1400:
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1322:
1320:
1319:
1314:
1284:
1282:
1281:
1276:
1227:
1222:
1210:
1209:
1178:
1176:
1175:
1172:{\displaystyle }
1170:
1146:
1144:
1143:
1138:
1123:
1121:
1120:
1115:
1103:
1101:
1100:
1095:
1025:
1023:
1022:
1017:
1015:
1007:
978:
976:
975:
970:
951:. The case of a
915:
913:
912:
907:
905:
904:
871:is used to mean
836:
834:
833:
828:
816:
814:
813:
808:
806:
791:
789:
788:
783:
765:
763:
762:
757:
755:
754:
742:
727:
725:
724:
719:
702:
676:
674:
673:
668:
656:
654:
653:
648:
636:
634:
633:
628:
626:
625:
609:
607:
606:
601:
599:
584:
582:
581:
576:
574:
573:
557:
555:
554:
549:
527:
525:
524:
519:
507:
505:
504:
499:
487:
485:
484:
479:
463:
461:
460:
455:
443:
441:
440:
435:
423:
421:
420:
415:
403:vector subspaces
348:
346:
345:
340:
338:
330:
318:
316:
315:
310:
305:
297:
278:
276:
275:
270:
255:
249:
191:hyperbolic angle
132:
130:
129:
124:
113:
105:
83:
81:
80:
75:
59:
53:
34:perpendicularity
5219:
5218:
5214:
5213:
5212:
5210:
5209:
5208:
5184:
5183:
5182:
5177:
5159:
5121:
5077:
5014:
4966:
4908:
4899:
4865:Change of basis
4855:Multilinear map
4793:
4775:
4770:
4740:
4728:
4724:
4711:Coxeter, H.S.M.
4709:
4708:
4704:
4697:
4682:
4681:
4677:
4670:
4655:
4654:
4647:
4640:
4625:
4624:
4620:
4610:
4609:
4605:
4596:
4595:
4591:
4584:
4571:
4570:
4566:
4562:
4492:
4468:
4467:
4446:
4441:
4440:
4419:
4414:
4413:
4412:then a line in
4370:
4369:
4348:
4343:
4342:
4321:
4316:
4315:
4272:
4271:
4252:
4251:
4230:
4225:
4224:
4203:
4198:
4197:
4154:
4153:
4132:
4127:
4126:
4105:
4100:
4099:
4070:
4069:
4050:
4049:
4030:
4029:
4010:
4009:
3988:
3983:
3982:
3961:
3956:
3955:
3930:
3929:
3907:
3906:
3884:
3883:
3861:
3860:
3838:
3837:
3815:
3814:
3764:
3763:
3717:
3716:
3673:
3672:
3643:
3642:
3623:
3622:
3603:
3602:
3583:
3582:
3563:
3562:
3543:
3542:
3523:
3522:
3503:
3502:
3483:
3482:
3463:
3462:
3453:of a Euclidean
3435:
3434:
3415:
3414:
3411:
3385:
3380:
3379:
3376:superimposition
3347:
3346:
3339:
3300:
3283:
3282:
3226:
3225:
3191:
3159:
3158:
3103:
3102:
3019:
3018:
2962:
2958:
2954:
2936:
2932:
2909:
2908:
2872:
2864:
2863:
2835:
2834:
2830:are orthogonal.
