6021:
160:
134:
6285:
32:
3227:
1988:
2678:
2830:
1718:
5369:. 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,
4525:, 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
4006:
5457:
2819:
2516:
3222:{\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}}}
4437:
3249:
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.
4293:
2417:
1537:
301:
4116:
3861:
5186:
1983:{\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}}}
1409:
3562:
932:
3690:
498:
2060:
5649:
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
2835:
1723:
5645:
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
689:
5280:
1580:
378:
341:
5497:
755:
4202:
4159:
5220:
607:
5643:
3407:
2332:
3725:
3457:
5041:
2269:
1257:
1147:
1704:
1295:
1205:
1064:
4044:
3896:
1666:
1026:
967:
2103:
222:
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.
5375:
4948:
4907:
5098:
2508:
5347:
5540:
2689:
2673:{\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}}.}
2200:
1627:
1605:
4473:
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
2459:
2152:
4800:
4724:
3759:
5845:
4977:
4325:
1095:
569:
3884:
3789:
3615:
853:
815:
3585:
3315:
1167:
402:
4744:
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
4333:
4210:
2339:
1435:
419:
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
239:
4575:, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold.
4052:
3797:
5520:. 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
4514:
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
5126:
3275:
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
1303:
3465:
5879:
5671:
5653:
of the initial matrix on the diagonal. It follows that, over the reals, if the matrix of an endomorphism is symmetric, then it is
6212:
6270:
5239:
is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is
861:
3622:
430:
96:
5702:
Although a basis is generally defined as a set of vectors (for example, as a spanning set that is linearly independent), the
1996:
68:
1422:
for the new one makes clearer the formulas that follows, and helps avoiding errors in proofs and explicit computations.)
1429:
expresses the coordinates over the old basis in terms of the coordinates over the new basis. With above notation, it is
614:
75:
5810:
5784:
5666:
115:
3279:
an isomorphism. As several bases of the same vector space are considered here, a more accurate wording is required.
6260:
5245:
1548:
346:
309:
49:
5465:
700:
6222:
6158:
4164:
4121:
501:
229:
which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using
82:
20:
5194:
574:
53:
5593:
are specially useful; this means that one generally prefer to restrict changes of basis to those that have an
5600:
5555:
4568:
if the multivariate function that represents it on some basis—and thus on every basis—has the same property.
3354:
2274:
3695:
3423:
64:
6000:
5872:
4994:
4001:{\displaystyle B_{\mathrm {new} }=\phi _{\mathrm {new} }(\phi _{\mathrm {old} }^{-1}(B_{\mathrm {old} })).}
2217:
1210:
1100:
6105:
5955:
1671:
1262:
1172:
1031:
5452:{\displaystyle (P^{\mathsf {T}}\mathbf {B} P)^{\mathsf {T}}=P^{\mathsf {T}}\mathbf {B} ^{\mathsf {T}}P,}
4014:
1636:
996:
937:
6010:
5904:
2076:
2062:
the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis.
6250:
5899:
4912:
4871:
2814:{\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.}
5054:
2464:
147:, these form a new basis. The linear combinations relating the first basis to the other extend to a
6242:
6125:
5802:
5682:
5509:
5296:
4470:
1414:(One could take the same summation index for the two sums, but choosing systematically the indexes
185:
5706:
notation is convenient here, since the indexing by the first positive integers makes the basis an
5523:
2157:
1610:
1588:
6309:
6288:
6217:
5995:
5865:
5286:
4547:
2424:
2112:
42:
6314:
6052:
5985:
5975:
4766:
4690:
4451:
4442:
which is the change-of-basis formula expressed in terms of linear maps instead of coordinates.
3737:
3258:
2203:
2071:
6067:
6062:
6057:
5990:
5935:
5654:
4953:
4459:
4301:
1071:
541:
306:
where "old" and "new" refer respectively to the initially defined basis and the other basis,
148:
89:
5516:
is not two, then for every symmetric bilinear form there is a basis for which the matrix is
3869:
3764:
3590:
828:
790:
6077:
6042:
6029:
5920:
5824:
5776:
4463:
4455:
3887:
3570:
3293:
3246:
3238:
761:
230:
203:
144:
8:
6255:
6135:
6110:
5960:
5797:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
4832:
4540:
4533:
4462:
whose variables are the coordinates on some basis of the vector on which the function is
4432:{\displaystyle \phi _{\mathrm {old} }^{-1}(v)=\psi _{A}(\phi _{\mathrm {new} }^{-1}(v)),}
3287:
508:
210:. If two different bases are considered, the coordinate vector that represents a vector
5965:
5795:
5676:
4760:
is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is
4288:{\displaystyle \phi _{\mathrm {old} }^{-1}=\psi _{A}\circ \phi _{\mathrm {new} }^{-1}.}
3418:
3328:-vector space whose addition and scalar multiplication are defined component-wise. Its
3262:
1152:
387:
140:
6163:
6120:
6047:
5940:
5828:
5806:
5780:
5594:
5590:
4806:
4561:
978:
192:
2412:{\displaystyle {\begin{bmatrix}\cos t&-\sin t\\\sin t&\cos t\end{bmatrix}}.}
6168:
6072:
5925:
5646:
5366:
4810:
4515:
1532:{\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad {\text{for }}i=1,\ldots ,n.}
214:
on one basis is, in general, different from the coordinate vector that represents
6227:
6020:
5970:
5517:
4554:
4522:
196:
3261:
of implied vector spaces, and to the fact that the choice of a basis induces an
6232:
6153:
5888:
3329:
2106:
296:{\displaystyle \mathbf {x} _{\mathrm {old} }=A\,\mathbf {x} _{\mathrm {new} },}
5851:
6303:
6265:
6188:
6148:
6115:
6095:
5707:
4823:
4749:
3567:
Conversely, such a linear isomorphism defines a basis, which is the image by
381:
177:
4813:
if and only if they represent the same endomorphism on two different bases.
4809:
can be used as a change-of-basis matrix, this implies that two matrices are
4111:{\displaystyle \phi _{\mathrm {new} }=\phi _{\mathrm {old} }\circ \psi _{A}}
3856:{\displaystyle \phi _{\mathrm {new} }=\phi _{\mathrm {old} }\circ \psi _{A}}
6198:
6087:
6037:
5930:
4737:
4601:
3732:
855:
such a convention is useful for avoiding errors in explicit computations.)
413:
181:
159:
5181:{\displaystyle B(v,w)=\mathbf {v} ^{\mathsf {T}}\mathbf {B} \mathbf {w} ,}
4484:
is the expression of the function in terms of the old coordinates, and if
133:
6178:
6143:
6100:
5945:
5650:
5543:
982:
207:
173:
4511:
is the expression of the same function in terms of the new coordinates.
143:
of one basis of vectors (purple) obtains new vectors (red). If they are
6207:
5950:
5586:
4850:
4584:
3242:
4729:
This is a straightforward consequence of the change-of-basis formula.
1404:{\displaystyle z=\sum _{i=1}^{n}x_{i}v_{i}=\sum _{j=1}^{n}y_{j}w_{j}.}
6005:
5223:
3557:{\displaystyle \phi (x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}v_{i}.}
412:), which is the matrix whose columns are the coordinates of the new
165:
A vector represented by two different bases (purple and red arrows).
31:
6173:
5570:
5566:
depends only on the bilinear form, and not of the change of basis.
5043:(the "old" basis in what follows) is the matrix whose entry of the
4572:
1712:
Using the above definition of the change-of basis matrix, one has
191:
allows representing uniquely any element of the vector space by a
5857:
5582:
5578:
5574:
5569:
Symmetric bilinear forms over the reals are often encountered in
3245:, and the product of a matrix and a column vector represents the
5832:
6183:
5703:
4011:
A straightforward verification shows that this definition of
3318:
3276:
384:
of the coordinates of the same vector on the two bases, and
3727:
the associated isomorphism. Given a change-of basis matrix
5112:
are the column vectors of the coordinates of two vectors
927:{\displaystyle B_{\mathrm {new} }=(w_{1},\ldots ,w_{n}),}
3685:{\displaystyle B_{\mathrm {old} }=(v_{1},\ldots ,v_{n})}
493:{\displaystyle B_{\mathrm {old} }=(v_{1},\ldots ,v_{n})}
2055:{\displaystyle z=\textstyle \sum _{i=1}^{n}x_{i}v_{i},}
2632:
2568:
2525:
2348:
2006:
5603:
5526:
5468:
5378:
5299:
5248:
5197:
5129:
5057:
4997:
4956:
4915:
4874:
4769:
4693:
4445:
4336:
4304:
4213:
4167:
4124:
4055:
4017:
3899:
3872:
3800:
3767:
3740:
3698:
3625:
3593:
3573:
3468:
3426:
3357:
3296:
2833:
2692:
2519:
2467:
2427:
2342:
2277:
2220:
2160:
2115:
2079:
1999:
1721:
1674:
1639:
1613:
1591:
1551:
1542:
In terms of matrices, the change of basis formula is
1438:
1306:
1265:
1213:
1175:
1155:
1103:
1074:
1034:
999:
940:
864:
831:
793:
703:
617:
577:
544:
433:
390:
349:
312:
242:
5685: — application in computational chemistry
5597:change-of-base matrix, that is, a matrix such that
5546:, these nonzero entries can be chosen to be either
56:. Unsourced material may be challenged and removed.
