6032:
171:
145:
6296:
43:
3238:
1999:
2689:
2841:
1729:
5380:. This implies that the property of being a symmetric matrix must be kept by the above change-of-base formula. One can also check this by noting that the transpose of a matrix product is the product of the transposes computed in the reverse order. In particular,
4536:, many function properties are kept by a change of basis. This allows defining these properties as properties of functions of a variable vector that are not related to any specific basis. So, a function whose domain is a vector space or a subset of it is
4017:
5468:
2830:
2527:
3233:{\displaystyle {\begin{aligned}x_{1}v_{1}+x_{2}v_{2}&=(y_{1}\cos t-y_{2}\sin t)v_{1}+(y_{1}\sin t+y_{2}\cos t)v_{2}\\&=y_{1}(\cos(t)v_{1}+\sin(t)v_{2})+y_{2}(-\sin(t)v_{1}+\cos(t)v_{2})\\&=y_{1}w_{1}+y_{2}w_{2}.\end{aligned}}}
4448:
3260:
of the corresponding linear map to the vector whose coordinates form the column vector. The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its definition and proof.
4304:
2428:
1548:
312:
4127:
3872:
5197:
1994:{\displaystyle {\begin{aligned}z&=\sum _{j=1}^{n}y_{j}w_{j}\\&=\sum _{j=1}^{n}\left(y_{j}\sum _{i=1}^{n}a_{i,j}v_{i}\right)\\&=\sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{i,j}y_{j}\right)v_{i}.\end{aligned}}}
1420:
3573:
943:
3701:
509:
2071:
5660:
asserts that, given such a symmetric matrix, there is an orthogonal change of basis such that the resulting matrix (of both the bilinear form and the endomorphism) is a diagonal matrix with the
2846:
1734:
5656:
Such matrices have the fundamental property that the change-of-base formula is the same for a symmetric bilinear form and the endomorphism that is represented by the same symmetric matrix. The
700:
5291:
1591:
389:
352:
5508:
766:
4213:
4170:
5231:
618:
5654:
3418:
2343:
3736:
3468:
5052:
2280:
1268:
1158:
1715:
1306:
1216:
1075:
4055:
3907:
1677:
1037:
978:
2114:
233:
consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.
5386:
4959:
4918:
5109:
2519:
5358:
5551:
2700:
2684:{\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}\cos t&-\sin t\\\sin t&\cos t\end{bmatrix}}\,{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}.}
2211:
1638:
1616:
4484:
of the function is changed. This change can be computed by substituting the "old" coordinates for their expressions in terms of the "new" coordinates. More precisely, if
2470:
2163:
4811:
4735:
3770:
5856:
4988:
4336:
1106:
580:
3895:
3800:
3626:
864:
826:
3596:
3326:
1178:
413:
4755:
to itself. For a change of basis, the formula of the preceding section applies, with the same change-of-basis matrix on both sides of the formula. That is, if
4344:
4221:
2350:
1446:
430:
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
250:
5890:
5682:
4586:, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold.
4063:
3808:
5531:. Moreover, the resulting nonzero entries on the diagonal are defined up to the multiplication by a square. So, if the ground field is the field
4525:
The fact that the change-of-basis formula expresses the old coordinates in terms of the new one may seem unnatural, but appears as useful, as no
5137:
6281:
3286:
is the field of scalars. When only one basis is considered for each vector space, it is worth to leave this isomorphism implicit, and to work
1314:
3476:
5821:
5795:
5677:
126:
5664:
of the initial matrix on the diagonal. It follows that, over the reals, if the matrix of an endomorphism is symmetric, then it is
6271:
60:
6233:
6169:
5250:
is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is
872:
512:
31:
3633:
441:
107:
5713:
Although a basis is generally defined as a set of vectors (for example, as a spanning set that is linearly independent), the
2007:
64:
79:
5566:
1433:
for the new one makes clearer the formulas that follows, and helps avoiding errors in proofs and explicit computations.)
6011:
5883:
1440:
expresses the coordinates over the old basis in terms of the coordinates over the new basis. With above notation, it is
625:
86:
6116:
5966:
3290:
an isomorphism. As several bases of the same vector space are considered here, a more accurate wording is required.
6021:
5915:
5256:
1559:
357:
320:
5476:
711:
53:
6261:
5910:
4175:
4132:
240:
which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using
93:
6136:
5205:
6253:
5813:
5693:
5520:
4481:
585:
196:
5604:
are specially useful; this means that one generally prefer to restrict changes of basis to those that have an
5611:
4579:
if the multivariate function that represents it on some basis—and thus on every basis—has the same property.
3365:
2285:
6320:
6299:
6006:
5876:
5297:
4558:
3706:
3434:
75:
5005:
4012:{\displaystyle B_{\mathrm {new} }=\phi _{\mathrm {new} }(\phi _{\mathrm {old} }^{-1}(B_{\mathrm {old} })).}
2228:
1221:
1111:
6325:
6063:
5996:
5986:
4462:
3269:
2214:
2082:
1682:
1273:
1183:
1042:
6223:
6078:
6073:
6068:
6001:
5946:
5665:
5463:{\displaystyle (P^{\mathsf {T}}\mathbf {B} P)^{\mathsf {T}}=P^{\mathsf {T}}\mathbf {B} ^{\mathsf {T}}P,}
4470:
4025:
1647:
1007:
948:
159:
2087:
2073:
the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis.
6088:
6053:
6040:
5931:
5835:
5787:
4923:
4882:
4474:
4466:
3898:
3257:
3249:
2825:{\displaystyle x_{1}=y_{1}\cos t-y_{2}\sin t\qquad {\text{and}}\qquad x_{2}=y_{1}\sin t+y_{2}\cos t.}
772:
241:
214:
155:
5065:
2475:
158:, these form a new basis. The linear combinations relating the first basis to the other extend to a
6266:
6146:
6121:
5971:
5307:
4843:
4551:
4544:
3298:
1425:(One could take the same summation index for the two sums, but choosing systematically the indexes
519:
5717:
notation is convenient here, since the indexing by the first positive integers makes the basis an
5534:
2168:
1621:
1599:
5976:
5687:
3429:
3273:
2435:
2123:
151:
4777:
4701:
4453:
which is the change-of-basis formula expressed in terms of linear maps instead of coordinates.
3748:
6174:
6131:
6058:
5951:
5839:
5817:
5791:
5605:
5601:
4817:
4572:
989:
203:
4964:
4312:
1082:
552:
317:
where "old" and "new" refer respectively to the initially defined basis and the other basis,
100:
6179:
6083:
5936:
5657:
5527:
is not two, then for every symmetric bilinear form there is a basis for which the matrix is
5377:
4821:
4526:
3880:
3775:
3601:
839:
801:
3581:
3304:
6238:
6031:
5981:
5528:
4565:
4533:
207:
170:
5808:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
4473:
whose variables are the coordinates on some basis of the vector on which the function is
4443:{\displaystyle \phi _{\mathrm {old} }^{-1}(v)=\psi _{A}(\phi _{\mathrm {new} }^{-1}(v)),}
221:. If two different bases are considered, the coordinate vector that represents a vector
144:
6243:
6228:
6164:
5899:
5806:
4771:
is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is
4299:{\displaystyle \phi _{\mathrm {old} }^{-1}=\psi _{A}\circ \phi _{\mathrm {new} }^{-1}.}
3340:
3339:-vector space whose addition and scalar multiplication are defined component-wise. Its
2117:
1163:
398:
6314:
6276:
6199:
6159:
6126:
6106:
5718:
4834:
4760:
392:
188:
17:
2423:{\displaystyle {\begin{bmatrix}\cos t&-\sin t\\\sin t&\cos t\end{bmatrix}}.}
6209:
6098:
6048:
5941:
4748:
4612:
3743:
1543:{\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad {\text{for }}i=1,\ldots ,n.}
424:
192:
225:
on one basis is, in general, different from the coordinate vector that represents
6189:
6154:
6111:
5956:
5661:
5554:
3272:
of implied vector spaces, and to the fact that the choice of a basis induces an
993:
218:
184:
42:
307:{\displaystyle \mathbf {x} _{\mathrm {old} }=A\,\mathbf {x} _{\mathrm {new} },}
6218:
5961:
5862:
5597:
4861:
4595:
3253:
3578:
Conversely, such a linear isomorphism defines a basis, which is the image by
6016:
5234:
4824:
if and only if they represent the same endomorphism on two different bases.
4820:
can be used as a change-of-basis matrix, this implies that two matrices are
4122:{\displaystyle \phi _{\mathrm {new} }=\phi _{\mathrm {old} }\circ \psi _{A}}
3867:{\displaystyle \phi _{\mathrm {new} }=\phi _{\mathrm {old} }\circ \psi _{A}}
866:
such a convention is useful for avoiding errors in explicit computations.)
5192:{\displaystyle B(v,w)=\mathbf {v} ^{\mathsf {T}}\mathbf {B} \mathbf {w} ,}
4495:
is the expression of the function in terms of the old coordinates, and if
6184:
5581:
4583:
4522:
is the expression of the same function in terms of the new coordinates.
154:
of one basis of vectors (purple) obtains new vectors (red). If they are
5593:
5589:
5585:
4740:
This is a straightforward consequence of the change-of-basis formula.