2813:
2812:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2767:
2761:
2760:
2742:
2741:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2696:
2690:
2689:
2671:
2670:
2665:
2660:
2655:
2650:
2645:
2640:
2635:
2625:
2619:
2618:
2581:
2580:
2561:
2560:
2537:
2501:
2490:
2469:
2464:
2463:
2430:
2429:
2408:
2373:
2372:
2351:
2316:
2315:
2227:
2226:
2129:
2128:
2038:
2037:
2016:
1982:
1948:
1925:
1924:
1920:
1905:Kronecker delta
1884:
1883:
1871:
1862:
1861:
1849:
1835:
1819:
1814:
1813:
1780:
1767:
1757:
1744:
1736:
1735:
1701:
1700:
1681:
1680:
1631:
1621:
1608:
1600:
1599:
1565:
1564:
1545:
1544:
1504:
1498:
1497:
1471:
1444:
1433:
1432:
1392:
1375:
1374:
1348:
1347:
1328:
1327:
1290:
1289:
1201:
1184:
1183:
1149:
1148:
1129:
1128:
1126:weight function
1106:
1105:
1086:
1085:
1071:
1065:
1001:
1000:
989:Euclidean space
985:
961:
960:
896:
885:
884:
819:
818:
799:
794:
793:
768:
767:
746:
735:
730:
729:
695:
687:
686:
683:natural pairing
659:
658:
639:
638:
637:and an element
617:
612:
611:
592:
587:
586:
565:
560:
559:
540:
539:
510:
509:
490:
489:
470:
469:
446:
445:
426:
425:
406:
405:
321:
320:
285:
284:
261:
260:
251:
245:
211:
172:Euclidean space
135:function spaces
90:
89:
66:
65:
55:
49:
21:
12:
11:
5:
5217:
5215:
5207:
5206:
5201:
5199:Linear algebra
5196:
5186:
5185:
5179:
5178:
5176:
5175:
5164:
5161:
5160:
5158:
5157:
5152:
5147:
5142:
5137:
5135:Floating-point
5131:
5129:
5123:
5122:
5120:
5119:
5117:Tensor product
5114:
5109:
5104:
5102:Function space
5099:
5094:
5088:
5086:
5079:
5078:
5076:
5075:
5070:
5065:
5060:
5055:
5050:
5045:
5040:
5038:Triple product
5035:
5030:
5024:
5022:
5016:
5015:
5013:
5012:
5007:
5002:
4997:
4992:
4987:
4982:
4976:
4974:
4968:
4967:
4965:
4964:
4959:
4954:
4952:Transformation
4949:
4944:
4942:Multiplication
4939:
4934:
4929:
4924:
4918:
4916:
4910:
4909:
4902:
4900:
4898:
4897:
4892:
4887:
4882:
4877:
4872:
4867:
4862:
4857:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4817:
4812:
4807:
4801:
4799:
4798:Basic concepts
4795:
4794:
4792:
4791:
4786:
4780:
4777:
4776:
4773:Linear algebra
4771:
4769:
4768:
4761:
4754:
4746:
4739:
4738:
4722:
4702:
4695:
4675:
4668:
4645:
4638:
4618:
4603:
4589:
4582:
4563:
4561:
4558:
4557:
4556:
4551:
4545:
4543:Orthonormality
4540:
4535:
4534:
4533:
4523:
4518:
4513:
4508:
4503:
4498:
4491:
4488:
4475:
4453:
4449:
4439:and a line in
4426:
4422:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4355:
4351:
4328:
4324:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4259:
4237:
4233:
4210:
4206:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4139:
4135:
4112:
4108:
4083:
4080:
4077:
4057:
4037:
4017:
4008:of dimensions
3995:
3991:
3968:
3964:
3940:
3937:
3917:
3914:
3894:
3891:
3871:
3868:
3848:
3845:
3825:
3822:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3650:
3630:
3610:
3590:
3570:
3550:
3530:
3510:
3490:
3470:
3442:
3422:
3410:
3407:
3392:
3388:
3360:
3357:
3354:
3338:
3335:
3334:
3333:
3326:
3315:
3307:
3303:
3299:
3296:
3292:
3275:
3260:
3248:
3245:
3242:
3239:
3236:
3233:
3214:
3195:mathematicians
3190:
3187:
3186:
3185:
3182:Fourier series
3169:
3166:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3085:
3081:
3078:
3075:
3071:
3068:
3065:
3062:
3058:
3055:
3052:
3048:
3045:
3042:
3039:
3035:
3032:
3029:
3026:
3017:The functions
3015:
3004:
3001:
2998:
2995:
2990:
2986:
2983:
2980:
2977:
2974:
2969:
2965:
2961:
2957:
2952:
2948:
2945:
2942:
2939:
2935:
2929:
2924:
2921:
2917:
2896:
2893:
2890:
2887:
2884:
2879:
2875:
2871:
2851:
2848:
2845:
2842:
2833:The functions
2831:
2817:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2772:
2770:
2746:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2701:
2699:
2675:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2630:
2628:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2568:
2546:
2541:
2534:
2530:
2526:
2519:
2516:
2513:
2510:
2507:
2504:
2499:
2496:
2493:
2487:
2483:
2478:
2473:
2449:
2444:
2439:
2411:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2354:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2312:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2214:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2117:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2019:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1985:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1951:
1947:
1944:
1941:
1938:
1935:
1932:
1919:
1916:
1901:
1900:
1888:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1841:
1838:
1834:
1829:
1826:
1822:
1807:
1806:
1795:
1790:
1787:
1783:
1779:
1774:
1770:
1764:
1760:
1756:
1751:
1747:
1743:
1720:
1717:
1714:
1711:
1708:
1688:
1673:
1672:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1638:
1634:
1628:
1624:
1620:
1615:
1611:
1607:
1584:
1581:
1578:
1575:
1572:
1552:
1526:
1522:
1519:
1516:
1511:
1507:
1494:
1493:
1478:
1474:
1470:
1467:
1464:
1461:
1456:
1451:
1447:
1443:
1440:
1419:
1418:
1407:
1404:
1399:
1395:
1391:
1388:
1385:
1382:
1355:
1335:
1312:
1309:
1306:
1303:
1300:
1297:
1286:
1285:
1274:
1271:
1268:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1226:
1221:
1217:
1213:
1208:
1204:
1200:
1197:
1194:
1191:
1168:
1165:
1162:
1159:
1156:
1136:
1113:
1093:
1067:Main article:
1064:
1061:
1013:
1010:
993:if and only if
984:
981:
968:
955:uses the term
903:
899:
895:
892:
862:orthogonal set
854:
853:
844:is said to be
838:
826:
805:
802:
781:
778:
775:
753:
749:
745:
741:
738:
717:
714:
711:
708:
705:
701:
698:
694:
685:is zero, i.e.