5794:
5793:Beauregard, Raymond A.; Fraleigh, John B. (1973),
5792:
5742:
5637:
5534:
5491:
5451:
5341:
5274:
5214:
5180:
5092:
5035:
4971:
4942:
4901:
4794:
4718:
4431:
4319:
4287:
4196:
4153:
4110:
4038:
4000:
3878:
3855:
3783:
3753:
3719:
3684:
3609:
3579:
3556:
3451:
3401:
3336:th element the tuple with all components equal to
3309:
3221:
2813:
2672:
2502:
2453:
2411:
2326:
2263:
2194:
2146:
2097:
2054:
1982:
1698:
1660:
1621:
1599:
1574:
1531:
1403:
1289:
1251:
1199:
1161:
1141:
1089:
1058:
1020:
961:
926:
847:
809:
749:
684:{\displaystyle w_{j}=\sum _{i=1}^{n}a_{i,j}v_{i}.}
683:
601:
563:
492:
396:
372:
335:
295:
6301:
5558:is a theorem that asserts that the numbers of
4046:is the same as that of the preceding section.
5873:
5846:MIT Linear Algebra Lecture on Change of Basis
5679:, the continuous analogue of change of basis.
5275:{\displaystyle P^{\mathsf {T}}\mathbf {B} P.}
4521:As the change-of-basis formula involves only
3692:be the "old basis" of a change of basis, and
2421:The change-of-basis formula asserts that, if
1629:are the column vectors of the coordinates of
1575:{\displaystyle \mathbf {x} =A\,\mathbf {y} ,}
373:{\displaystyle \mathbf {x} _{\mathrm {new} }}
336:{\displaystyle \mathbf {x} _{\mathrm {old} }}
5492:{\displaystyle P^{\mathsf {T}}\mathbf {B} P}
750:{\displaystyle A=\left(a_{i,j}\right)_{i,j}}
5462:and the two members of this equation equal
4197:{\displaystyle \phi _{\mathrm {new} }^{-1}}
4154:{\displaystyle \phi _{\mathrm {old} }^{-1}}
5880:
5866:
5215:{\displaystyle \mathbf {v} ^{\mathsf {T}}}
4571:This is specially useful in the theory of
3232:
768:th column is formed by the coordinates of
422:
5528:
3731:, one could consider it the matrix of an
2626:
2082:
1563:
602:{\displaystyle B_{\mathrm {old} }\colon }
269:
116:Learn how and when to remove this message
5672:Covariance and contravariance of vectors
5852:Khan Academy Lecture on Change of Basis
5638:{\displaystyle P^{\mathsf {T}}=P^{-1}.}
3402:{\displaystyle B=(v_{1},\ldots ,v_{n})}
3257:a linear map, one refers implicitly to
2327:{\displaystyle w_{2}=(-\sin t,\cos t).}
779:. (Here and in what follows, the index
6302:
6271:Comparison of linear algebra libraries
5818:
5754:
5610:
5475:
5437:
5423:
5408:
5388:
5255:
5206:
5159:
4740:, are linear maps from a vector space
4619:. It is represented on "old" bases of
3720:{\displaystyle \phi _{\mathrm {old} }}
3452:{\displaystyle \phi \colon F^{n}\to V}
981:, or equivalently if it has a nonzero
5861:
5770:
5730:
5036:{\displaystyle (v_{1},\ldots ,v_{n})}
4641:. A change of bases is defined by an
2264:{\displaystyle w_{1}=(\cos t,\sin t)}
1252:{\displaystyle (y_{1},\ldots ,y_{n})}
1142:{\displaystyle (x_{1},\ldots ,x_{n})}
4497:is the change-of-base formula, then
2461:are the new coordinates of a vector
54:adding citations to reliable sources
25:
1699:{\displaystyle B_{\mathrm {new} },}
1290:{\displaystyle B_{\mathrm {new} };}
1200:{\displaystyle B_{\mathrm {old} },}
1059:{\displaystyle B_{\mathrm {new} }.}
225:Such a conversion results from the
16:Coordinate change in linear algebra
13:
5887:
4680:On the "new" bases, the matrix of
4446:Function defined on a vector space
4400:
4397:
4394:
4349:
4346:
4343:
4268:
4265:
4262:
4226:
4223:
4220:
4180:
4177:
4174:
4137:
4134:
4131:
4089:
4086:
4083:
4068:
4065:
4062:
4039:{\displaystyle B_{\mathrm {new} }}
4030:
4027:
4024:
3983:
3980:
3977:
3954:
3951:
3948:
3933:
3930:
3927:
3912:
3909:
3906:
3834:
3831:
3828:
3813:
3810:
3807:
3711:
3708:
3705:
3638:
3635:
3632:
2336:So, the change-of-basis matrix is
1687:
1684:
1681:
1661:{\displaystyle B_{\mathrm {old} }}
1652:
1649:
1646:
1278:
1275:
1272:
1188:
1185:
1182:
1047:
1044:
1041:
1021:{\displaystyle B_{\mathrm {old} }}
1012:
1009:
1006:
962:{\displaystyle B_{\mathrm {new} }}
953:
950:
947:
877:
874:
871:
590:
587:
584:
446:
443:
440:
364:
361:
358:
327:
324:
321:
284:
281:
278:
257:
254:
251:
14:
6326:
5839:
5667:Active and passive transformation
4816:
2098:{\displaystyle \mathbb {R} ^{2}.}
6284:
6283:
6261:Basic Linear Algebra Subprograms
6019:
5821:Linear Algebra and Matrix Theory
5482:
5431:
5395:
5361:. It follows that the matrix of
5262:
5200:
5171:
5166:
5153:
4732:
2824:This may be verified by writing
1615:
1593:
1565:
1553:
821:refers always to the columns of
352:
315:
272:
245:
158:
132:
30:
6159:Seven-dimensional cross product
5764:
5743:Beauregard & Fraleigh (1973
4943:{\displaystyle v\mapsto B(w,v)}
4902:{\displaystyle v\mapsto B(v,w)}
4469:When the basis is changed, the
4454:that has a vector space as its
4049:Now, by composing the equation
2753:
2747:
1499:
502:finite-dimensional vector space
41:needs additional citations for
5748:
5736:
5724:
5696:
5403:
5379:
5336:
5324:
5315:
5303:
5145:
5133:
5093:{\displaystyle B(v_{i},v_{j})}
5087:
5061:
5030:
4998:
4937:
4925:
4919:
4896:
4884:
4878:
4578:
4423:
4420:
4414:
4385:
4369:
4363:
3992:
3989:
3968:
3939:
3679:
3647:
3504:
3472:
3443:
3396:
3364:
3156:
3143:
3137:
3115:
3109:
3097:
3081:
3068:
3062:
3040:
3034:
3025:
2992:
2948:
2932:
2888:
2503:{\displaystyle (x_{1},x_{2}),}
2494:
2468:
2318:
2291:
2258:
2234:
2186:
2174:
2141:
2129:
1246:
1214:
1136:
1104:
918:
886:
487:
455:
233:, this formula can be written
1:
5717:
5342:{\displaystyle B(v,w)=B(w,v)}
3332:is the basis that has as its
783:refers always to the rows of
151:, called the change of basis.