1415:{\displaystyle z=\sum _{i=1}^{n}x_{i}v_{i}=\sum _{j=1}^{n}y_{j}w_{j}.}
6194:
3568:{\displaystyle \phi (x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}v_{i}.}
423:), which is the matrix whose columns are the coordinates of the new
176:
A vector represented by two different bases (purple and red arrows).
5577:
depends only on the bilinear form, and not of the change of basis.
5054:(the "old" basis in what follows) is the matrix whose entry of the
5714:
3329:
3287:
1723:
Using the above definition of the change-of basis matrix, one has
202:
allows representing uniquely any element of the vector space by a
5868:
5580:
Symmetric bilinear forms over the reals are often encountered in
3256:, and the product of a matrix and a column vector represents the
5843:
5872:
36:
4022:
A straightforward verification shows that this definition of
395:
of the coordinates of the same vector on the two bases, and
3738:
the associated isomorphism. Given a change-of basis matrix
5123:
are the column vectors of the coordinates of two vectors
938:{\displaystyle B_{\mathrm {new} }=(w_{1},\ldots ,w_{n}),}
3696:{\displaystyle B_{\mathrm {old} }=(v_{1},\ldots ,v_{n})}
504:{\displaystyle B_{\mathrm {old} }=(v_{1},\ldots ,v_{n})}
2066:{\displaystyle z=\textstyle \sum _{i=1}^{n}x_{i}v_{i},}
2643:
2579:
2536:
2359:
2017:
5614:
5537:
5479:
5389:
5310:
5259:
5208:
5140:
5068:
5008:
4967:
4926:
4885:
4780:
4704:
4347:
4315:
4224:
4178:
4135:
4066:
4028:
3910:
3883:
3811:
3778:
3751:
3709:
3636:
3604:
3584:
3479:
3437:
3368:
3307:
2844:
2703:
2530:
2478:
2438:
2353:
2288:
2231:
2171:
2126:
2090:
2010:
1732:
1685:
1650:
1624:
1602:
1562:
1553:
In terms of matrices, the change of basis formula is
1449:
1317:
1276:
1224:
1186:
1166:
1114:
1085:
1045:
1010:
951:
875:
842:
804:
714:
628:
588:
555:
444:
401:
360:
323:
253:
5696: — application in computational chemistry
6252:
6208:
6145:
6097:
6039:
5924:
5608:change-of-base matrix, that is, a matrix such that
5557:, these nonzero entries can be chosen to be either
67:. Unsourced material may be challenged and removed.
5805:
5804:Beauregard, Raymond A.; Fraleigh, John B. (1973),
5753:
5648:
5545:
5502:
5462:
5352:
5285:
5225:
5191:
5103:
5046:
4982:
4953:
4912:
4805:
4729:
4442:
4330:
4298:
4207:
4164:
4121:
4049:
4011:
3889:
3866:
3794:
3764:
3730:
3695:
3620:
3590:
3567:
3462:
3412:
3347:th element the tuple with all components equal to
3320:
3232:
2824:
2683:
2513:
2464:
2422:
2337:
2274:
2205:
2157:
2108:
2065:
1993:
1709:
1671:
1632:
1610:
1585:
1542:
1414:
1300:
1262:
1210:
1172:
1152:
1100:
1069:
1031:
972:
937:
858:
820:
760:
695:{\displaystyle w_{j}=\sum _{i=1}^{n}a_{i,j}v_{i}.}
694:
612:
574:
503:
407:
383:
346:
306:
5569:is a theorem that asserts that the numbers of
4057:is the same as that of the preceding section.
5884:
5857:MIT Linear Algebra Lecture on Change of Basis
5690:, the continuous analogue of change of basis.
5286:{\displaystyle P^{\mathsf {T}}\mathbf {B} P.}
4532:As the change-of-basis formula involves only
3703:be the "old basis" of a change of basis, and
2432:The change-of-basis formula asserts that, if
1640:are the column vectors of the coordinates of
1586:{\displaystyle \mathbf {x} =A\,\mathbf {y} ,}
384:{\displaystyle \mathbf {x} _{\mathrm {new} }}
347:{\displaystyle \mathbf {x} _{\mathrm {old} }}
8:
5503:{\displaystyle P^{\mathsf {T}}\mathbf {B} P}
761:{\displaystyle A=\left(a_{i,j}\right)_{i,j}}
5473:and the two members of this equation equal
4208:{\displaystyle \phi _{\mathrm {new} }^{-1}}
4165:{\displaystyle \phi _{\mathrm {old} }^{-1}}
5891:
5877:
5869:
5226:{\displaystyle \mathbf {v} ^{\mathsf {T}}}
4582:This is specially useful in the theory of
779:th column is formed by the coordinates of
5634:
5620:
5619:
5613:
5539:
5538:
5536:
5492:
5485:
5484:
5478:
5447:
5446:
5441:
5433:
5432:
5418:
5417:
5405:
5398:
5397:
5388:
5309:
5272:
5265:
5264:
5258:
5216:
5215:
5210:
5207:
5181:
5176:
5169:
5168:
5163:
5139:
5092:
5079:
5067:
5035:
5016:
5007:
4966:
4925:
4884:
4785:
4779:
4709:
4703:
4416:
4404:
4403:
4390:
4365:
4353:
4352:
4346:
4314:
4284:
4272:
4271:
4258:
4242:
4230:
4229:
4223:
4196:
4184:
4183:
4177:
4153:
4141:
4140:
4134:
4113:
4093:
4092:
4072:
4071:
4065:
4034:
4033:
4027:
3987:
3986:
3970:
3958:
3957:
3937:
3936:
3916:
3915:
3909:
3882:
3858:
3838:
3837:
3817:
3816:
3810:
3783:
3777:
3756:
3750:
3742:, one could consider it the matrix of an
3715:
3714:
3708:
3684:
3665:
3642:
3641:
3635:
3609:
3603:
3583:
3556:
3546:
3536:
3525:
3509:
3490:
3478:
3448:
3436:
3401:
3382:
3367:
3312:
3306:
3217:
3207:
3194:
3184:
3161:
3133:
3102:
3086:
3058:
3030:
3010:
2988:
2966:
2950:
2928:
2906:
2886:
2876:
2863:
2853:
2845:
2843:
2804:
2782:
2769:
2759:
2743:
2721:
2708:
2702:
2664:
2650:
2638:
2637:
2574:
2557:
2543:
2531:
2529:
2499:
2486:
2477:
2456:
2443:
2437:
2354:
2352:
2293:
2287:
2236:
2230:
2176:
2170:
2131:
2125:
2097:
2093:
2092:
2089:
2053:
2043:
2033:
2022:
2009:
1978:
1963:
1947:
1937:
1926:
1911:
1900:
1875:
1859:
1849:
1838:
1828:
1813:
1802:
1782:
1772:
1762:
1751:
1733:
1731:
1691:
1690:
1684:
1656:
1655:
1649:
1625:
1623:
1603:
1601:
1575:
1574:
1563:
1561:
1511:
1504:
1488:
1478:
1467:
1454:
1448:
1403:
1393:
1383:
1372:
1359:
1349:
1339:
1328:
1316:
1282:
1281:
1275:
1251:
1232:
1223:
1192:
1191:
1185:
1165:
1141:
1122:
1113:
1084:
1051:
1050:
1044:
1016:
1015:
1009:
957:
956:
950:
923:
904:
881:
880:
874:
847:
841:
809:
803:
746:
730:
713:
683:
667:
657:
646:
633:
627:
613:{\displaystyle B_{\mathrm {old} }\colon }
594:
593:
587:
560:
554:
492:
473:
450:
449:
443:
400:
368:
367:
362:
359:
331:
330:
325:
322:
288:
287:
282:
280:
261:
260:
255:
252:
127:Learn how and when to remove this message
5683:Covariance and contravariance of vectors
5863:Khan Academy Lecture on Change of Basis
5734:
5706:
5649:{\displaystyle P^{\mathsf {T}}=P^{-1}.}
3413:{\displaystyle B=(v_{1},\ldots ,v_{n})}
3268:a linear map, one refers implicitly to
2338:{\displaystyle w_{2}=(-\sin t,\cos t).}
790:. (Here and in what follows, the index
6282:Comparison of linear algebra libraries
5765:
5621:
5486:
5448:
5434:
5419:
5399:
5266:
5217:
5170:
4751:, are linear maps from a vector space
4630:. It is represented on "old" bases of
3731:{\displaystyle \phi _{\mathrm {old} }}
3463:{\displaystyle \phi \colon F^{n}\to V}
992:, or equivalently if it has a nonzero
5741:
5047:{\displaystyle (v_{1},\ldots ,v_{n})}
4652:. A change of bases is defined by an
2275:{\displaystyle w_{1}=(\cos t,\sin t)}
1263:{\displaystyle (y_{1},\ldots ,y_{n})}
1153:{\displaystyle (x_{1},\ldots ,x_{n})}
7:
4508:is the change-of-base formula, then
2472:are the new coordinates of a vector
65:adding citations to reliable sources
1710:{\displaystyle B_{\mathrm {new} },}
1301:{\displaystyle B_{\mathrm {new} };}
1211:{\displaystyle B_{\mathrm {old} },}
1070:{\displaystyle B_{\mathrm {new} }.}
236:Such a conversion results from the
27:Coordinate change in linear algebra
4691:On the "new" bases, the matrix of
4457:Function defined on a vector space
4411:
4408:
4405:
4360:
4357:
4354:
4279:
4276:
4273:
4237:
4234:
4231:
4191:
4188:
4185:
4148:
4145:
4142:
4100:
4097:
4094:
4079:
4076:
4073:
4050:{\displaystyle B_{\mathrm {new} }}
4041:
4038:
4035:
3994:
3991:
3988:
3965:
3962:
3959:
3944:
3941:
3938:
3923:
3920:
3917:
3845:
3842:
3839:
3824:
3821:
3818:
3722:
3719:
3716:
3649:
3646:
3643:
2347:So, the change-of-basis matrix is
1698:
1695:
1692:
1672:{\displaystyle B_{\mathrm {old} }}
1663:
1660:
1657:
1289:
1286:
1283:
1199:
1196:
1193:
1058:
1055:
1052:
1032:{\displaystyle B_{\mathrm {old} }}
1023:
1020:
1017:
973:{\displaystyle B_{\mathrm {new} }}
964:
961:
958:
888:
885:
882:
601:
598:
595:
457:
454:
451:
375:
372:
369:
338:
335:
332:
295:
292:
289:
268:
265:
262:
25:
5678:Active and passive transformation
2109:{\displaystyle \mathbb {R} ^{2}.}
6295:
6294:
6272:Basic Linear Algebra Subprograms
6030:
5832:Linear Algebra and Matrix Theory
5493:
5442:
5406:
5372:. It follows that the matrix of
5273:
5211:
5182:
5177:
5164:
2835:This may be verified by writing
1626:
1604:
1576:
1564:
832:refers always to the columns of
363:
326:
283:
256:
169:
143:
41:
6170:Seven-dimensional cross product
5754:Beauregard & Fraleigh (1973
4954:{\displaystyle v\mapsto B(w,v)}
4913:{\displaystyle v\mapsto B(v,w)}
4480:When the basis is changed, the
4465:that has a vector space as its
4060:Now, by composing the equation
2764:
2758:
1510:
513:finite-dimensional vector space
52:needs additional citations for
5414:
5390:
5347:
5335:
5326:
5314:
5156:
5144:
5104:{\displaystyle B(v_{i},v_{j})}
5098:
5072:
5041:
5009:
4948:
4936:
4930:
4907:
4895:
4889:
4434:
4431:
4425:
4396:
4380:
4374:
4003:
4000:
3979:
3950:
3690:
3658:
3515:
3483:
3454:
3407:
3375:
3167:
3154:
3148:
3126:
3120:
3108:
3092:
3079:
3073:
3051:
3045:
3036:
3003:
2959:
2943:
2899:
2514:{\displaystyle (x_{1},x_{2}),}
2505:
2479:
2329:
2302:
2269:
2245:
2197:
2185:
2152:
2140:
1257:
1225:
1147:
1115:
929:
897:
498:
466:
244:, this formula can be written
1:
5353:{\displaystyle B(v,w)=B(w,v)}
3343:is the basis that has as its
794:refers always to the rows of
162:, called the change of basis.