666:
646:
624:
620:
598:
595:
572:
568:
547:
533:
517:
497:
477:
453:
433:
413:
399:
384:
365:
364:to each other.
350:
337:
333:
329:
308:
304:
300:
296:
292:
268:
239:
210:
207:
137:, families of
122:
119:
116:
112:
108:
104:
100:
97:
73:
43:bilinear forms
39:linear algebra
13:
10:
9:
6:
4:
3:
2:
5216:
5205:
5204:Orthogonality
5202:
5200:
5197:
5195:
5192:
5191:
5189:
5174:
5166:
5165:
5162:
5156:
5153:
5151:
5150:Sparse matrix
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5132:
5130:
5128:
5124:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5089:
5087:
5085:constructions
5084:
5080:
5074:
5073:Outermorphism
5071:
5069:
5066:
5064:
5061:
5059:
5056:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5033:Cross product
5031:
5029:
5026:
5025:
5023:
5021:
5017:
5011:
5008:
5006:
5003:
5001:
5000:Outer product
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4980:Orthogonality
4978:
4977:
4975:
4973:
4969:
4963:
4960:
4958:
4957:Cramer's rule
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4928:
4927:Decomposition
4925:
4923:
4920:
4919:
4917:
4915:
4911:
4906:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4811:
4808:
4806:
4803:
4802:
4800:
4796:
4790:
4787:
4785:
4782:
4781:
4778:
4774:
4767:
4762:
4760:
4755:
4753:
4748:
4747:
4744:
4735:
4731:
4726:
4723:
4718:
4717:
4712:
4706:
4703:
4698:
4692:
4688:
4687:
4679:
4676:
4671:
4665:
4661:
4660:
4652:
4650:
4646:
4641:
4635:
4631:
4630:
4622:
4619:
4615:, p. 234
4614:
4607:
4604:
4599:
4593:
4590:
4585:
4583:0-7167-0344-0
4579:
4575:
4568:
4565:
4559:
4555:
4552:
4550:
4549:coquaternions
4546:
4544:
4541:
4539:
4536:
4532:
4529:
4528:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4493:
4489:
4487:
4473:
4451:
4447:
4424:
4420:
4399:
4396:
4393:
4390:
4384:
4378:
4375:
4353:
4349:
4326:
4322:
4301:
4298:
4295:
4292:
4286:
4280:
4277:
4257:
4235:
4231:
4208:
4204:
4183:
4180:
4177:
4174:
4168:
4162:
4159:
4137:
4133:
4110:
4106:
4097:
4081:
4078:
4075:
4055:
4035:
4015:
3993:
3989:
3966:
3962:
3952:
3938:
3935:
3915:
3912:
3892:
3889:
3869:
3866:
3846:
3843:
3823:
3820:
3812:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3669:
3667:
3664:
3648:
3628:
3608:
3588:
3568:
3548:
3528:
3508:
3488:
3468:
3460:
3456:
3440:
3420:
3408:
3406:
3390:
3386:
3377:
3373:
3372:Latin squares
3358:
3355:
3352:
3344:
3343:combinatorics
3337:Combinatorics
3336:
3331:
3327:
3313:
3305:
3301:
3297:
3294:
3290:
3280:
3276:
3273:
3269:
3265:
3261:
3243:
3240:
3237:
3234:
3223:
3219:
3215:
3212:
3208:
3204:
3203:
3202:
3200:
3196:
3188:
3183:
3167:
3164:
3141:
3138:
3135:
3132:
3126:
3120:
3117:
3114:
3111:
3100:
3079:
3076:
3073:
3066:
3063:
3056:
3053:
3050:
3043:
3040:
3033:
3030:
3027:
3024:
3016:
3002:
2999:
2996:
2993:
2988:
2984:
2981:
2978:
2975:
2972:
2967:
2963:
2959:
2955:
2950:
2946:
2943:
2940:
2937:
2933:
2927:
2922:
2919:
2915:
2894:
2891:
2888:
2885:
2882:
2877:
2873:
2869:
2849:
2846:
2843:
2840:
2832:
2815:
2809:
2804:
2799:
2794:
2789:
2784:
2779:
2774:
2768:
2744:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2703:
2697:
2673:
2667:
2662:
2657:
2652:
2647:
2642:
2637:
2632:
2626:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2566:
2544:
2532:
2528:
2524:
2517:
2514:
2511:
2508:
2505:
2502:
2497:
2494:
2491:
2485:
2481:
2476:
2447:
2442:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2313:
2298:
2295:
2289:
2286:
2277:
2271:
2265:
2256:
2250:
2244:
2235:
2212:
2206:
2203:
2197:
2194:
2185:
2179:
2173:
2164:
2161:
2155:
2149:
2140:
2115:
2109:
2106:
2100:
2091:
2085:
2079:
2076:
2067:
2061:
2055:
2046:
2013:
2010:
2007:
2004:
2001:
1998:
1992:
1979:
1976:
1973:
1970:
1967:
1964:
1958:
1945:
1942:
1939:
1936:
1933:
1922:
1921:
1917:
1915:
1913:
1908:
1906:
1880:
1877:
1874:
1868:
1865:
1858:
1855:
1852:
1846:
1843:
1836:
1832:
1827:
1824:
1820:
1812:
1811:
1810:
1793:
1788:
1785:
1781:
1777:
1772:
1762:
1758:
1754:
1749:
1745:
1734:
1733:
1732:
1715:
1712:
1709:
1686:
1678:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1636:
1626:
1622:
1618:
1613:
1609:
1598:
1597:
1596:
1579:
1576:
1573:
1550:
1542:
1520:
1517:
1514:
1509:
1505:
1476:
1468:
1465:
1462:
1454:
1449:
1441:
1431:
1430:
1429:
1427:
1424:We write the
1422:
1405:
1402:
1397:
1389:
1386:
1383:
1373:
1372:
1371:
1369:
1353:
1333:
1324:
1310:
1307:
1301:
1295:
1272:
1269:
1266:
1259:
1253:
1247:
1241:
1235:
1229:
1224:
1219:
1215:
1211:
1206:
1198:
1195:
1192:
1182:
1181:
1180:
1163:
1160:
1157:
1134:
1127:
1111:
1091:
1084:
1080:
1079:inner product
1076:
1070:
1062:
1060:
1058:
1053:
1049:
1047:
1043:
1039:
1034:
1032:
1031:perpendicular
1028:
1011:
1008:
998:
994:
990:
982:
980:
966:
958:
954:
950:
946:
945:bilinear form
941:
939:
935:
931:
927:
923:
919:
901:
897:
893:
890:
882:
878:
874:
870:
865:
863:
859:
851:
847:
843:
839:
824:
803:
800:
779:
776:
773:
751:
747:
743:
739:
736:
715:
712:
706:
703:
699:
696:
684:
680:
664:
644:
622:
618:
596:
593:
585:, an element
570:
566:
558:and its dual
545:
538:
534:
531:
515:
495:
475:
467:
451:
431:
411:
404:
400:
397:
393:
389:
385:
382:
378:
374:
370:
366:
363:
359:
355:
351:
331:
298:
282:
266:
259:
254:
248:
244:
240:
237:
233:
229:
228:perpendicular
225:
221:
217:
213:
212:
208:
203:
199:
195:
192:
188:
184:
180:
177:
173:
170:
165:
161:
159:
158:combinatorics
155:
151:
146:
144:
140:
136:
120:
117:
106:
95:
87:
71:
63:
58:
52:
48:Two elements
46:
44:
40:
36:
35:
30:
29:orthogonality
26:
19:
18:Orthogonality
5083:Vector space
4815:Vector space
4733:
4725:
4715:
4705:
4685:
4678:
4657:
4628:
4621:
4612:
4606:
4592:
4573:
4567:
4095:
4068:of at least
3953:
3810:
3670:
3662:
3458:
3412:
3340:
3192:
2314:The vectors
1923:The vectors
1909:
1902:
1808:
1676:
1674:
1540:
1495:
1423:
1420:
1367:
1325:
1287:
1072:
1054:
1050:
1035:
986:
948:
942:
929:
926:unit vectors
917:
880:
872:
868:
866:
861:
857:
855:
678:
465:
395:
391:
381:unit vectors
376:
280:
252:
246:
234:they form a
231:
226:if they are
223:
197:
193:
182:
178:
168:
147:
85:
62:vector space
56:
50:
47:
32:
28:
22:
5063:Multivector
5028:Determinant
4985:Dot product
4830:Linear span
4730:P.H.Schoute
4574:Gravitation
3457:are called
1677:orthonormal
997:dot product
934:probability
922:orthonormal
728:. Two sets
464:are called
362:orthonormal
236:right angle
209:Definitions
200:denote the
25:mathematics
5188:Categories
5097:Direct sum
4932:Invertible
4835:Linear map
4560:References
2579:, and for
1541:orthogonal
1368:orthogonal
1057:hyperplane
949:orthogonal
938:statistics
873:orthogonal
846:orthogonal
679:orthogonal
396:orthogonal
392:orthogonal
377:orthogonal
281:orthogonal
224:orthogonal
202:worldlines
86:orthogonal
5127:Numerical
4890:Transpose
4713:(1973) .
4613:Algebra I
4379:
4281:
4163:
3356:×
3298:−
3235:−
3168:π
3142:π
3136:π
3133:−
3121:π
3080:∈
3074:∣
3057:
3034:
2982:−
2920:−
2916:∫
2892:−
2602:−
2596:≤
2590:≤
2486:∑
2406:…
2349:…
2287:−
2195:−
2162:−
2077:−
2011:−
1971:−
1878:≠
1821:δ
1782:δ
1769:⟩
1742:⟨
1654:≠
1648:∣
1633:⟩
1606:⟨
1521:∈
1515:∣
1473:⟩
1460:⟨
1446:‖
1439:‖
1394:⟩
1381:⟨
1216:∫
1203:⟩
1190:⟨
1083:functions
1073:By using
1009:π
967:ϕ
850:confluent
777:⊆
752:∗
744:⊆
710:⟩
693:⟨
681:if their
623:∗
571:∗
332:⊥
307:⟩
291:⟨
5173:Category
5112:Subspace
5107:Quotient
5058:Bivector
4972:Bilinear
4914:Matrices
4789:Glossary
4490:See also
1918:Examples
804:′
740:′
700:′
597:′
535:Given a
216:geometry
189:through
4784:Outline
4554:Up tack
1903:is the
1081:of two
1027:radians
243:vectors
37:to the
5068:Tensor
4880:Kernel
4810:Vector
4805:Scalar
4693:
4666:
4636:
4580:
3345:, two
2210:
2135:
2113:
1809:where
995:their
930:normal
918:normal
869:normal
537:module
358:matrix
256:in an
218:, two
183:right:
156:, and
4937:Minor
4922:Block
4860:Basis
4314:then
4270:. If
4196:then
4152:. If
1046:plane
373:basis
371:is a
356:is a
176:angle
169:left:
143:basis
88:when
60:of a
5092:Dual
4947:Rank
4691:ISBN
4664:ISBN
4634:ISBN
4578:ISBN
4341:and
4293:>
4223:and
4028:and
3981:and
3928:and
3882:and
3836:and
3641:and
3521:and
3433:and
3277:The
3262:The
3216:The
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2862:and
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2371:and
2225:and
1539:are
1426:norm
1366:are
1346:and
1104:and
1042:line
1036:The
936:and
766:and
677:are
424:and
401:Two
279:are
250:and
241:Two
232:i.e.
222:are
84:are
54:and
4376:dim
4278:dim
4160:dim
3663:and
3341:In
3054:cos
3031:sin
1731:if
1595:if
987:In
657:of
610:of
367:An
352:An
214:In
185:in
41:of
23:In
5190::
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4648:^
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