6001:Eigenvalues and eigenvectors
5577:, typically in the study of
5535:{\displaystyle \mathbb {R} }
4853:in both arguments. That is,
3253:When one says that a matrix
2195:{\displaystyle v_{2}=(0,1).}
1622:{\displaystyle \mathbf {y} }
1600:{\displaystyle \mathbf {x} }
7:
5660:
4950:are linear for every fixed
4458:is commonly specified as a
3265:between a vector space and
2454:{\displaystyle y_{1},y_{2}}
2147:{\displaystyle v_{1}=(1,0)}
10:
6331:
5823:(2nd ed.), New York:
5775:(5th ed.), New York:
5556:Sylvester's law of inertia
2065:
973:if and only if the matrix
527:, one can define a vector
18:
6279:
6241:
6197:
6134:
6086:
6028:
6017:
5913:
5895:
5848:, from MIT OpenCourseWare
5773:Elementary Linear Algebra
4795:{\displaystyle P^{-1}MP.}
4756:over an "old" basis, and
4719:{\displaystyle P^{-1}MQ.}
3754:{\displaystyle \psi _{A}}
3587:of the standard basis of
5819:Nering, Evar D. (1970),
5803:Houghton Mifflin Company
5689:
5683:Chirgwin-Coulson weights
4868:is bilinear if the maps
4204:on the right, one gets
2109:consists of the vectors
19:Not to be confused with
5287:symmetric bilinear form
4972:{\displaystyle w\in V.}
4669:change-of-basis matrix
4651:change-of-basis matrix
4548:differentiable function
4320:{\displaystyle v\in V,}
3233:In terms of linear maps
1427:change-of-basis formula
1090:{\displaystyle z\in V,}
564:{\displaystyle a_{i,j}}
423:Change of basis formula
227:change-of-basis formula
5986:Row and column vectors
5771:Anton, Howard (1987),
5639:
5536:
5493:
5453:
5343:
5276:
5216:
5182:
5094:
5037:
4973:
4944:
4903:
4796:
4752:of an endomorphism of
4720:
4433:
4321:
4289:
4198:
4155:
4112:
4040:
4002:
3880:
3879:{\displaystyle \circ }
3857:
3785:
3784:{\displaystyle F^{n}.}
3755:
3721:
3686:
3611:
3610:{\displaystyle F^{n}.}
3581:
3558:
3530:
3453:
3403:
3311:
3223:
2815:
2674:
2504:
2455:
2413:
2328:
2265:
2196:
2148:
2099:
2072:Euclidean vector space
2056:
2027:
1984:
1931:
1905:
1843:
1807:
1756:
1700:
1662:
1623:
1601:
1576:
1533:
1472:
1418:for the old basis and
1405:
1377:
1333:
1291:
1253:
1201:
1163:
1149:be the coordinates of
1143:
1091:
1060:
1022:
991:change-of-basis matrix
963:
928:
849:
848:{\displaystyle w_{j};}
811:
810:{\displaystyle v_{i},}
751:
685:
651:
603:
565:
494:
406:change-of-basis matrix
398:
374:
337:
297:
218:on the other basis. A
5991:Row and column spaces
5936:Scalar multiplication
5640:
5537:
5494:
5454:
5344:
5277:
5217:
5183:
5100:. It follows that if
5095:
5038:
4974:
4945:
4904:
4797:
4721:
4460:multivariate function
4434:
4322:
4298:It follows that, for
4290:
4199:
4156:
4113:
4041:
4003:
3881:
3858:
3786:
3756:
3722:
3687:
3612:
3582:
3580:{\displaystyle \phi }
3559:
3510:
3454:
3404:
3312:
3310:{\displaystyle F^{n}}
3224:
2816:
2675:
2505:
2456:
2414:
2329:
2266:
2197:
2149:
2100:
2057:
2007:
1985:
1911:
1885:
1823:
1787:
1736:
1701:
1663:
1624:
1602:
1577:
1534:
1452:
1406:
1357:
1313:
1292:
1259:its coordinates over
1254:
1202:
1164:
1144:
1092:
1061:
1023:
964:
929:
850:
812:
752:
686:
631:
604:
566:
495:
399:
375:
338:
298:
149:linear transformation
6126:Gram–Schmidt process
6078:Gaussian elimination
5601:
5524:
5512:of the ground field
5466:
5376:
5297:
5246:
5195:
5127:
5055:
4995:
4954:
4913:
4872:
4767:
4691:
4334:
4302:
4211:
4165:
4122:
4053:
4015:
3897:
3888:function composition
3870:
3798:
3765:
3738:
3696:
3623:
3591:
3571:
3466:
3424:
3355:
3294:
3247:function application
2831:
2690:
2517:
2465:
2425:
2340:
2275:
2218:
2206:them by an angle of
2158:
2113:
2077:
1997:
1719:
1672:
1637:
1611:
1589:
1549:
1436:
1304:
1263:
1211:
1173:
1153:
1101:
1072:
1032:
997:
938:
862:
829:
791:
701:
615:
575:
542:
431:
388:
347:
310:
240:
145:linearly independent
50:improve this article
6256:Numerical stability
6136:Multilinear algebra
6111:Inner product space
5961:Linear independence
5854:, from Khan Academy
5745:, pp. 240–243)
5733:, pp. 221–237)
5289:is a bilinear form
4987:of a bilinear form
4541:continuous function
4534:polynomial function
4413:
4362:
4281:
4239:
4193:
4150:
3967:
538:by its coordinates
5966:Linear combination
5677:Integral transform
5635:
5589:. In these cases,
5532:
5489:
5449:
5339:
5272:
5212:
5178:
5090:
5033:
4969:
4940:
4899:
4827:on a vector space
4792:
4716:
4611:to a vector space
4529:a linear function,
4429:
4388:
4337:
4317:
4285:
4256:
4214:
4194:
4168:
4151:
4125:
4108:
4036:
3998:
3942:
3876:
3853:
3781:
3751:
3717:
3682:
3607:
3577:
3554:
3449:
3419:linear isomorphism
3399:
3307:
3219:
3217:
2811:
2670:
2661:
2620:
2554:
2500:
2451:
2409:
2400:
2324:
2261:
2192:
2144:
2095:
2052:
2051:
1980:
1978:
1696:
1658:
1619:
1597:
1572:
1529:
1401:
1287:
1249:
1197:
1159:
1139:
1087:
1056:
1018:
989:is said to be the
959:
924:
845:
807:
747:
681:
599:
561:
490:
416:on the old basis.
394:
370:
333:
293:
141:linear combination
6297:
6296:
6164:Geometric algebra
6121:Kronecker product
5956:Linear projection
5941:Vector projection
5757:, pp. 50–52)
5591:orthonormal bases
4807:invertible matrix
4562:analytic function
2751:
1503:
1162:{\displaystyle z}
410:transition matrix
397:{\displaystyle A}
193:coordinate vector
126:
125:
118:
100:
65:"Change of basis"
6322:
6287:
6286:
6169:Exterior algebra
6106:Hadamard product
6023:
6011:Linear equations
5882:
5875:
5868:
5859:
5858:
5835:
5815:
5800:
5789:
5758:
5752:
5746:
5740:
5734:
5728:
5711:
5700:
5647:Spectral theorem
5644:
5642:
5641:
5636:
5631:
5630:
5615:
5614:
5613:
5565:
5561:
5553:
5549:
5541:
5539:
5538:
5533:
5531:
5515:
5504:
5498:
5496:
5495:
5490:
5485:
5480:
5479:
5478:
5458:
5456:
5455:
5450:
5442:
5441:
5440:
5434:
5428:
5427:
5426:
5413:
5412:
5411:
5398:
5393:
5392:
5391:
5365:on any basis is
5364:
5360:
5356:
5352:
5348:
5346:
5345:
5340:
5292:
5281:
5279:
5278:
5273:
5265:
5260:
5259:
5258:
5238:
5231:
5221:
5219:
5218:
5213:
5211:
5210:
5209:
5203:
5187:
5185:
5184:
5179:
5174:
5169:
5164:
5163:
5162:
5156:
5119:
5115:
5111:
5105:
5099:
5097:
5096:
5091:
5086:
5085:
5073:
5072:
5050:
5046:
5042:
5040:
5039:
5034:
5029:
5028:
5010:
5009:
4990:
4986:
4978:
4976:
4975:
4970:
4949:
4947:
4946:
4941:
4908:
4906:
4905:
4900:
4867:
4848:
4837:
4801:
4799:
4798:
4793:
4782:
4781:
4759:
4755:
4747:
4743:
4725:
4723:
4722:
4717:
4706:
4705:
4683:
4676:
4672:
4668:
4658:
4654:
4650:
4640:
4636:
4626:
4622:
4618:
4614:
4610:
4606:
4599:
4523:linear functions
4518:is needed here.