6012:Eigenvalues and eigenvectors
5588:, typically in the study of
5546:{\displaystyle \mathbb {R} }
4864:in both arguments. That is,
3264:When one says that a matrix
2206:{\displaystyle v_{2}=(0,1).}
1633:{\displaystyle \mathbf {y} }
1611:{\displaystyle \mathbf {x} }
4961:are linear for every fixed
4469:is commonly specified as a
3276:between a vector space and
2465:{\displaystyle y_{1},y_{2}}
2158:{\displaystyle v_{1}=(1,0)}
6342:
5834:(2nd ed.), New York:
5786:(5th ed.), New York:
5567:Sylvester's law of inertia
984:if and only if the matrix
538:, one can define a vector
29:
6290:
6028:
5906:
5859:, from MIT OpenCourseWare
5784:Elementary Linear Algebra
4806:{\displaystyle P^{-1}MP.}
4767:over an "old" basis, and
4730:{\displaystyle P^{-1}MQ.}
3765:{\displaystyle \psi _{A}}
3598:of the standard basis of
5830:Nering, Evar D. (1970),
5814:Houghton Mifflin Company
5694:Chirgwin-Coulson weights
4879:is bilinear if the maps
4215:on the right, one gets
2120:consists of the vectors
30:Not to be confused with
5298:symmetric bilinear form
4983:{\displaystyle w\in V.}
4680:change-of-basis matrix
4662:change-of-basis matrix
4559:differentiable function
4331:{\displaystyle v\in V,}
3244:In terms of linear maps
1438:change-of-basis formula
1101:{\displaystyle z\in V,}
575:{\displaystyle a_{i,j}}
434:Change of basis formula
238:change-of-basis formula
5997:Row and column vectors
5782:Anton, Howard (1987),
5650:
5547:
5504:
5464:
5354:
5287:
5227:
5193:
5105:
5048:
4984:
4955:
4914:
4807:
4763:of an endomorphism of
4731:
4444:
4332:
4300:
4209:
4166:
4123:
4051:
4013:
3891:
3890:{\displaystyle \circ }
3868:
3796:
3795:{\displaystyle F^{n}.}
3766:
3732:
3697:
3622:
3621:{\displaystyle F^{n}.}
3592:
3569:
3541:
3464:
3414:
3322:
3234:
2826:
2685:
2515:
2466:
2424:
2339:
2276:
2207:
2159:
2110:
2083:Euclidean vector space
2067:
2038:
1995:
1942:
1916:
1854:
1818:
1767:
1711:
1673:
1634:
1612:
1587:
1544:
1483:
1429:for the old basis and
1416:
1388:
1344:
1302:
1264:
1212:
1174:
1160:be the coordinates of
1154:
1102:
1071:
1033:
1002:change-of-basis matrix
974:
939:
860:
859:{\displaystyle w_{j};}
822:
821:{\displaystyle v_{i},}
762:
696:
662:
614:
576:
505:
417:change-of-basis matrix
409:
385:
348:
308:
229:on the other basis. A
6002:Row and column spaces
5947:Scalar multiplication
5651:
5548:
5505:
5465:
5355:
5288:
5228:
5194:
5111:. It follows that if
5106:
5049:
4985:
4956:
4915:
4808:
4732:
4471:multivariate function
4445:
4333:
4309:It follows that, for
4301:
4210:
4167:
4124:
4052:
4014:
3892:
3869:
3797:
3767:
3733:
3698:
3623:
3593:
3591:{\displaystyle \phi }
3570:
3521:
3465:
3415:
3323:
3321:{\displaystyle F^{n}}
3235:
2827:
2686:
2516:
2467:
2425:
2340:
2277:
2208:
2160:
2111:
2068:
2018:
1996:
1922:
1896:
1834:
1798:
1747:
1712:
1674:
1635:
1613:
1588:
1545:
1463:
1417:
1368:
1324:
1303:
1270:its coordinates over
1265:
1213:
1175:
1155:
1103:
1072:
1034:
975:
940:
861:
823:
763:
697:
642:
615:
577:
506:
410:
386:
349:
309:
160:linear transformation
18:Change of coordinates
6137:Gram–Schmidt process
6089:Gaussian elimination
5612:
5535:
5523:of the ground field
5477:
5387:
5308:
5257:
5206:
5138:
5066:
5006:
4965:
4924:
4883:
4778:
4702:
4345:
4313:
4222:
4176:
4133:
4064:
4026:
3908:
3899:function composition
3881:
3809:
3776:
3749:
3707:
3634:
3602:
3582:
3477:
3435:
3366:
3305:
3258:function application
2842:
2701:
2528:
2476:
2436:
2351:
2286:
2229:
2217:them by an angle of
2169:
2124:
2088:
2008:
1730:
1683:
1648:
1622:
1600:
1560:
1447:
1315:
1274:
1222:
1184:
1164:
1112:
1083:
1043:
1008:
949:
873:
840:
802:
712:
626:
586:
553:
442:
399:
358:
321:
251:
156:linearly independent
61:improve this article
6267:Numerical stability
6147:Multilinear algebra
6122:Inner product space
5972:Linear independence
5865:, from Khan Academy
5756:, pp. 240–243)
5744:, pp. 221–237)
5300:is a bilinear form
4998:of a bilinear form
4552:continuous function
4545:polynomial function
4424:
4373:
4292:
4250:
4204:
4161:
3978:
549:by its coordinates
5977:Linear combination
5688:Integral transform
5646:
5600:. In these cases,
5543:
5500:
5460:
5350:
5283:
5223:
5189:
5101:
5044:
4980:
4951:
4910:
4838:on a vector space
4803:
4727:
4622:to a vector space
4540:a linear function,
4440:
4399:
4348:
4328:
4296:
4267:
4225:
4205:
4179:
4162:
4136:
4119:
4047:
4009:
3953:
3887:
3864:
3792:
3762:
3728:
3693:
3618:
3588:
3565:
3460:
3430:linear isomorphism
3410:
3318:
3230:
3228:
2822:
2681:
2672:
2631:
2565:
2511:
2462:
2420:
2411:
2335:
2272:
2203:
2155:
2106:
2063:
2062:
1991:
1989:
1707:
1669:
1630:
1608:
1583:
1540:
1412:
1298:
1260:
1208:
1170:
1150:
1098:
1067:
1029:
1000:is said to be the
970:
935:
856:
818:
758:
692:
610:
572:
501:
427:on the old basis.