4516:matrix inversion
4510:
4496:
4483:
4438:
4436:
4435:
4430:
4412:
4404:
4403:
4384:
4383:
4361:
4353:
4352:
4326:
4324:
4323:
4318:
4294:
4292:
4291:
4286:
4280:
4272:
4271:
4252:
4251:
4238:
4230:
4229:
4203:
4201:
4200:
4195:
4192:
4184:
4183:
4161:on the left and
4160:
4158:
4157:
4152:
4149:
4141:
4140:
4117:
4115:
4114:
4109:
4107:
4106:
4094:
4093:
4092:
4073:
4072:
4071:
4045:
4043:
4042:
4037:
4035:
4034:
4033:
4007:
4005:
4004:
3999:
3988:
3987:
3986:
3966:
3958:
3957:
3938:
3937:
3936:
3917:
3916:
3915:
3885:
3883:
3882:
3877:
3862:
3860:
3859:
3854:
3852:
3851:
3839:
3838:
3837:
3818:
3817:
3816:
3791:Finally, define
3790:
3788:
3787:
3782:
3777:
3776:
3760:
3758:
3757:
3752:
3750:
3749:
3730:
3726:
3724:
3723:
3718:
3716:
3715:
3714:
3691:
3689:
3688:
3683:
3678:
3677:
3659:
3658:
3643:
3642:
3641:
3616:
3614:
3613:
3608:
3603:
3602:
3586:
3584:
3583:
3578:
3563:
3561:
3560:
3555:
3550:
3549:
3540:
3539:
3529:
3524:
3503:
3502:
3484:
3483:
3458:
3456:
3455:
3450:
3442:
3441:
3416:
3412:
3408:
3406:
3405:
3400:
3395:
3394:
3376:
3375:
3347:
3343:
3339:
3335:
3327:
3321:
3316:
3314:
3313:
3308:
3306:
3305:
3285:
3274:
3270:
3228:
3226:
3225:
3220:
3218:
3211:
3210:
3201:
3200:
3188:
3187:
3178:
3177:
3162:
3155:
3154:
3127:
3126:
3096:
3095:
3080:
3079:
3052:
3051:
3024:
3023:
3008:
3004:
3003:
2982:
2981:
2960:
2959:
2944:
2943:
2922:
2921:
2900:
2899:
2880:
2879:
2870:
2869:
2857:
2856:
2847:
2846:
2820:
2818:
2817:
2812:
2798:
2797:
2776:
2775:
2763:
2762:
2752:
2749:
2737:
2736:
2715:
2714:
2702:
2701:
2679:
2677:
2676:
2671:
2666:
2665:
2658:
2657:
2644:
2643:
2625:
2624:
2559:
2558:
2551:
2550:
2537:
2536:
2509:
2507:
2506:
2501:
2493:
2492:
2480:
2479:
2460:
2458:
2457:
2452:
2450:
2449:
2437:
2436:
2418:
2416:
2415:
2410:
2405:
2404:
2333:
2331:
2330:
2325:
2287:
2286:
2270:
2268:
2267:
2262:
2230:
2229:
2209:
2201:
2199:
2198:
2193:
2170:
2169:
2153:
2151:
2150:
2145:
2125:
2124:
2104:
2102:
2101:
2096:
2091:
2090:
2085:
2061:
2059:
2058:
2053:
2047:
2046:
2037:
2036:
2026:
2021:
1989:
1987:
1986:
1981:
1979:
1972:
1971:
1962:
1958:
1957:
1956:
1947:
1946:
1930:
1925:
1904:
1899:
1878:
1874:
1870:
1869:
1868:
1859:
1858:
1842:
1837:
1822:
1821:
1806:
1801:
1780:
1776:
1775:
1766:
1765:
1755:
1750:
1705:
1703:
1702:
1697:
1692:
1691:
1690:
1667:
1665:
1664:
1659:
1657:
1656:
1655:
1632:
1628:
1626:
1625:
1620:
1618:
1606:
1604:
1603:
1598:
1596:
1581:
1579:
1578:
1573:
1568:
1556:
1538:
1536:
1535:
1530:
1504:
1501:
1498:
1497:
1488:
1487:
1471:
1466:
1448:
1447:
1421:
1417:
1410:
1408:
1407:
1402:
1397:
1396:
1387:
1386:
1376:
1371:
1353:
1352:
1343:
1342:
1332:
1327:
1296:
1294:
1293:
1288:
1283:
1282:
1281:
1258:
1256:
1255:
1250:
1245:
1244:
1226:
1225:
1206:
1204:
1203:
1198:
1193:
1192:
1191:
1168:
1166:
1165:
1160:
1148:
1146:
1145:
1140:
1135:
1134:
1116:
1115:
1096:
1094:
1093:
1088:
1065:
1063:
1062:
1057:
1052:
1051:
1050:
1027:
1025:
1024:
1019:
1017:
1016:
1015:
988:
985:. In this case,
976:
972:
968:
966:
965:
960:
958:
957:
956:
933:
931:
930:
925:
917:
916:
898:
897:
882:
881:
880:
854:
852:
851:
846:
841:
840:
824:
820:
817:while the index
816:
814:
813:
808:
803:
802:
786:
782:
778:
767:
756:
754:
753:
748:
746:
745:
734:
730:
729:
690:
688:
687:
682:
677:
676:
667:
666:
650:
645:
627:
626:
608:
606:
605:
600:
595:
594:
593:
570:
568:
567:
562:
560:
559:
537:
526:
513:
506:
500:be a basis of a
499:
497:
496:
491:
486:
485:
467:
466:
451:
450:
449:
403:
401:
400:
395:
379:
377:
376:
371:
369:
368:
367:
355:
342:
340:
339:
334:
332:
331:
330:
318:
302:
300:
299:
294:
289:
288:
287:
275:
262:
261:
260:
248:
217:
213:
202:
190:
162:
136:
121:
114:
110:
107:
101:
99:
58:
34:
26:
6330:
6329:
6325:
6324:
6323:
6321:
6320:
6319:
6300:
6299:
6298:
6293:
6275:
6237:
6193:
6130:
6082:
6024:
6015:
5981:Change of basis
5971:Multilinear map
5909:
5891:
5886:
5842:
5813:
5787:
5767:
5762:
5761:
5753:
5749:
5741:
5737:
5729:
5725:
5720:
5715:
5714:
5701:
5697:
5692:
5663:
5623:
5619:
5609:
5608:
5604:
5602:
5599:
5598:
5563:
5559:
5551:
5547:
5527:
5525:
5522:
5521:
5513:
5500:
5481:
5474:
5473:
5469:
5467:
5464:
5463:
5436:
5435:
5430:
5429:
5422:
5421:
5417:
5407:
5406:
5402:
5394:
5387:
5386:
5382:
5377:
5374:
5373:
5362:
5358:
5354:
5350:
5298:
5295:
5294:
5290:
5261:
5254:
5253:
5249:
5247:
5244:
5243:
5236:
5227:
5205:
5204:
5199:
5198:
5196:
5193:
5192:
5170:
5165:
5158:
5157:
5152:
5151:
5128:
5125:
5124:
5117:
5113:
5107:
5101:
5081:
5077:
5068:
5064:
5056:
5053:
5052:
5048:
5044:
5024:
5020:
5005:
5001:
4996:
4993:
4992:
4988:
4982:
4955:
4952:
4951:
4914:
4911:
4910:
4873:
4870:
4869:
4854:
4839:
4835:
4819:
4774:
4770:
4768:
4765:
4764:
4757:
4753:
4745:
4741:
4735:
4698:
4694:
4692:
4689:
4688:
4681:
4674:
4670:
4660:
4656:
4652:
4642:
4638:
4628:
4624:
4620:
4616:
4612:
4608:
4604:
4587:
4581:
4555:smooth function
4498:
4485:
4474:
4448:
4405:
4393:
4392:
4379:
4375:
4354:
4342:
4341:
4335:
4332:
4331:
4303:
4300:
4299:
4273:
4261:
4260:
4247:
4243:
4231:
4219:
4218:
4212:
4209:
4208:
4185:
4173:
4172:
4166:
4163:
4162:
4142:
4130:
4129:
4123:
4120:
4119:
4102:
4098:
4082:
4081:
4077:
4061:
4060:
4056:
4054:
4051:
4050:
4023:
4022:
4018:
4016:
4013:
4012:
3976:
3975:
3971:
3959:
3947:
3946:
3926:
3925:
3921:
3905:
3904:
3900:
3898:
3895:
3894:
3871:
3868:
3867:
3847:
3843:
3827:
3826:
3822:
3806:
3805:
3801:
3799:
3796:
3795:
3772:
3768:
3766:
3763:
3762:
3745:
3741:
3739:
3736:
3735:
3728:
3704:
3703:
3699:
3697:
3694:
3693:
3673:
3669:
3654:
3650:
3631:
3630:
3626:
3624:
3621:
3620:
3598:
3594:
3592:
3589:
3588:
3572:
3569:
3568:
3545:
3541:
3535:
3531:
3525:
3514:
3498:
3494:
3479:
3475:
3467:
3464:
3463:
3437:
3433:
3425:
3422:
3421:
3414:
3410:
3390:
3386:
3371:
3367:
3356:
3353:
3352:
3345:
3341:
3337:
3333:
3325:
3319:
3301:
3297:
3295:
3292:
3291:
3283:
3272:
3266:
3235:
3216:
3215:
3206:
3202:
3196:
3192:
3183:
3179:
3173:
3169:
3160:
3159:
3150:
3146:
3122:
3118:
3091:
3087:
3075:
3071:
3047:
3043:
3019:
3015:
3006:
3005:
2999:
2995:
2977:
2973:
2955:
2951:
2939:
2935:
2917:
2913:
2895:
2891:
2881:
2875:
2871:
2865:
2861:
2852:
2848:
2842:
2838:
2834:
2832:
2829:
2828:
2793:
2789:
2771:
2767:
2758:
2754:
2748:
2732:
2728:
2710:
2706:
2697:
2693:
2691:
2688:
2687:
2660:
2659:
2653:
2649:
2646:
2645:
2639:
2635:
2628:
2627:
2619:
2618:
2607:
2595:
2594:
2580:
2564:
2563:
2553:
2552:
2546:
2542:
2539:
2538:
2532:
2528:
2521:
2520:
2518:
2515:
2514:
2488:
2484:
2475:
2471:
2466:
2463:
2462:
2445:
2441:
2432:
2428:
2426:
2423:
2422:
2399:
2398:
2387:
2375:
2374:
2360:
2344:
2343:
2341:
2338:
2337:
2282:
2278:
2276:
2273:
2272:
2225:
2221:
2219:
2216:
2215:
2207:
2165:
2161:
2159:
2156:
2155:
2120:
2116:
2114:
2111:
2110:
2086:
2081:
2080:
2078:
2075:
2074:
2068:
2042:
2038:
2032:
2028:
2022:
2011:
1998:
1995:
1994:
1977:
1976:
1967:
1963:
1952:
1948:
1936:
1932:
1926:
1915:
1910:
1906:
1900:
1889:
1876:
1875:
1864:
1860:
1848:
1844:
1838:
1827:
1817:
1813:
1812:
1808:
1802:
1791:
1778:
1777:
1771:
1767:
1761:
1757:
1751:
1740:
1729:
1722:
1720:
1717:
1716:
1680:
1679:
1675:
1673:
1670:
1669:
1645:
1644:
1640:
1638:
1635:
1634:
1630:
1614:
1612:
1609:
1608:
1592:
1590:
1587:
1586:
1564:
1552:
1550:
1547:
1546:
1500:
1493:
1489:
1477:
1473:
1467:
1456:
1443:
1439:
1437:
1434:
1433:
1419:
1415:
1392:
1388:
1382:
1378:
1372:
1361:
1348:
1344:
1338:
1334:
1328:
1317:
1305:
1302:
1301:
1271:
1270:
1266:
1264:
1261:
1260:
1240:
1236:
1221:
1217:
1212:
1209:
1208:
1181:
1180:
1176:
1174:
1171:
1170:
1154:
1151:
1150:
1130:
1126:
1111:
1107:
1102:
1099:
1098:
1073:
1070:
1069:
1068:Given a vector
1040:
1039:
1035:
1033:
1030:
1029:
1005:
1004:
1000:
998:
995:
994:
993:from the basis
986:
974:
970:
946:
945:
941:
939:
936:
935:
912:
908:
893:
889:
870:
869:
865:
863:
860:
859:
836:
832:
830:
827:
826:
822:
818:
798:
794:
792:
789:
788:
784:
780:
777:
769:
765:
735:
719:
715:
711:
710:
702:
699:
698:
672:
668:
656:
652:
646:
635:
622:
618:
616:
613:
612:
583:
582:
578:
576:
573:
572:
549:
545:
543:
540:
539:
536:
528:
518:
511:
504:
481:
477:
462:
458:
439:
438:
434:
432:
429:
428:
425:
389:
386:
385:
357:
356:
351:
350:
348:
345:
344:
320:
319:
314:
313:
311:
308:
307:
277:
276:
271:
270:
250:
249:
244:
243:
241:
238:
237:
220:change of basis
215:
211:
200:
188:
170:
169:
168:
167:
166:
163:
154:
153:
152:
137:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
6328:
6318:
6317:
6312:
6310:Linear algebra
6295:
6294:
6292:
6291:
6280:
6277:
6276:
6274:
6273:
6268:
6263:
6258:
6253:
6251:Floating-point
6247:
6245:
6239:
6238:
6236:
6235:
6233:Tensor product
6230:
6225:
6220:
6218:Function space
6215:
6210:
6204:
6202:
6195:
6194:
6192:
6191:
6186:
6181:
6176:
6171:
6166:
6161:
6156:
6154:Triple product
6151:
6146:
6140:
6138:
6132:
6131:
6129:
6128:
6123:
6118:
6113:
6108:
6103:
6098:
6092:
6090:
6084:
6083:
6081:
6080:
6075:
6070:
6068:Transformation
6065:
6060:
6058:Multiplication
6055:
6050:
6045:
6040:
6034:
6032:
6026:
6025:
6018:
6016:
6014:
6013:
6008:
6003:
5998:
5993:
5988:
5983:
5978:
5973:
5968:
5963:
5958:
5953:
5948:
5943:
5938:
5933:
5928:
5923:
5917:
5915:
5914:Basic concepts
5911:
5910:
5908:
5907:
5902:
5896:
5893:
5892:
5889:Linear algebra
5885:
5884:
5877:
5870:
5862:
5856:
5855:
5849:
5841:
5840:External links
5838:
5837:
5836:
5816:
5811:
5790:
5785:
5766:
5763:
5760:
5759:
5747:
5735:
5722:
5721:
5719:
5716:
5713:
5712:
5694:
5693:
5691:
5688:
5687:
5686:
5680:
5674:
5669:
5662:
5659:
5655:diagonalizable
5634:
5629:
5626:
5622:
5618:
5612:
5607:
5530:
5510:characteristic
5505:is symmetric.
5499:if the matrix
5488:
5484:
5477:
5472:
5460:
5459:
5448:
5445:
5439:
5433:
5425:
5420:
5416:
5410:
5405:
5401:
5397:
5390:
5385:
5381:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5283:
5282:
5271:
5268:
5264:
5257:
5252:
5226:of the matrix
5208:
5202:
5189:
5188:
5177:
5173:
5168:
5161:
5155:
5150:
5147:
5144:
5141:
5138:
5135:
5132:
5089:
5084:
5080:
5076:
5071:
5067:
5063:
5060:
5032:
5027:
5023:
5019:
5016:
5013:
5008:
5004:
5000:
4968:
4965:
4962:
4959:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4918:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4877:
4838:is a function
4818:
4817:Bilinear forms
4815:
4803:
4802:
4791:
4788:
4785:
4780:
4777:
4773:
4734:
4731:
4727:
4726:
4715:
4712:
4709:
4704:
4701:
4697:
4580:
4577:
4566:
4565:
4558:
4551:
4544:
4537:
4530:
4447:
4444:
4440:
4439:
4428:
4425:
4422:
4419:
4416:
4411:
4408:
4402:
4399:
4396:
4391:
4387:
4382:
4378:
4374:
4371:
4368:
4365:
4360:
4357:
4351:
4348:
4345:
4340:
4316:
4313:
4310:
4307:
4296:
4295:
4284:
4279:
4276:
4270:
4267:
4264:
4259:
4255:
4250:
4246:
4242:
4237:
4234:
4228:
4225:
4222:
4217:
4191:
4188:
4182:
4179:
4176:
4171:
4148:
4145:
4139:
4136:
4133:
4128:
4105:
4101:
4097:
4091:
4088:
4085:
4080:
4076:
4070:
4067:
4064:
4059:
4032:
4029:
4026:
4021:
4009:
4008:
3997:
3994:
3991:
3985:
3982:
3979:
3974:
3970:
3965:
3962:
3956:
3953:
3950:
3945:
3941:
3935:
3932:
3929:
3924:
3920:
3914:
3911:
3908:
3903:
3875:
3864:
3863:
3850:
3846:
3842:
3836:
3833:
3830:
3825:
3821:
3815:
3812:
3809:
3804:
3780:
3775:
3771:
3748:
3744:
3713:
3710:
3707:
3702:
3681:
3676:
3672:
3668:
3665:
3662:
3657:
3653:
3649:
3646:
3640:
3637:
3634:
3629:
3606:
3601:
3597:
3576:
3565:
3564:
3553:
3548:
3544:
3538:
3534:
3528:
3523:
3520:
3517:
3513:
3509:
3506:
3501:
3497:
3493:
3490:
3487:
3482:
3478:
3474:
3471:
3448:
3445:
3440:
3436:
3432:
3429:
3413:-vector space
3398:
3393:
3389:
3385:
3382:
3379:
3374:
3370:
3366:
3363:
3360:
3330:standard basis
3304:
3300:
3234:
3231:
3230:
3229:
3214:
3209:
3205:
3199:
3195:
3191:
3186:
3182:
3176:
3172:
3168:
3165:
3163:
3161:
3158:
3153:
3149:
3145:
3142:
3139:
3136:
3133:
3130:
3125:
3121:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3094:
3090:
3086:
3083:
3078:
3074:
3070:
3067:
3064:
3061:
3058:
3055:
3050:
3046:
3042:
3039:
3036:
3033:
3030:
3027:
3022:
3018:
3014:
3011:
3009:
3007:
3002:
2998:
2994:
2991:
2988:
2985:
2980:
2976:
2972:
2969:
2966:
2963:
2958:
2954:
2950:
2947:
2942:
2938:
2934:
2931:
2928:
2925:
2920:
2916:
2912:
2909:
2906:
2903:
2898:
2894:
2890:
2887:
2884:
2882:
2878:
2874:
2868:
2864:
2860:
2855:
2851:
2845:
2841:
2837:
2836:
2822:
2821:
2810:
2807:
2804:
2801:
2796:
2792:
2788:
2785:
2782:
2779:
2774:
2770:
2766:
2761:
2757:
2746:
2743:
2740:
2735:
2731:
2727:
2724:
2721:
2718:
2713:
2709:
2705:
2700:
2696:
2681:
2680:
2669:
2664:
2656:
2652:
2648:
2647:
2642:
2638:
2634:
2633:
2631:
2623:
2617:
2614:
2611:
2608:
2606:
2603:
2600:
2597:
2596:
2593:
2590:
2587:
2584:
2581:
2579:
2576:
2573:
2570:
2569:
2567:
2562:
2557:
2549:
2545:
2541:
2540:
2535:
2531:
2527:
2526:
2524:
2499:
2496:
2491:
2487:
2483:
2478:
2474:
2470:
2448:
2444:
2440:
2435:
2431:
2408:
2403:
2397:
2394:
2391:
2388:
2386:
2383:
2380:
2377:
2376:
2373:
2370:
2367:
2364:
2361:
2359:
2356:
2353:
2350:
2349:
2347:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2285:
2281:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2228:
2224:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2168:
2164:
2143:
2140:
2137:
2134:
2131:
2128:
2123:
2119:
2107:standard basis
2094:
2089:
2084:
2067:
2064:
2050:
2045:
2041:
2035:
2031:
2025:
2020:
2017:
2014:
2010:
2005:
2002:
1991:
1990:
1975:
1970:
1966:
1961:
1955:
1951:
1945:
1942:
1939:
1935:
1929:
1924:
1921:
1918:
1914:
1909:
1903:
1898:
1895:
1892:
1888:
1884:
1881:
1879:
1877:
1873:
1867:
1863:
1857:
1854:
1851:
1847:
1841:
1836:
1833:
1830:
1826:
1820:
1816:
1811:
1805:
1800:
1797:
1794:
1790:
1786:
1783:
1781:
1779:
1774:
1770:
1764:
1760:
1754:
1749:
1746:
1743:
1739:
1735:
1732:
1730:
1728:
1725:
1724:
1706:respectively.