405:
381:
344:
304:
152:linear combination
6308:
6307:
6175:Geometric algebra
6132:Kronecker product
5967:Linear projection
5952:Vector projection
5768:, pp. 50–52)
5602:orthonormal bases
4818:invertible matrix
4573:analytic function
2762:
1514:
1173:{\displaystyle z}
421:transition matrix
408:{\displaystyle A}
204:coordinate vector
137:
136:
129:
111:
76:"Change of basis"
16:(Redirected from
6333:
6298:
6297:
6180:Exterior algebra
6117:Hadamard product
6034:
6022:Linear equations
5893:
5886:
5879:
5870:
5846:
5826:
5811:
5800:
5769:
5763:
5757:
5751:
5745:
5739:
5722:
5711:
5658:Spectral theorem
5655:
5653:
5652:
5647:
5642:
5641:
5626:
5625:
5624:
5576:
5572:
5564:
5560:
5552:
5550:
5549:
5544:
5542:
5526:
5515:
5509:
5507:
5506:
5501:
5496:
5491:
5490:
5489:
5469:
5467:
5466:
5461:
5453:
5452:
5451:
5445:
5439:
5438:
5437:
5424:
5423:
5422:
5409:
5404:
5403:
5402:
5376:on any basis is
5375:
5371:
5367:
5363:
5359:
5357:
5356:
5351:
5303:
5292:
5290:
5289:
5284:
5276:
5271:
5270:
5269:
5249:
5242:
5232:
5230:
5229:
5224:
5222:
5221:
5220:
5214:
5198:
5196:
5195:
5190:
5185:
5180:
5175:
5174:
5173:
5167:
5130:
5126:
5122:
5116:
5110:
5108:
5107:
5102:
5097:
5096:
5084:
5083:
5061:
5057:
5053:
5051:
5050:
5045:
5040:
5039:
5021:
5020:
5001:
4997:
4989:
4987:
4986:
4981:
4960:
4958:
4957:
4952:
4919:
4917:
4916:
4911:
4878:
4859:
4848:
4812:
4810:
4809:
4804:
4793:
4792:
4770:
4766:
4758:
4754:
4736:
4734:
4733:
4728:
4717:
4716:
4694:
4687:
4683:
4679:
4669:
4665:
4661:
4651:
4647:
4637:
4633:
4629:
4625:
4621:
4617:
4610:
4534:linear functions
4529:is needed here.
4527:matrix inversion
4521:
4507:
4494:
4449:
4447:
4446:
4441:
4423:
4415:
4414:
4395:
4394:
4372:
4364:
4363:
4337:
4335:
4334:
4329:
4305:
4303:
4302:
4297:
4291:
4283:
4282:
4263:
4262:
4249:
4241:
4240:
4214:
4212:
4211:
4206:
4203:
4195:
4194:
4172:on the left and
4171:
4169:
4168:
4163:
4160:
4152:
4151:
4128:
4126:
4125:
4120:
4118:
4117:
4105:
4104:
4103:
4084:
4083:
4082:
4056:
4054:
4053:
4048:
4046:
4045:
4044:
4018:
4016:
4015:
4010:
3999:
3998:
3997:
3977:
3969:
3968:
3949:
3948:
3947:
3928:
3927:
3926:
3896:
3894:
3893:
3888:
3873:
3871:
3870:
3865:
3863:
3862:
3850:
3849:
3848:
3829:
3828:
3827:
3802:Finally, define
3801:
3799:
3798:
3793:
3788:
3787:
3771:
3769:
3768:
3763:
3761:
3760:
3741:
3737:
3735:
3734:
3729:
3727:
3726:
3725:
3702:
3700:
3699:
3694:
3689:
3688:
3670:
3669:
3654:
3653:
3652:
3627:
3625:
3624:
3619:
3614:
3613:
3597:
3595:
3594:
3589:
3574:
3572:
3571:
3566:
3561:
3560:
3551:
3550:
3540:
3535:
3514:
3513:
3495:
3494:
3469:
3467:
3466:
3461:
3453:
3452:
3427:
3423:
3419:
3417:
3416:
3411:
3406:
3405:
3387:
3386:
3358:
3354:
3350:
3346:
3338:
3332:
3327:
3325:
3324:
3319:
3317:
3316:
3296:
3285:
3281:
3239:
3237:
3236:
3231:
3229:
3222:
3221:
3212:
3211:
3199:
3198:
3189:
3188:
3173:
3166:
3165:
3138:
3137:
3107:
3106:
3091:
3090:
3063:
3062:
3035:
3034:
3019:
3015:
3014:
2993:
2992:
2971:
2970:
2955:
2954:
2933:
2932:
2911:
2910:
2891:
2890:
2881:
2880:
2868:
2867:
2858:
2857:
2831:
2829:
2828:
2823:
2809:
2808:
2787:
2786:
2774:
2773:
2763:
2760:
2748:
2747:
2726:
2725:
2713:
2712:
2690:
2688:
2687:
2682:
2677:
2676:
2669:
2668:
2655:
2654:
2636:
2635:
2570:
2569:
2562:
2561:
2548:
2547:
2520:
2518:
2517:
2512:
2504:
2503:
2491:
2490:
2471:
2469:
2468:
2463:
2461:
2460:
2448:
2447:
2429:
2427:
2426:
2421:
2416:
2415:
2344:
2342:
2341:
2336:
2298:
2297:
2281:
2279:
2278:
2273:
2241:
2240:
2220:
2212:
2210:
2209:
2204:
2181:
2180:
2164:
2162:
2161:
2156:
2136:
2135:
2115:
2113:
2112:
2107:
2102:
2101:
2096:
2072:
2070:
2069:
2064:
2058:
2057:
2048:
2047:
2037:
2032:
2000:
1998:
1997:
1992:
1990:
1983:
1982:
1973:
1969:
1968:
1967:
1958:
1957:
1941:
1936:
1915:
1910:
1889:
1885:
1881:
1880:
1879:
1870:
1869:
1853:
1848:
1833:
1832:
1817:
1812:
1791:
1787:
1786:
1777:
1776:
1766:
1761:
1716:
1714:
1713:
1708:
1703:
1702:
1701:
1678:
1676:
1675:
1670:
1668:
1667:
1666:
1643:
1639:
1637:
1636:
1631:
1629:
1617:
1615:
1614:
1609:
1607:
1592:
1590:
1589:
1584:
1579:
1567:
1549:
1547:
1546:
1541:
1515:
1512:
1509:
1508:
1499:
1498:
1482:
1477:
1459:
1458:
1432:
1428:
1421:
1419:
1418:
1413:
1408:
1407:
1398:
1397:
1387:
1382:
1364:
1363:
1354:
1353:
1343:
1338:
1307:
1305:
1304:
1299:
1294:
1293:
1292:
1269:
1267:
1266:
1261:
1256:
1255:
1237:
1236:
1217:
1215:
1214:
1209:
1204:
1203:
1202:
1179:
1177:
1176:
1171:
1159:
1157:
1156:
1151:
1146:
1145:
1127:
1126:
1107:
1105:
1104:
1099:
1076:
1074:
1073:
1068:
1063:
1062:
1061:
1038:
1036:
1035:
1030:
1028:
1027:
1026:
999:
996:. In this case,
987:
983:
979:
977:
976:
971:
969:
968:
967:
944:
942:
941:
936:
928:
927:
909:
908:
893:
892:
891:
865:
863:
862:
857:
852:
851:
835:
831:
828:while the index
827:
825:
824:
819:
814:
813:
797:
793:
789:
778:
767:
765:
764:
759:
757:
756:
745:
741:
740:
701:
699:
698:
693:
688:
687:
678:
677:
661:
656:
638:
637:
619:
617:
616:
611:
606:
605:
604:
581:
579:
578:
573:
571:
570:
548:
537:
524:
517:
511:be a basis of a
510:
508:
507:
502:
497:
496:
478:
477:
462:
461:
460:
414:
412:
411:
406:
390:
388:
387:
382:
380:
379:
378:
366:
353:
351:
350:
345:
343:
342:
341:
329:
313:
311:
310:
305:
300:
299:
298:
286:
273:
272:
271:
259:
228:
224:
213:
201:
173:
147:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
6341:
6340:
6336:
6335:
6334:
6332:
6331:
6330:
6311:
6310:
6309:
6304:
6286:
6248:
6204:
6141:
6093:
6035:
6026:
5992:Change of basis
5982:Multilinear map
5920:
5902:
5897:
5853:
5829:
5824:
5803:
5798:
5781:
5778:
5773:
5772:
5764:
5760:
5752:
5748:
5740:
5736:
5731:
5726:
5725:
5712:
5708:
5703:
5674:
5630:
5615:
5610:
5609:
5574:
5570:
5562:
5558:
5533:
5532:
5524:
5511:
5480:
5475:
5474:
5440:
5428:
5413:
5393:
5385:
5384:
5373:
5369:
5365:
5361:
5306:
5305:
5301:
5260:
5255:
5254:
5247:
5238:
5209:
5204:
5203:
5162:
5136:
5135:
5128:
5124:
5118:
5112:
5088:
5075:
5064:
5063:
5059:
5055:
5031:
5012:
5004:
5003:
4999:
4993:
4963:
4962:
4922:
4921:
4881:
4880:
4865:
4850:
4846:
4830:
4781:
4776:
4775:
4768:
4764:
4756:
4752:
4746:
4705:
4700:
4699:
4692:
4685:
4681:
4671:
4667:
4663:
4653:
4649:
4639:
4635:
4631:
4627:
4623:
4619:
4615:
4598:
4592:
4566:smooth function
4509:
4496:
4485:
4459:
4386:
4343:
4342:
4311:
4310:
4254:
4220:
4219:
4174:
4173:
4131:
4130:
4109:
4088:
4067:
4062:
4061:
4029:
4024:
4023:
3982:
3932:
3911:
3906:
3905:
3879:
3878:
3854:
3833:
3812:
3807:
3806:
3779:
3774:
3773:
3752:
3747:
3746:
3739:
3710:
3705:
3704:
3680:
3661:
3637:
3632:
3631:
3605:
3600:
3599:
3580:
3579:
3552:
3542:
3505:
3486:
3475:
3474:
3444:
3433:
3432:
3425:
3421:
3397:
3378:
3364:
3363:
3356:
3352:
3348:
3344:
3336:
3330:
3308:
3303:
3302:
3294:
3283:
3277:
3246:
3227:
3226:
3213:
3203:
3190:
3180:
3171:
3170:
3157:
3129:
3098:
3082:
3054:
3026:
3017:
3016:
3006:
2984:
2962:
2946:
2924:
2902:
2892:
2882:
2872:
2859:
2849:
2840:
2839:
2800:
2778:
2765:
2739:
2717:
2704:
2699:
2698:
2671:
2670:
2660:
2657:
2656:
2646:
2639:
2630:
2629:
2618:
2606:
2605:
2591:
2575:
2564:
2563:
2553:
2550:
2549:
2539:
2532:
2526:
2525:
2495:
2482:
2474:
2473:
2452:
2439:
2434:
2433:
2410:
2409:
2398:
2386:
2385:
2371:
2355:
2349:
2348:
2289:
2284:
2283:
2232:
2227:
2226:
2218:
2172:
2167:
2166:
2127:
2122:
2121:
2091:
2086:
2085:
2079:
2049:
2039:
2006:
2005:
1988:
1987:
1974:
1959:
1943:
1921:
1917:
1887:
1886:
1871:
1855:
1824:
1823:
1819:
1789:
1788:
1778:
1768:
1740:
1728:
1727:
1686:
1681:
1680:
1651:
1646:
1645:
1641:
1620:
1619:
1598:
1597:
1558:
1557:
1500:
1484:
1450:
1445:
1444:
1430:
1426:
1399:
1389:
1355:
1345:
1313:
1312:
1277:
1272:
1271:
1247:
1228:
1220:
1219:
1187:
1182:
1181:
1162:
1161:
1137:
1118:
1110:
1109:
1081:
1080:
1079:Given a vector
1046:
1041:
1040:
1011:
1006:
1005:
1004:from the basis
997:
985:
981:
952:
947:
946:
919:
900:
876:
871:
870:
843:
838:
837:
833:
829:
805:
800:
799:
795:
791:
788:
780:
776:
726:
722:
721:
710:
709:
679:
663:
629:
624:
623:
589:
584:
583:
556:
551:
550:
547:
539:
529:
522:
515:
488:
469:
445:
440:
439:
436:
397:
396:
361:
356:
355:
324:
319:
318:
281:
254:
249:
248:
231:change of basis
226:
222:
211:
199:
181:
180:
179:
178:
177:
174:
165:
164:
163:
148:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
6339:
6337:
6329:
6328:
6323:
6321:Linear algebra
6313:
6312:
6306:
6305:
6303:
6302:
6291:
6288:
6287:
6285:
6284:
6279:
6274:
6269:
6264:
6262:Floating-point
6258:
6256:
6250:
6249:
6247:
6246:
6244:Tensor product
6241:
6236:
6231:
6229:Function space
6226:
6221:
6215:
6213:
6206:
6205:
6203:
6202:
6197:
6192:
6187:
6182:
6177:
6172:
6167:
6165:Triple product
6162:
6157:
6151:
6149:
6143:
6142:
6140:
6139:
6134:
6129:
6124:
6119:
6114:
6109:
6103:
6101:
6095:
6094:
6092:
6091:
6086:
6081:
6079:Transformation
6076:
6071:
6069:Multiplication
6066:
6061:
6056:
6051:
6045:
6043:
6037:
6036:
6029:
6027:
6025:
6024:
6019:
6014:
6009:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5969:
5964:
5959:
5954:
5949:
5944:
5939:
5934:
5928:
5926:
5925:Basic concepts
5922:
5921:
5919:
5918:
5913:
5907:
5904:
5903:
5900:Linear algebra
5898:
5896:
5895:
5888:
5881:
5873:
5867:
5866:
5860:
5852:
5851:External links
5849:
5848:
5847:
5827:
5822:
5801:
5796:
5777:
5774:
5771:
5770:
5758:
5746:
5733:
5732:
5730:
5727:
5724:
5723:
5705:
5704:
5702:
5699:
5698:
5697:
5691:
5685:
5680:
5673:
5670:
5666:diagonalizable
5645:
5640:
5637:
5633:
5629:
5623:
5618:
5541:
5521:characteristic
5516:is symmetric.
5510:if the matrix
5499:
5495:
5488:
5483:
5471:
5470:
5459:
5456:
5450:
5444:
5436:
5431:
5427:
5421:
5416:
5412:
5408:
5401:
5396:
5392:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5294:
5293:
5282:
5279:
5275:
5268:
5263:
5237:of the matrix
5219:
5213:
5200:
5199:
5188:
5184:
5179:
5172:
5166:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5100:
5095:
5091:
5087:
5082:
5078:
5074:
5071:
5043:
5038:
5034:
5030:
5027:
5024:
5019:
5015:
5011:
4979:
4976:
4973:
4970:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4849:is a function
4829:
4828:Bilinear forms
4826:
4814:
4813:
4802:
4799:
4796:
4791:
4788:
4784:
4745:
4742:
4738:
4737:
4726:
4723:
4720:
4715:
4712:
4708:
4591:
4588:
4577:
4576:
4569:
4562:
4555:
4548:
4541:
4458:
4455:
4451:
4450:
4439:
4436:
4433:
4430:
4427:
4422:
4419:
4413:
4410:
4407:
4402:
4398:
4393:
4389:
4385:
4382:
4379:
4376:
4371:
4368:
4362:
4359:
4356:
4351:
4327:
4324:
4321:
4318:
4307:
4306:
4295:
4290:
4287:
4281:
4278:
4275:
4270:
4266:
4261:
4257:
4253:
4248:
4245:
4239:
4236:
4233:
4228:
4202:
4199:
4193:
4190:
4187:
4182:
4159:
4156:
4150:
4147:
4144:
4139:
4116:
4112:
4108:
4102:
4099:
4096:
4091:
4087:
4081:
4078:
4075:
4070:
4043:
4040:
4037:
4032:
4020:
4019:
4008:
4005:
4002:
3996:
3993:
3990:
3985:
3981:
3976:
3973:
3967:
3964:
3961:
3956:
3952:
3946:
3943:
3940:
3935:
3931:
3925:
3922:
3919:
3914:
3886:
3875:
3874:
3861:
3857:
3853:
3847:
3844:
3841:
3836:
3832:
3826:
3823:
3820:
3815:
3791:
3786:
3782:
3759:
3755:
3724:
3721:
3718:
3713:
3692:
3687:
3683:
3679:
3676:
3673:
3668:
3664:
3660:
3657:
3651:
3648:
3645:
3640:
3617:
3612:
3608:
3587:
3576:
3575:
3564:
3559:
3555:
3549:
3545:
3539:
3534:
3531:
3528:
3524:
3520:
3517:
3512:
3508:
3504:
3501:
3498:
3493:
3489:
3485:
3482:
3459:
3456:
3451:
3447:
3443:
3440:
3424:-vector space
3409:
3404:
3400:
3396:
3393:
3390:
3385:
3381:
3377:
3374:
3371:
3341:standard basis
3315:
3311:
3245:
3242:
3241:
3240:
3225:
3220:
3216:
3210:
3206:
3202:
3197:
3193:
3187:
3183:
3179:
3176:
3174:
3172:
3169:
3164:
3160:
3156:
3153:
3150:
3147:
3144:
3141:
3136:
3132:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3105:
3101:
3097:
3094:
3089:
3085:
3081:
3078:
3075:
3072:
3069:
3066:
3061:
3057:
3053:
3050:
3047:
3044:
3041:
3038:
3033:
3029:
3025:
3022:
3020:
3018:
3013:
3009:
3005:
3002:
2999:
2996:
2991:
2987:
2983:
2980:
2977:
2974:
2969:
2965:
2961:
2958:
2953:
2949:
2945:
2942:
2939:
2936:
2931:
2927:
2923:
2920:
2917:
2914:
2909:
2905:
2901:
2898:
2895:
2893:
2889:
2885:
2879:
2875:
2871:
2866:
2862:
2856:
2852:
2848:
2847:
2833:
2832:
2821:
2818:
2815:
2812:
2807:
2803:
2799:
2796:
2793:
2790:
2785:
2781:
2777:
2772:
2768:
2757:
2754:
2751:
2746:
2742:
2738:
2735:
2732:
2729:
2724:
2720:
2716:
2711:
2707:
2692:
2691:
2680:
2675:
2667:
2663:
2659:
2658:
2653:
2649:
2645:
2644:
2642:
2634:
2628:
2625:
2622:
2619:
2617:
2614:
2611:
2608:
2607:
2604:
2601:
2598:
2595:
2592:
2590:
2587:
2584:
2581:
2580:
2578:
2573:
2568:
2560:
2556:
2552:
2551:
2546:
2542:
2538:
2537:
2535:
2510:
2507:
2502:
2498:
2494:
2489:
2485:
2481:
2459:
2455:
2451:
2446:
2442:
2419:
2414:
2408:
2405:
2402:
2399:
2397:
2394:
2391:
2388:
2387:
2384:
2381:
2378:
2375:
2372:
2370:
2367:
2364:
2361:
2360:
2358:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2296:
2292:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2239:
2235:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2179:
2175:
2154:
2151:
2148:
2145:
2142:
2139:
2134:
2130:
2118:standard basis
2105:
2100:
2095:
2078:
2075:
2061:
2056:
2052:
2046:
2042:
2036:
2031:
2028:
2025:
2021:
2016:
2013:
2002:
2001:
1986:
1981:
1977:
1972:
1966:
1962:
1956:
1953:
1950:
1946:
1940:
1935:
1932:
1929:
1925:
1920:
1914:
1909:
1906:
1903:
1899:
1895:
1892:
1890:
1888:
1884:
1878:
1874:
1868:
1865:
1862:
1858:
1852:
1847:
1844:
1841:
1837:
1831:
1827:
1822:
1816:
1811:
1808:
1805:
1801:
1797:
1794:
1792:
1790:
1785:
1781:
1775:
1771:
1765:
1760:
1757:
1754:
1750:
1746:
1743:
1741:
1739:
1736:
1735:
1717:respectively.
1706:
1700:
1697:
1694:
1689:
1665:
1662:
1659:
1654:
1628:
1606:
1594:
1593:
1582:
1578:
1573:
1570:
1566:
1551:
1550:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1507:
1503:
1497:
1494:
1491:
1487:
1481:
1476:
1473:
1470:
1466:
1462:
1457:
1453:
1423:
1422:
1411:
1406:
1402:
1396:
1392:
1386:
1381:
1378:
1375:
1371:
1367:
1362:
1358:
1352:
1348:
1342:
1337:
1334:
1331:
1327:
1323:
1320:
1297:
1291:
1288:
1285:
1280:
1259:
1254:
1250:
1246:
1243:
1240:
1235:
1231:
1227:
1207:
1201:
1198:
1195:
1190:
1169:
1149:
1144:
1140:
1136:
1133:
1130:
1125:
1121:
1117:
1097:
1094:
1091:
1088:
1066:
1060:
1057:
1054:
1049:
1025:
1022:
1019:
1014:
980:is a basis of
966:
963:
960:
955:
934:
931:
926:
922:
918:
915:
912:
907:
903:
899:
896:
890:
887:
884:
879:
855:
850:
846:
817:
812:
808:
784:
769:
768:
755:
752:
749:
744:
739:
736:
733:
729:
725:
720:
717:
703:
702:
691:
686:
682:
676:
673:
670:
666:
660:
655:
652:
649:
645:
641:
636:
632:
609:
603:
600:
597:
592:
569:
566:
563:
559:
543:
500:
495:
491:
487:
484:
481:
476:
472:
468:
465:
459:
456:
453:
448:
435:
432:
404:
393:column vectors
377:
374:
371:
365:
340:
337:
334:
328:
315:
314:
303:
297:
294:
291:
285:
279:
276:
270:
267:
264:
258:
175:
168:
167:
166:
149:
142:
141:
140:
139:
138:
135:
134:
49:
47:
40:
32:Change of base
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6338:
6327:
6326:Matrix theory
6324:
6322:
6319:
6318:
6316:
6301:
6293:
6292:
6289:
6283:
6280:
6278:
6277:Sparse matrix
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6259:
6257:
6255:
6251:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6220:
6217:
6216:
6214:
6212:constructions
6211:
6207:
6201:
6200:Outermorphism
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6178:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6160:Cross product
6158:
6156:
6153:
6152:
6150:
6148:
6144:
6138:
6135:
6133:
6130:
6128:
6127:Outer product
6125:
6123:
6120:
6118:
6115:
6113:
6110:
6108:
6107:Orthogonality
6105:
6104:
6102:
6100:
6096:
6090:
6087:
6085:
6084:Cramer's rule
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6055:
6054:Decomposition
6052:
6050:
6047:
6046:
6044:
6042:
6038:
6033:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5973:
5970:
5968:
5965:
5963:
5960:
5958:
5955:
5953:
5950:
5948:
5945:
5943:
5940:
5938:
5935:
5933:
5930:
5929:
5927:
5923:
5917:
5914:
5912:
5909:
5908:
5905:
5901:
5894:
5889:
5887:
5882:
5880:
5875:
5874:
5871:
5864:
5861:
5858:
5855:
5854:
5850:
5845:
5841:
5837:
5833:
5828:
5825:
5823:0-395-14017-X
5819:
5815:
5810:
5809:
5802:
5799:
5797:0-471-84819-0
5793:
5789:
5785:
5780:
5779:
5775:
5767:
5762:
5759:
5755:
5750:
5747:
5743:
5738:
5735:
5728:
5720:
5719:ordered basis
5716:
5710:
5707:
5700:
5695:
5692:
5689:
5686:
5684:
5681:
5679:
5676:
5675:
5671:
5669:
5667:
5663:
5659:
5643:
5638:
5635:
5631:
5627:
5616:
5607:
5603:
5599:
5595:
5591:
5587:
5583:
5578:
5568:
5556:
5530:
5522:
5517:
5514:
5497:
5481:
5457:
5454:
5429:
5425:
5410:
5394:
5383:
5382:
5381:
5379:
5344:
5341:
5338:
5332:
5329:
5323:
5320:
5317:
5311:
5299:
5280:
5277:
5261:
5253:
5252:
5251:
5244:
5241:
5236:
5186:
5159:
5153:
5150:
5147:
5141:
5134:
5133:
5132:
5121:
5115:
5093:
5089:
5085:
5080:
5076:
5069:
5062:th column is
5036:
5032:
5028:
5025:
5022:
5017:
5013:
4996:
4990:
4977:
4974:
4971:
4968:
4945:
4942:
4939:
4933:
4927:
4904:
4901:
4898:
4892:
4886:
4876:
4872:
4868:
4863:
4857:
4853:
4845:
4841:
4837:
4836:
4835:bilinear form
4827:
4825:
4823:
4819:
4800:
4797:
4794:
4789:
4786:
4782:
4774:
4773:
4772:
4762:
4761:square matrix
4750:
4749:Endomorphisms
4744:Endomorphisms
4743:
4741:
4724:
4721:
4718:
4713:
4710:
4706:
4698:
4697:
4696:
4689:
4678:
4674:
4660:
4656:
4646:
4642:
4626:of dimension
4618:of dimension
4614:
4609:
4605:
4601:
4597:
4589:
4587:
4585:
4580:
4574:
4570:
4567:
4563:
4560:
4556:
4553:
4549:
4546:
4542:
4539:
4538:
4537:
4535:
4530:
4528:
4523:
4519:
4516:
4512:
4506:
4503:
4499:
4492:
4488:
4483:
4478:
4476:
4472:
4468:
4464:
4456:
4454:
4437:
4428:
4420:
4417:
4400:
4391:
4387:
4383:
4377:
4369:
4366:
4349:
4341:
4340:
4339:
4325:
4322:
4319:
4316:
4293:
4288:
4285:
4268:
4264:
4259:
4255:
4251:
4246:
4243:
4226:
4218:
4217:
4216:
4200:
4197:
4180:
4157:
4154:
4137:
4114:
4110:
4106:
4089:
4085:
4068:
4058:
4030:
4006:
3983:
3974:
3971:
3954:
3933:
3929:
3912:
3904:
3903:
3902:
3900:
3884:
3859:
3855:
3851:
3834:
3830:
3813:
3805:
3804:
3803:
3789:
3784:
3780:
3757:
3753:
3745:
3711:
3685:
3681:
3677:
3674:
3671:
3666:
3662:
3655:
3638:
3628:
3615:
3610:
3606:
3585:
3562:
3557:
3553:
3547:
3543:
3537:
3532:
3529:
3526:
3522:
3518:
3510:
3506:
3502:
3499:
3496:
3491:
3487:
3480:
3473:
3472:
3471:
3457:
3449:
3445:
3441:
3438:
3431:
3402:
3398:
3394:
3391:
3388:
3383:
3379:
3372:
3369:
3360:
3342:
3334:
3313:
3309:
3300:
3291:
3289:
3280:
3275:
3271:
3267:
3262:
3259:
3255:
3252:represents a
3251:
3243:
3223:
3218:
3214:
3208:
3204:
3200:
3195:
3191:
3185:
3181:
3177:
3175:
3162:
3158:
3151:
3145:
3142:
3139:
3134:
3130:
3123:
3117:
3114:
3111:
3103:
3099:
3095:
3087:
3083:
3076:
3070:
3067:
3064:
3059:
3055:
3048:
3042:
3039:
3031:
3027:
3023:
3021:
3011:
3007:
3000:
2997:
2994:
2989:
2985:
2981:
2978:
2975:
2972:
2967:
2963:
2956:
2951:
2947:
2940:
2937:
2934:
2929:
2925:
2921:
2918:
2915:
2912:
2907:
2903:
2896:
2894:
2887:
2883:
2877:
2873:
2869:
2864:
2860:
2854:
2850:
2838:
2837:
2836:
2819:
2816:
2813:
2810:
2805:
2801:
2797:
2794:
2791:
2788:
2783:
2779:
2775:
2770:
2766:
2755:
2752:
2749:
2744:
2740:
2736:
2733:
2730:
2727:
2722:
2718:
2714:
2709:
2705:
2697:
2696:
2695:
2678:
2673:
2665:
2661:
2651:
2647:
2640:
2632:
2626:
2623:
2620:
2615:
2612:
2609:
2602:
2599:
2596:
2593:
2588:
2585:
2582:
2576:
2571:
2566:
2558:
2554:
2544:
2540:
2533:
2524:
2523:
2522:
2521:then one has
2508:
2500:
2496:
2492:
2487:
2483:
2457:
2453:
2449:
2444:
2440:
2430:
2417:
2412:
2406:
2403:
2400:
2395:
2392:
2389:
2382:
2379:
2376:
2373:
2368:
2365:
2362:
2356:
2345:
2332:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2299:
2294:
2290:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2242:
2237:
2233:
2224:
2221:, one gets a
2216:
2200:
2194:
2191:
2188:
2182:
2177:
2173:
2149:
2146:
2143:
2137:
2132:
2128:
2119:
2103:
2098:
2084:
2081:Consider the
2076:
2074:
2059:
2054:
2050:
2044:
2040:
2034:
2029:
2026:
2023:
2019:
2014:
2011:
1984:
1979:
1975:
1970:
1964:
1960:
1954:
1951:
1948:
1944:
1938:
1933:
1930:
1927:
1923:
1918:
1912:
1907:
1904:
1901:
1897:
1893:
1891:
1882:
1876:
1872:
1866:
1863:
1860:
1856:
1850:
1845:
1842:
1839:
1835:
1829:
1825:
1820:
1814:
1809:
1806:
1803:
1799:
1795:
1793:
1783:
1779:
1773:
1769:
1763:
1758:
1755:
1752:
1748:
1744:
1742:
1737:
1726:
1725:
1724:
1722:
1718:
1704:
1687:
1652:
1580:
1571:
1568:
1556:
1555:
1554:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1505:
1501:
1495:
1492:
1489:
1485:
1479:
1474:
1471:
1468:
1464:
1460:
1455:
1451:
1443:
1442:
1441:
1439:
1434:
1409:
1404:
1400:
1394:
1390:
1384:
1379:
1376:
1373:
1369:
1365:
1360:
1356:
1350:
1346:
1340:
1335:
1332:
1329:
1325:
1321:
1318:
1311:
1310:
1309:
1295:
1278:
1252:
1248:
1244:
1241:
1238:
1233:
1229:
1205:
1188:
1167:
1142:
1138:
1134:
1131:
1128:
1123:
1119:
1095:
1092:
1089:
1086:
1077:
1064:
1047:
1039:to the basis
1012:
1003:
995:
991:
953:
945:one has that
932:
924:
920:
916:
913:
910:
905:
901:
894:
877:
867:
853:
848:
844:
815:
810:
806:
787:
783:
774:
753:
750:
747:
742:
737:
734:
731:
727:
723:
718:
715:
708:
707:
706:
689:
684:
680:
674:
671:
668:
664:
658:
653:
650:
647:
643:
639:
634:
630:
622:
621:
620:
607:
590:
567:
564:
561:
557:
546:
542:
536:
532:
526:
521:
514:
493:
489:
485:
482:
479:
474:
470:
463:
446:
433:
431:
428:
426:
425:basis vectors
422:
419:(also called
418:
402:
394:
301:
277:
274:
247:
246:
245:
243:
239:
234:
232:
220:
216:
209:
206:, which is a
205:
198:
194:
190:
189:ordered basis
186:
172:
161:
157:
153:
146:
131:
128:
120:
117:November 2017
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
6210:Vector space
5991:
5942:Vector space
5831:
5807:
5783:
5776:Bibliography
5766:Nering (1970
5761:
5749:
5737:
5709:
5579:
5555:real numbers
5518:
5512:
5472:
5295:
5245:
5239:
5233:denotes the
5201:
5119:
5113:
4994:
4991:
4874:
4870:
4866:
4855:
4851:
4839:
4833:
4831:
4815:
4747:
4739:
4690:
4676:
4672:
4658:
4654:
4644:
4640:
4613:vector space
4607:
4603:
4599:
4593:
4581:
4578:
4531:
4524:
4517:
4514:
4510:
4504:
4501:
4497:
4490:
4486:
4479:
4460:
4452:
4308:
4059:
4021:
3876:
3744:endomorphism
3629:
3577:
3361:
3292:
3278:
3265:
3263:
3248:Normally, a
3247:
2834:
2693:
2431:
2346:
2222:
2080:
2003:
1720:
1719:
1595:
1552:
1437:
1435:
1424:
1078:
1001:
868:
785:
781:
770:
704:
544:
540:
534:
530:
527:
437:
429:
420:
416:
316:
237:
235:
230:
193:vector space
182:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
6190:Multivector
6155:Determinant
6112:Dot product
5957:Linear span
5742:Anton (1987
5662:eigenvalues
5592:and of the
5058:th row and
5002:on a basis
4992:The matrix
4594:Consider a
4590:Linear maps
3355:th that is
3351:except the
3274:isomorphism
994:determinant
219:coordinates
185:mathematics
6315:Categories
6224:Direct sum
6059:Invertible
5962:Linear map
5812:, Boston:
5729:References
5606:orthogonal
5598:rigid body
5360:for every
5304:such that
5131:, one has
4596:linear map
4482:expression
3428:defines a
3301:, the set
3266:represents
3254:linear map
2694:That is,
2225:formed by
990:invertible
533:= 1, ...,
195:of finite
87:newspapers
6254:Numerical
6017:Transpose
5636:−
5378:symmetric
5235:transpose
5026:…
4972:∈
4931:↦
4890:↦
4860:which is
4816:As every
4787:−
4711:−
4670:, and an
4584:manifolds
4418:−
4401:ϕ
4388:ψ
4367:−
4350:ϕ
4338:one has
4320:∈
4286:−
4269:ϕ
4265:∘
4256:ψ
4244:−
4227:ϕ
4198:−
4181:ϕ
4155:−
4138:ϕ
4111:ψ
4107:∘
4090:ϕ
4069:ϕ
3972:−
3955:ϕ
3934:ϕ
3885:∘
3856:ψ
3852:∘
3835:ϕ
3814:ϕ
3754:ψ
3712:ϕ
3675:…
3586:ϕ
3523:∑
3500:…
3481:ϕ
3455:→
3442::
3439:ϕ
3392:…
3146:
3118:
3112:−
3071:
3043:
2998:
2976:
2938:
2922:−
2916:
2814:
2792:
2753:
2737:−
2731:
2624:
2613:
2600:
2594:−
2586:
2404:
2393:
2380:
2374:−
2366:
2324:
2312:
2306:−
2264:
2252:
2223:new basis
2020:∑
1924:∑
1898:∑
1836:∑
1800:∑
1749:∑
1529:…
1513:for
1465:∑
1370:∑
1326:∑
1308:that is
1242:…
1132:…
1090:∈
914:…
644:∑
608::
483:…
197:dimension
6300:Category
6239:Subspace
6234:Quotient
6185:Bivector
6099:Bilinear
6041:Matrices
5916:Glossary
5844:76091646
5672:See also
5590:quadrics
5582:geometry
5529:diagonal
4869: :
4463:function
3897:denotes
3362:A basis
3282:, where
869:Setting
836:and the
798:and the
391:are the
242:matrices
208:sequence
5911:Outline
5594:inertia
5586:physics
5573:and of
5553:of the
5519:If the
4842:over a
4822:similar
4759:is the
4648:matrix
4611:from a
4475:applied
3901:), and
3877:(where
3333:-tuples
3328:of the
2215:rotates
2213:If one
2077:Example
771:be the
518:over a
415:is the
217:called
215:scalars
101:scholar
6195:Tensor
6007:Kernel
5937:Vector
5932:Scalar
5842:
5820:
5794:
5202:where
4862:linear
4467:domain
3250:matrix
1721:Proof:
1596:where
775:whose
773:matrix
103:
96:
89:
82:
74:
6064:Minor
6049:Block
5987:Basis
5836:Wiley
5788:Wiley
5715:tuple
5701:Notes
5596:of a
4844:field
4638:by a
4129:with
3420:of a
3335:is a
3299:field
3297:be a
3288:up to
3270:bases
1644:over
1180:over
582:over
520:field
191:of a
187:, an
108:JSTOR
94:books
6219:Dual
6074:Rank
5840:LCCN
5818:ISBN
5792:ISBN
5584:and
5364:and
5127:and
5117:and
4920:and
4695:is
4684:for
4666:for
4634:and
3630:Let
3470:by
3293:Let
2282:and
2165:and
2116:Its
1679:and
1618:and
1436:The
1218:and
1108:let
705:Let
528:For
438:Let
354:and
80:news
5561:or
5368:in
5246:If
4877:→ F
4858:→ F
4571:an
3772:of
3143:cos
3115:sin
3068:sin
3040:cos
2995:cos
2973:sin
2935:sin
2913:cos
2811:cos
2789:sin
2761:and
2750:sin
2728:cos
2621:cos
2610:sin
2597:sin
2583:cos
2401:cos
2390:sin
2377:sin
2363:cos
2321:cos
2309:sin
2261:sin
2249:cos
2004:As
988:is
210:of
183:In
63:by
6317::
5838:,
5816:,
5790:,
5668:.
5575:–1
5565:.