1695:
1689:
1686:
1683:
1678:
1654:
1651:
1648:
1643:
1617:
1595:
1583:
1582:
1571:
1567:
1562:
1559:
1555:
1540:
1539:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1496:
1492:
1486:
1483:
1480:
1476:
1470:
1465:
1462:
1459:
1455:
1451:
1446:
1442:
1412:
1411:
1400:
1395:
1391:
1385:
1381:
1375:
1370:
1367:
1364:
1360:
1356:
1351:
1347:
1341:
1337:
1331:
1326:
1323:
1320:
1316:
1312:
1309:
1286:
1280:
1277:
1274:
1269:
1248:
1243:
1239:
1235:
1232:
1229:
1224:
1220:
1216:
1196:
1190:
1187:
1184:
1179:
1158:
1138:
1133:
1129:
1125:
1122:
1119:
1114:
1110:
1106:
1086:
1083:
1080:
1077:
1055:
1049:
1046:
1043:
1038:
1014:
1011:
1008:
1003:
969:is a basis of
955:
952:
949:
944:
923:
920:
915:
911:
907:
904:
901:
896:
892:
888:
885:
879:
876:
873:
868:
844:
839:
835:
806:
801:
797:
773:
758:
757:
744:
741:
738:
733:
728:
725:
722:
718:
714:
709:
706:
692:
691:
680:
675:
671:
665:
662:
659:
655:
649:
644:
641:
638:
634:
630:
625:
621:
598:
592:
589:
586:
581:
558:
555:
552:
548:
532:
489:
484:
480:
476:
473:
470:
465:
461:
457:
454:
448:
445:
442:
437:
424:
421:
393:
382:column vectors
366:
363:
360:
354:
329:
326:
323:
317:
304:
303:
292:
286:
283:
280:
274:
268:
265:
259:
256:
253:
247:
164:
157:
156:
155:
138:
131:
130:
129:
128:
127:
124:
123:
38:
36:
29:
21:Change of base
15:
9:
6:
4:
3:
2:
6327:
6316:
6315:Matrix theory
6313:
6311:
6308:
6307:
6305:
6290:
6282:
6281:
6278:
6272:
6269:
6267:
6266:Sparse matrix
6264:
6262:
6259:
6257:
6254:
6252:
6249:
6248:
6246:
6244:
6240:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6205:
6203:
6201:constructions
6200:
6196:
6190:
6189:Outermorphism
6187:
6185:
6182:
6180:
6177:
6175:
6172:
6170:
6167:
6165:
6162:
6160:
6157:
6155:
6152:
6150:
6149:Cross product
6147:
6145:
6142:
6141:
6139:
6137:
6133:
6127:
6124:
6122:
6119:
6117:
6116:Outer product
6114:
6112:
6109:
6107:
6104:
6102:
6099:
6097:
6096:Orthogonality
6094:
6093:
6091:
6089:
6085:
6079:
6076:
6074:
6073:Cramer's rule
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6049:
6046:
6044:
6043:Decomposition
6041:
6039:
6036:
6035:
6033:
6031:
6027:
6022:
6012:
6009:
6007:
6004:
6002:
5999:
5997:
5994:
5992:
5989:
5987:
5984:
5982:
5979:
5977:
5974:
5972:
5969:
5967:
5964:
5962:
5959:
5957:
5954:
5952:
5949:
5947:
5944:
5942:
5939:
5937:
5934:
5932:
5929:
5927:
5924:
5922:
5919:
5918:
5916:
5912:
5906:
5903:
5901:
5898:
5897:
5894:
5890:
5883:
5878:
5876:
5871:
5869:
5864:
5863:
5860:
5853:
5850:
5847:
5844:
5843:
5834:
5830:
5826:
5822:
5817:
5814:
5812:0-395-14017-X
5808:
5804:
5799:
5798:
5791:
5788:
5786:0-471-84819-0
5782:
5778:
5774:
5769:
5768:
5756:
5751:
5744:
5739:
5732:
5727:
5723:
5709:
5708:ordered basis
5705:
5699:
5695:
5684:
5681:
5678:
5675:
5673:
5670:
5668:
5665:
5664:
5658:
5656:
5652:
5648:
5632:
5627:
5624:
5620:
5616:
5605:
5596:
5592:
5588:
5584:
5580:
5576:
5572:
5567:
5557:
5545:
5519:
5511:
5506:
5503:
5486:
5470:
5446:
5443:
5418:
5414:
5399:
5383:
5372:
5371:
5370:
5368:
5333:
5330:
5327:
5321:
5318:
5312:
5309:
5306:
5300:
5288:
5269:
5266:
5250:
5242:
5241:
5240:
5233:
5230:
5225:
5175:
5148:
5142:
5139:
5136:
5130:
5123:
5122:
5121:
5110:
5104:
5082:
5078:
5074:
5069:
5065:
5058:
5051:th column is
5025:
5021:
5017:
5014:
5011:
5006:
5002:
4985:
4979:
4966:
4963:
4960:
4957:
4934:
4931:
4928:
4922:
4916:
4893:
4890:
4887:
4881:
4875:
4865:
4861:
4857:
4852:
4846:
4842:
4834:
4830:
4826:
4825:
4824:bilinear form
4814:
4812:
4808:
4789:
4786:
4783:
4778:
4775:
4771:
4763:
4762:
4761:
4751:
4750:square matrix
4739:
4738:Endomorphisms
4733:Endomorphisms
4730:
4713:
4710:
4707:
4702:
4699:
4695:
4687:
4686:
4685:
4678:
4667:
4663:
4649:
4645:
4635:
4631:
4615:of dimension
4607:of dimension
4603:
4598:
4594:
4590:
4586:
4576:
4574:
4569:
4563:
4559:
4556:
4552:
4549:
4545:
4542:
4538:
4535:
4531:
4528:
4527:
4526:
4524:
4519:
4517:
4512:
4508:
4505:
4501:
4495:
4492:
4488:
4481:
4477:
4472:
4467:
4465:
4461:
4457:
4453:
4443:
4426:
4417:
4409:
4406:
4389:
4380:
4376:
4372:
4366:
4358:
4355:
4338:
4330:
4329:
4328:
4314:
4311:
4308:
4305:
4282:
4277:
4274:
4257:
4253:
4248:
4244:
4240:
4235:
4232:
4215:
4207:
4206:
4205:
4189:
4186:
4169:
4146:
4143:
4126:
4103:
4099:
4095:
4078:
4074:
4057:
4047:
4019:
3995:
3972:
3963:
3960:
3943:
3922:
3918:
3901:
3893:
3892:
3891:
3889:
3873:
3848:
3844:
3840:
3823:
3819:
3802:
3794:
3793:
3792:
3778:
3773:
3769:
3746:
3742:
3734:
3700:
3674:
3670:
3666:
3663:
3660:
3655:
3651:
3644:
3627:
3617:
3604:
3599:
3595:
3574:
3551:
3546:
3542:
3536:
3532:
3526:
3521:
3518:
3515:
3511:
3507:
3499:
3495:
3491:
3488:
3485:
3480:
3476:
3469:
3462:
3461:
3460:
3446:
3438:
3434:
3430:
3427:
3420:
3391:
3387:
3383:
3380:
3377:
3372:
3368:
3361:
3358:
3349:
3331:
3323:
3302:
3298:
3289:
3280:
3278:
3269:
3264:
3260:
3256:
3251:
3248:
3244:
3241:represents a
3240:
3212:
3207:
3203:
3197:
3193:
3189:
3184:
3180:
3174:
3170:
3166:
3164:
3151:
3147:
3140:
3134:
3131:
3128:
3123:
3119:
3112:
3106:
3103:
3100:
3092:
3088:
3084:
3076:
3072:
3065:
3059:
3056:
3053:
3048:
3044:
3037:
3031:
3028:
3020:
3016:
3012:
3010:
3000:
2996:
2989:
2986:
2983:
2978:
2974:
2970:
2967:
2964:
2961:
2956:
2952:
2945:
2940:
2936:
2929:
2926:
2923:
2918:
2914:
2910:
2907:
2904:
2901:
2896:
2892:
2885:
2883:
2876:
2872:
2866:
2862:
2858:
2853:
2849:
2843:
2839:
2827:
2826:
2825:
2808:
2805:
2802:
2799:
2794:
2790:
2786:
2783:
2780:
2777:
2772:
2768:
2764:
2759:
2755:
2744:
2741:
2738:
2733:
2729:
2725:
2722:
2719:
2716:
2711:
2707:
2703:
2698:
2694:
2686:
2685:
2684:
2667:
2662:
2654:
2650:
2640:
2636:
2629:
2621:
2615:
2612:
2609:
2604:
2601:
2598:
2591:
2588:
2585:
2582:
2577:
2574:
2571:
2565:
2560:
2555:
2547:
2543:
2533:
2529:
2522:
2513:
2512:
2511:
2510:then one has
2497:
2489:
2485:
2481:
2476:
2472:
2446:
2442:
2438:
2433:
2429:
2419:
2406:
2401:
2395:
2392:
2389:
2384:
2381:
2378:
2371:
2368:
2365:
2362:
2357:
2354:
2351:
2345:
2334:
2321:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2288:
2283:
2279:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2231:
2226:
2222:
2213:
2210:, one gets a
2205:
2189:
2183:
2180:
2177:
2171:
2166:
2162:
2138:
2135:
2132:
2126:
2121:
2117:
2108:
2092:
2087:
2073:
2070:Consider the
2063:
2048:
2043:
2039:
2033:
2029:
2023:
2018:
2015:
2012:
2008:
2003:
2000:
1973:
1968:
1964:
1959:
1953:
1949:
1943:
1940:
1937:
1933:
1927:
1922:
1919:
1916:
1912:
1907:
1901:
1896:
1893:
1890:
1886:
1882:
1880:
1871:
1865:
1861:
1855:
1852:
1849:
1845:
1839:
1834:
1831:
1828:
1824:
1818:
1814:
1809:
1803:
1798:
1795:
1792:
1788:
1784:
1782:
1772:
1768:
1762:
1758:
1752:
1747:
1744:
1741:
1737:
1733:
1731:
1726:
1715:
1714:
1713:
1711:
1707:
1693:
1676:
1641:
1569:
1560:
1557:
1545:
1544:
1543:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1494:
1490:
1484:
1481:
1478:
1474:
1468:
1463:
1460:
1457:
1453:
1449:
1444:
1440:
1432:
1431:
1430:
1428:
1423:
1398:
1393:
1389:
1383:
1379:
1373:
1368:
1365:
1362:
1358:
1354:
1349:
1345:
1339:
1335:
1329:
1324:
1321:
1318:
1314:
1310:
1307:
1300:
1299:
1298:
1284:
1267:
1241:
1237:
1233:
1230:
1227:
1222:
1218:
1194:
1177:
1156:
1131:
1127:
1123:
1120:
1117:
1112:
1108:
1084:
1081:
1078:
1075:
1066:
1053:
1036:
1028:to the basis
1001:
992:
984:
980:
942:
934:one has that
921:
913:
909:
905:
902:
899:
894:
890:
883:
866:
856:
842:
837:
833:
804:
799:
795:
776:
772:
763:
742:
739:
736:
731:
726:
723:
720:
716:
712:
707:
704:
697:
696:
695:
678:
673:
669:
663:
660:
657:
653:
647:
642:
639:
636:
632:
628:
623:
619:
611:
610:
609:
596:
579:
556:
553:
550:
546:
535:
531:
525:
521:
515:
510:
503:
482:
478:
474:
471:
468:
463:
459:
452:
435:
420:
417:
415:
414:basis vectors
411:
408:(also called
407:
391:
383:
290:
266:
263:
236:
235:
234:
232:
228:
223:
221:
209:
205:
198:
195:, which is a
194:
187:
183:
179:
178:ordered basis
175:
161:
150:
146:
142:
135:
120:
117:
109:
106:November 2017
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: –
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
6199:Vector space
5980:
5931:Vector space
5820:
5796:
5772:
5765:Bibliography
5755:Nering (1970
5750:
5738:
5726:
5698:
5568:
5544:real numbers
5507:
5501:
5461:
5284:
5234:
5228:
5222:denotes the
5190:
5108:
5102:
4983:
4980:
4863:
4859:
4855:
4844:
4840:
4828:
4822:
4820:
4804:
4736:
4728:
4679:
4665:
4661:
4647:
4643:
4633:
4629:
4602:vector space
4596:
4592:
4588:
4582:
4570:
4567:
4520:
4513:
4506:
4503:
4499:
4493:
4490:
4486:
4479:
4475:
4468:
4449:
4441:
4297:
4048:
4010:
3865:
3733:endomorphism
3618:
3566:
3350:
3281:
3267:
3254:
3252:
3237:Normally, a
3236:
2823:
2682:
2420:
2335:
2211:
2069:
1992:
1709:
1708:
1584:
1541:
1426:
1424:
1413:
1067:
990:
857:
774:
770:
759:
693:
533:
529:
523:
519:
516:
426:
418:
409:
405:
305:
226:
224:
219:
182:vector space
171:
112:
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
6179:Multivector
6144:Determinant
6101:Dot product
5946:Linear span
5731:Anton (1987
5651:eigenvalues
5581:and of the
5047:th row and
4991:on a basis
4981:The matrix
4583:Consider a
4579:Linear maps
3344:th that is
3340:except the
3263:isomorphism
983:determinant
208:coordinates
174:mathematics
6304:Categories
6213:Direct sum
6048:Invertible
5951:Linear map
5801:, Boston:
5718:References
5595:orthogonal
5587:rigid body
5349:for every
5293:such that
5120:, one has
4585:linear map
4471:expression
3417:defines a
3290:, the set
3255:represents
3243:linear map
2683:That is,
2214:formed by
979:invertible
522:= 1, ...,
184:of finite
76:newspapers
6243:Numerical
6006:Transpose
5625:−
5367:symmetric
5224:transpose
5015:…
4961:∈
4920:↦
4879:↦
4849:which is
4805:As every
4776:−
4700:−
4659:, and an
4573:manifolds
4407:−
4390:ϕ
4377:ψ
4356:−
4339:ϕ
4327:one has
4309:∈
4275:−
4258:ϕ
4254:∘
4245:ψ
4233:−
4216:ϕ
4187:−
4170:ϕ
4144:−
4127:ϕ
4100:ψ
4096:∘
4079:ϕ
4058:ϕ
3961:−
3944:ϕ
3923:ϕ
3874:∘
3845:ψ
3841:∘
3824:ϕ
3803:ϕ
3743:ψ
3701:ϕ
3664:…
3575:ϕ
3512:∑
3489:…
3470:ϕ
3444:→
3431::
3428:ϕ
3381:…
3135:
3107:
3101:−
3060:
3032:
2987:
2965:
2927:
2911:−
2905:
2803:
2781:
2742:
2726:−
2720:
2613:
2602:
2589:
2583:−
2575:
2393:
2382:
2369:
2363:−
2355:
2313:
2301:
2295:−
2253:
2241:
2212:new basis
2009:∑
1913:∑
1887:∑
1825:∑
1789:∑
1738:∑
1518:…
1502:for
1454:∑
1359:∑
1315:∑
1297:that is
1231:…
1121:…
1079:∈
903:…
633:∑
597::
472:…
186:dimension
6289:Category
6228:Subspace
6223:Quotient
6174:Bivector
6088:Bilinear
6030:Matrices
5905:Glossary
5833:76091646
5661:See also
5579:quadrics
5571:geometry
5518:diagonal
4858: :
4452:function
3886:denotes
3351:A basis
3271:, where
858:Setting
825:and the
787:and the
380:are the
231:matrices
197:sequence
5900:Outline
5583:inertia
5575:physics
5562:and of
5542:of the
5508:If the
4831:over a
4811:similar
4748:is the
4637:matrix
4600:from a
4464:applied
3890:), and
3866:(where
3322:-tuples
3317:of the
2204:rotates
2202:If one
2066:Example
760:be the
507:over a
404:is the
206:called
204:scalars
90:scholar
6184:Tensor
5996:Kernel
5926:Vector
5921:Scalar
5831:
5809:
5783:
5191:where
4851:linear
4456:domain
3239:matrix
1710:Proof:
1585:where
764:whose
762:matrix
92:
85:
78:
71:
63:
6053:Minor
6038:Block
5976:Basis
5825:Wiley
5777:Wiley
5704:tuple
5690:Notes
5585:of a
4833:field
4627:by a
4118:with
3409:of a
3324:is a
3288:field
3286:be a
3277:up to
3259:bases
1633:over
1169:over
571:over
509:field
180:of a
176:, an
97:JSTOR
83:books
6208:Dual
6063:Rank
5829:LCCN
5807:ISBN
5781:ISBN
5573:and
5353:and
5116:and
5106:and
4909:and
4684:is
4673:for
4655:for
4623:and
3619:Let
3459:by
3282:Let
2271:and
2154:and
2105:Its
1668:and
1607:and
1425:The
1207:and
1097:let
694:Let
517:For
427:Let
343:and
69:news
5550:or
5357:in
5235:If
4866:→ F
4847:→ F
4560:an
3761:of
3132:cos
3104:sin
3057:sin
3029:cos
2984:cos
2962:sin
2924:sin
2902:cos
2800:cos
2778:sin
2750:and
2739:sin
2717:cos
2610:cos
2599:sin
2586:sin
2572:cos
2390:cos
2379:sin
2366:sin
2352:cos
2310:cos
2298:sin
2250:sin
2238:cos
1993:As
977:is
199:of
172:In
52:by
6306::
5827:,
5805:,
5779:,
5657:.
5564:–1
5554:.
5552:–1
5285:A
5232:.
4862:×
4843:×
4821:A
4677:.
4595:→
4591::
4553:a
4546:a
4539:a
4532:a
4489:=
4466:.
4450:A
3348:.
514:.
139:A
5881:e
5874:t
5867:v
5710:.
5633:.