5563:–1
5296:A
5243:.
4873:×
4854:×
4832:A
4688:.
4606:→
4602::
4564:a
4557:a
4550:a
4543:a
4500:=
4477:.
4461:A
3359:.
525:.
150:A
5892:e
5885:t
5878:v
5721:.
5644:.
5639:1
5632:P
5628:=
5622:T
5617:P
5571:1
5559:1
5540:R
5525:F
5513:B
5498:P
5494:B
5487:T
5482:P
5458:,
5455:P
5449:T
5443:B
5435:T
5430:P
5426:=
5420:T
5415:)
5411:P
5407:B
5400:T
5395:P
5391:(
5374:B
5370:V
5366:w
5362:v
5348:)
5345:v
5342:,
5339:w
5336:(
5333:B
5330:=
5327:)
5324:w
5321:,
5318:v
5315:(
5312:B
5302:B
5281:.
5278:P
5274:B
5267:T
5262:P
5248:P
5240:v
5218:T
5212:v
5187:,
5183:w
5178:B
5171:T
5165:v
5160:=
5157:)
5154:w
5151:,
5148:v
5145:(
5142:B
5129:w
5125:v
5120:w
5114:v
5099:)
5094:j
5090:v
5086:,
5081:i
5077:v
5073:(
5070:B
5060:j
5056:i
5042:)
5037:n
5033:v
5029:,
5023:,
5018:1
5014:v
5010:(
5000:B
4995:B
4978:.
4975:V
4969:w
4949:)
4946:v
4943:,
4940:w
4937:(
4934:B
4928:v
4908:)
4905:w
4902:,
4899:v
4896:(
4893:B
4887:v
4875:V
4871:V
4867:B
4856:V
4852:V
4847:F
4840:V
4801:.
4798:P
4795:M
4790:1
4783:P
4769:P
4765:V
4757:M
4753:V
4725:.
4722:Q
4719:M
4714:1
4707:P
4693:T
4686:W
4682:Q
4677:n
4675:×
4673:n
4668:V
4664:P
4659:m
4657:×
4655:m
4650:M
4645:n
4643:×
4641:m
4636:W
4632:V
4628:m
4624:V
4620:n
4616:W
4608:V
4604:W
4600:T
4575:,
4568:,
4561:,
4554:,
4547:,
4520:)
4518:y
4515:A
4513:(
4511:f
4505:y
4502:A
4498:x
4493:)
4491:x
4489:(
4487:f
4438:,
4435:)
4432:)
4429:v
4426:(
4421:1
4412:w
4409:e
4406:n
4397:(
4392:A
4384:=
4381:)
4378:v
4375:(
4370:1
4361:d
4358:l
4355:o
4326:,
4323:V
4317:v
4294:.
4289:1
4280:w
4277:e
4274:n
4260:A
4252:=
4247:1
4238:d
4235:l
4232:o
4201:1
4192:w
4189:e
4186:n
4158:1
4149:d
4146:l
4143:o
4115:A
4101:d
4098:l
4095:o
4086:=
4080:w
4077:e
4074:n
4042:w
4039:e
4036:n
4031:B
4007:.
4004:)
4001:)
3995:d
3992:l
3989:o
3984:B
3980:(
3975:1
3966:d
3963:l
3960:o
3951:(
3945:w
3942:e
3939:n
3930:=
3924:w
3921:e
3918:n
3913:B
3860:A
3846:d
3843:l
3840:o
3831:=
3825:w
3822:e
3819:n
3790:.
3785:n
3781:F
3758:A
3740:A
3723:d
3720:l
3717:o
3691:)
3686:n
3682:v
3678:,
3672:,
3667:1
3663:v
3659:(
3656:=
3650:d
3647:l
3644:o
3639:B
3616:.
3611:n
3607:F
3563:.
3558:i
3554:v
3548:i
3544:x
3538:n
3533:1
3530:=
3527:i
3519:=
3516:)
3511:n
3507:x
3503:,
3497:,
3492:1
3488:x
3484:(
3458:V
3450:n
3446:F
3426:V
3422:F
3408:)
3403:n
3399:v
3395:,
3389:,
3384:1
3380:v
3376:(
3373:=
3370:B
3357:1
3353:i
3349:0
3345:i
3337:F
3331:n
3314:n
3310:F
3295:F
3284:F
3279:F
3224:.
3219:2
3215:w
3209:2
3205:y
3201:+
3196:1
3192:w
3186:1
3182:y
3178:=
3168:)
3163:2
3159:v
3155:)
3152:t
3149:(
3140:+
3135:1
3131:v
3127:)
3124:t
3121:(
3109:(
3104:2
3100:y
3096:+
3093:)
3088:2
3084:v
3080:)
3077:t
3074:(
3065:+
3060:1
3056:v
3052:)
3049:t
3046:(
3037:(
3032:1
3028:y
3024:=
3012:2
3008:v
3004:)
3001:t
2990:2
2986:y
2982:+
2979:t
2968:1
2964:y
2960:(
2957:+
2952:1
2948:v
2944:)
2941:t
2930:2
2926:y
2919:t
2908:1
2904:y
2900:(
2897:=
2888:2
2884:v
2878:2
2874:x
2870:+
2865:1
2861:v
2855:1
2851:x
2820:.
2817:t
2806:2
2802:y
2798:+
2795:t
2784:1
2780:y
2776:=
2771:2
2767:x
2756:t
2745:2
2741:y
2734:t
2723:1
2719:y
2715:=
2710:1
2706:x
2679:.
2674:]
2666:2
2662:y
2652:1
2648:y
2641:[
2633:]
2627:t
2616:t
2603:t
2589:t
2577:[
2572:=
2567:]
2559:2
2555:x
2545:1
2541:x
2534:[
2509:,
2506:)
2501:2
2497:x
2493:,
2488:1
2484:x
2480:(
2458:2
2454:y
2450:,
2445:1
2441:y
2418:.
2413:]
2407:t
2396:t
2383:t
2369:t
2357:[
2333:.
2330:)
2327:t
2318:,
2315:t
2303:(
2300:=
2295:2
2291:w
2270:)
2267:t
2258:,
2255:t
2246:(
2243:=
2238:1
2234:w
2219:t
2201:.
2198:)
2195:1
2192:,
2189:0
2186:(
2183:=
2178:2
2174:v
2153:)
2150:0
2147:,
2144:1
2141:(
2138:=
2133:1
2129:v
2104:.
2099:2
2094:R
2060:,
2055:i
2051:v
2045:i
2041:x
2035:n
2030:1
2027:=
2024:i
2015:=
2012:z
1985:.
1980:i
1976:v
1971:)
1965:j
1961:y
1955:j
1952:,
1949:i
1945:a
1939:n
1934:1
1931:=
1928:j
1919:(
1913:n
1908:1
1905:=
1902:i
1894:=
1883:)
1877:i
1873:v
1867:j
1864:,
1861:i
1857:a
1851:n
1846:1
1843:=
1840:i
1830:j
1826:y
1821:(
1815:n
1810:1
1807:=
1804:j
1796:=
1784:j
1780:w
1774:j
1770:y
1764:n
1759:1
1756:=
1753:j
1745:=
1738:z
1705:,
1699:w
1696:e
1693:n
1688:B
1664:d
1661:l
1658:o
1653:B
1642:z
1627:y
1605:x
1581:,
1577:y
1572:A
1569:=
1565:x
1538:.
1535:n
1532:,
1526:,
1523:1
1520:=
1517:i
1506:j
1502:y
1496:j
1493:,
1490:i
1486:a
1480:n
1475:1
1472:=
1469:j
1461:=
1456:i
1452:x
1431:j
1427:i
1410:.
1405:j
1401:w
1395:j
1391:y
1385:n
1380:1
1377:=
1374:j
1366:=
1361:i
1357:v
1351:i
1347:x
1341:n
1336:1
1333:=
1330:i
1322:=
1319:z
1296:;
1290:w
1287:e
1284:n
1279:B
1258:)
1253:n
1249:y
1245:,
1239:,
1234:1
1230:y
1226:(
1206:,
1200:d
1197:l
1194:o
1189:B
1168:z
1148:)
1143:n
1139:x
1135:,
1129:,
1124:1
1120:x
1116:(
1096:,
1093:V
1087:z
1065:.
1059:w
1056:e
1053:n
1048:B
1024:d
1021:l
1018:o
1013:B
998:A
986:A
982:V
965:w
962:e
959:n
954:B
933:,
930:)
925:n
921:w
917:,
911:,
906:1
902:w
898:(
895:=
889:w
886:e
883:n
878:B
854:;
849:j
845:w
834:A
830:j
816:,
811:i
807:v
796:A
792:i
786:j
782:w
777:j
754:j
751:,
748:i
743:)
738:j
735:,
732:i
728:a
724:(
719:=
716:A
690:.
685:i
681:v
675:j
672:,
669:i
665:a
659:n
654:1
651:=
648:i
640:=
635:j
631:w
602:d
599:l
596:o
591:B
568:j
565:,
562:i
558:a
545:j
541:w
535:n
531:j
523:F
516:V
499:)
494:n
490:v
486:,
480:,
475:1
471:v
467:(
464:=
458:d
455:l
452:o
447:B
403:A
376:w
373:e
370:n
364:x
339:d
336:l
333:o
327:x
302:,
296:w
293:e
290:n
284:x
278:A
275:=
269:d
266:l
263:o
257:x
227:v
223:v
212:n
200:n
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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