5628:1
5621:P
5617:=
5611:T
5606:P
5560:1
5548:1
5529:R
5514:F
5502:B
5487:P
5483:B
5476:T
5471:P
5447:,
5444:P
5438:T
5432:B
5424:T
5419:P
5415:=
5409:T
5404:)
5400:P
5396:B
5389:T
5384:P
5380:(
5363:B
5359:V
5355:w
5351:v
5337:)
5334:v
5331:,
5328:w
5325:(
5322:B
5319:=
5316:)
5313:w
5310:,
5307:v
5304:(
5301:B
5291:B
5270:.
5267:P
5263:B
5256:T
5251:P
5237:P
5229:v
5207:T
5201:v
5176:,
5172:w
5167:B
5160:T
5154:v
5149:=
5146:)
5143:w
5140:,
5137:v
5134:(
5131:B
5118:w
5114:v
5109:w
5103:v
5088:)
5083:j
5079:v
5075:,
5070:i
5066:v
5062:(
5059:B
5049:j
5045:i
5031:)
5026:n
5022:v
5018:,
5012:,
5007:1
5003:v
4999:(
4989:B
4984:B
4967:.
4964:V
4958:w
4938:)
4935:v
4932:,
4929:w
4926:(
4923:B
4917:v
4897:)
4894:w
4891:,
4888:v
4885:(
4882:B
4876:v
4864:V
4860:V
4856:B
4845:V
4841:V
4836:F
4829:V
4790:.
4787:P
4784:M
4779:1
4772:P
4758:P
4754:V
4746:M
4742:V
4714:.
4711:Q
4708:M
4703:1
4696:P
4682:T
4675:W
4671:Q
4666:n
4664:×
4662:n
4657:V
4653:P
4648:m
4646:×
4644:m
4639:M
4634:n
4632:×
4630:m
4625:W
4621:V
4617:m
4613:V
4609:n
4605:W
4597:V
4593:W
4589:T
4564:,
4557:,
4550:,
4543:,
4536:,
4509:)
4507:y
4504:A
4502:(
4500:f
4494:y
4491:A
4487:x
4482:)
4480:x
4478:(
4476:f
4427:,
4424:)
4421:)
4418:v
4415:(
4410:1
4401:w
4398:e
4395:n
4386:(
4381:A
4373:=
4370:)
4367:v
4364:(
4359:1
4350:d
4347:l
4344:o
4315:,
4312:V
4306:v
4283:.
4278:1
4269:w
4266:e
4263:n
4249:A
4241:=
4236:1
4227:d
4224:l
4221:o
4190:1
4181:w
4178:e
4175:n
4147:1
4138:d
4135:l
4132:o
4104:A
4090:d
4087:l
4084:o
4075:=
4069:w
4066:e
4063:n
4031:w
4028:e
4025:n
4020:B
3996:.
3993:)
3990:)
3984:d
3981:l
3978:o
3973:B
3969:(
3964:1
3955:d
3952:l
3949:o
3940:(
3934:w
3931:e
3928:n
3919:=
3913:w
3910:e
3907:n
3902:B
3849:A
3835:d
3832:l
3829:o
3820:=
3814:w
3811:e
3808:n
3779:.
3774:n
3770:F
3747:A
3729:A
3712:d
3709:l
3706:o
3680:)
3675:n
3671:v
3667:,
3661:,
3656:1
3652:v
3648:(
3645:=
3639:d
3636:l
3633:o
3628:B
3605:.
3600:n
3596:F
3552:.
3547:i
3543:v
3537:i
3533:x
3527:n
3522:1
3519:=
3516:i
3508:=
3505:)
3500:n
3496:x
3492:,
3486:,
3481:1
3477:x
3473:(
3447:V
3439:n
3435:F
3415:V
3411:F
3397:)
3392:n
3388:v
3384:,
3378:,
3373:1
3369:v
3365:(
3362:=
3359:B
3346:1
3342:i
3338:0
3334:i
3326:F
3320:n
3303:n
3299:F
3284:F
3273:F
3268:F
3213:.
3208:2
3204:w
3198:2
3194:y
3190:+
3185:1
3181:w
3175:1
3171:y
3167:=
3157:)
3152:2
3148:v
3144:)
3141:t
3138:(
3129:+
3124:1
3120:v
3116:)
3113:t
3110:(
3098:(
3093:2
3089:y
3085:+
3082:)
3077:2
3073:v
3069:)
3066:t
3063:(
3054:+
3049:1
3045:v
3041:)
3038:t
3035:(
3026:(
3021:1
3017:y
3013:=
3001:2
2997:v
2993:)
2990:t
2979:2
2975:y
2971:+
2968:t
2957:1
2953:y
2949:(
2946:+
2941:1
2937:v
2933:)
2930:t
2919:2
2915:y
2908:t
2897:1
2893:y
2889:(
2886:=
2877:2
2873:v
2867:2
2863:x
2859:+
2854:1
2850:v
2844:1
2840:x
2809:.
2806:t
2795:2
2791:y
2787:+
2784:t
2773:1
2769:y
2765:=
2760:2
2756:x
2745:t
2734:2
2730:y
2723:t
2712:1
2708:y
2704:=
2699:1
2695:x
2668:.
2663:]
2655:2
2651:y
2641:1
2637:y
2630:[
2622:]
2616:t
2605:t
2592:t
2578:t
2566:[
2561:=
2556:]
2548:2
2544:x
2534:1
2530:x
2523:[
2498:,
2495:)
2490:2
2486:x
2482:,
2477:1
2473:x
2469:(
2447:2
2443:y
2439:,
2434:1
2430:y
2407:.
2402:]
2396:t
2385:t
2372:t
2358:t
2346:[
2322:.
2319:)
2316:t
2307:,
2304:t
2292:(
2289:=
2284:2
2280:w
2259:)
2256:t
2247:,
2244:t
2235:(
2232:=
2227:1
2223:w
2208:t
2190:.
2187:)
2184:1
2181:,
2178:0
2175:(
2172:=
2167:2
2163:v
2142:)
2139:0
2136:,
2133:1
2130:(
2127:=
2122:1
2118:v
2093:.
2088:2
2083:R
2049:,
2044:i
2040:v
2034:i
2030:x
2024:n
2019:1
2016:=
2013:i
2004:=
2001:z
1974:.
1969:i
1965:v
1960:)
1954:j
1950:y
1944:j
1941:,
1938:i
1934:a
1928:n
1923:1
1920:=
1917:j
1908:(
1902:n
1897:1
1894:=
1891:i
1883:=
1872:)
1866:i
1862:v
1856:j
1853:,
1850:i
1846:a
1840:n
1835:1
1832:=
1829:i
1819:j
1815:y
1810:(
1804:n
1799:1
1796:=
1793:j
1785:=
1773:j
1769:w
1763:j
1759:y
1753:n
1748:1
1745:=
1742:j
1734:=
1727:z
1694:,
1688:w
1685:e
1682:n
1677:B
1653:d
1650:l
1647:o
1642:B
1631:z
1616:y
1594:x
1570:,
1566:y
1561:A
1558:=
1554:x
1527:.
1524:n
1521:,
1515:,
1512:1
1509:=
1506:i
1495:j
1491:y
1485:j
1482:,
1479:i
1475:a
1469:n
1464:1
1461:=
1458:j
1450:=
1445:i
1441:x
1420:j
1416:i
1399:.
1394:j
1390:w
1384:j
1380:y
1374:n
1369:1
1366:=
1363:j
1355:=
1350:i
1346:v
1340:i
1336:x
1330:n
1325:1
1322:=
1319:i
1311:=
1308:z
1285:;
1279:w
1276:e
1273:n
1268:B
1247:)
1242:n
1238:y
1234:,
1228:,
1223:1
1219:y
1215:(
1195:,
1189:d
1186:l
1183:o
1178:B
1157:z
1137:)
1132:n
1128:x
1124:,
1118:,
1113:1
1109:x
1105:(
1085:,
1082:V
1076:z
1054:.
1048:w
1045:e
1042:n
1037:B
1013:d
1010:l
1007:o
1002:B
987:A
975:A
971:V
954:w
951:e
948:n
943:B
922:,
919:)
914:n
910:w
906:,
900:,
895:1
891:w
887:(
884:=
878:w
875:e
872:n
867:B
843:;
838:j
834:w
823:A
819:j
805:,
800:i
796:v
785:A
781:i
775:j
771:w
766:j
743:j
740:,
737:i
732:)
727:j
724:,
721:i
717:a
713:(
708:=
705:A
679:.
674:i
670:v
664:j
661:,
658:i
654:a
648:n
643:1
640:=
637:i
629:=
624:j
620:w
591:d
588:l
585:o
580:B
557:j
554:,
551:i
547:a
534:j
530:w
524:n
520:j
512:F
505:V
488:)
483:n
479:v
475:,
469:,
464:1
460:v
456:(
453:=
447:d
444:l
441:o
436:B
392:A
365:w
362:e
359:n
353:x
328:d
325:l
322:o
316:x
291:,
285:w
282:e
279:n
273:x
267:A
264:=
258:d
255:l
252:o
246:x
216:v
212:v
201:n
189:n
119:)
113:(
108:)
104:(
94:·
87:·
80:·
73:·
46:.
23:.
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