10642:
11851:
10906:
43:
140:
176:
2497:
10165:
6314:
6523:
6096:
8108:
5052:
4359:
4849:
4678:
4517:
1728:
A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of
253:
A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the
6677:
1136:
If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is
5581:
5712:
2309:
5447:
9917:
7144:
8510:
6107:
6325:
5895:
7913:
4860:
700:
8351:
982:
510:
4198:
1736:
In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "5 miles northeast" vector is a
4689:
4531:
4370:
9318:
1061:
This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of
7630:
6538:
8176:
3712:
then (1) is true if and only if (2) is true; that is, in this particular case either both (1) and (2) are true (and the vectors are linearly dependent) or else both (1) and (2) are false (and the vectors are linearly
869:
325:
3286:
5458:
2492:{\displaystyle a_{1}\mathbf {v} _{1}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} +\cdots +\mathbf {0} +a_{i}\mathbf {v} _{i}+\mathbf {0} +\cdots +\mathbf {0} =a_{i}\mathbf {v} _{i}=a_{i}\mathbf {0} =\mathbf {0} .}
2231:
5592:
2301:
7030:
10160:{\textstyle \sum _{k\neq i}M_{k}={\Big \{}m_{1}+\cdots +m_{i-1}+m_{i+1}+\cdots +m_{d}:m_{k}\in M_{k}{\text{ for all }}k{\Big \}}=\operatorname {span} \bigcup _{k\in \{1,\ldots ,i-1,i+1,\ldots ,d\}}M_{k}.}
7724:
5325:
6309:{\displaystyle {\begin{bmatrix}1&7&-2\\4&10&1\\2&-4&5\\-3&-1&-4\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}={\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}.}
9462:
7041:
8362:
2643:
2583:
1889:
1817:
6518:{\displaystyle {\begin{bmatrix}1&7&-2\\0&-18&9\\0&0&0\\0&0&0\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}={\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}.}
6091:{\displaystyle \mathbf {v} _{1}={\begin{bmatrix}1\\4\\2\\-3\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}7\\10\\-4\\-1\end{bmatrix}},\mathbf {v} _{3}={\begin{bmatrix}-2\\1\\5\\-4\end{bmatrix}}.}
3650:
8103:{\displaystyle {\begin{matrix}\mathbf {e} _{1}&=&(1,0,0,\ldots ,0)\\\mathbf {e} _{2}&=&(0,1,0,\ldots ,0)\\&\vdots \\\mathbf {e} _{n}&=&(0,0,0,\ldots ,1).\end{matrix}}}
7248:
5047:{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}={\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}=-a_{3}{\begin{bmatrix}16/5\\2/5\end{bmatrix}}.}
6858:
2876:
9889:
579:
6781:
1987:
1219:
9612:
3710:
3680:
3580:
3522:
2805:
769:
3374:
3748:
3492:
3174:
3075:
8242:
3778:
3404:
3338:
3208:
401:
5317:
5211:
4140:
5848:
4190:
4088:
10292:
5798:
5264:
5158:
5111:
1718:
9391:. Any affine combination is a linear combination; therefore every affinely dependent set is linearly dependent. Conversely, every linearly independent set is affinely independent.
10237:
9797:
7881:
8890:
7839:
5888:
209:
8826:
6944:
6887:
2836:
2747:
2718:
170:
10472:
1947:
9751:
8587:
8250:
881:
409:
4354:{\displaystyle a_{1}{\begin{bmatrix}1\\1\end{bmatrix}}+a_{2}{\begin{bmatrix}-3\\2\end{bmatrix}}+a_{3}{\begin{bmatrix}2\\4\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}},}
1059:
4027:
4005:
3981:
3959:
3933:
3911:
3886:
3864:
3842:
3820:
3426:
3308:
3121:
3099:
3022:
3000:
2971:
2949:
2927:
2905:
1839:
1131:
1082:
535:
3607:
2533:
571:
1671:
1642:
1610:
1581:
1552:
1523:
1494:
1465:
1434:
1405:
1374:
1345:
1316:
1281:
1252:
9011:
7801:, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.
5751:
2166:
2020:
8546:
7394:
3548:
2133:
1018:
732:
9170:
9041:
8700:
7799:
7322:
9140:
8670:
7287:
6811:
4844:{\displaystyle {\begin{bmatrix}1&0&16/5\\0&1&2/5\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}
2047:
9087:
8972:
3456:
3234:
795:
9638:
9113:
8946:
7773:
2689:
9912:
8920:
1109:
10185:
9817:
9706:
9686:
9666:
9502:
9482:
9412:
9061:
8768:
8748:
8720:
8643:
8615:
7901:
7747:
7653:
7554:
7514:
7494:
7474:
7454:
7426:
7365:
7345:
7177:
6915:
4673:{\displaystyle {\begin{bmatrix}1&-3&2\\0&5&2\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}
4512:{\displaystyle {\begin{bmatrix}1&-3&2\\1&2&4\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}
3798:
3141:
3042:
2767:
2663:
2107:
2087:
2067:
1909:
1144:
A sequence of vectors is linearly independent if and only if it does not contain the same vector twice and the set of its vectors is linearly independent.
1164:
if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.
9242:
1612:
is not a linear combination of them or, equivalently, because they do not belong to a common plane. The three vectors define a three-dimensional space.
11509:
6672:{\displaystyle {\begin{bmatrix}1&7\\0&-18\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}=-a_{3}{\begin{bmatrix}-2\\9\end{bmatrix}}.}
7562:
8116:
809:
265:
10324:
5576:{\displaystyle {\begin{bmatrix}1&-3\\1&2\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}
7809:
If there are more vectors than dimensions, the vectors are linearly dependent. This is illustrated in the example above of three vectors in
5707:{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}
3239:
2691:). A collection of vectors that consists of exactly one vector is linearly dependent if and only if that vector is zero. Explicitly, if
2171:
2236:
1175:
if it does not contain the same vector twice, and if the set of its vectors is linearly independent. Otherwise, the family is said to be
6970:
1141:
if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.
5442:{\displaystyle a_{1}{\begin{bmatrix}1\\1\end{bmatrix}}+a_{2}{\begin{bmatrix}-3\\2\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}},}
4683:
Continue the row reduction by (i) dividing the second row by 5, and then (ii) multiplying by 3 and adding to the first row, that is
1748:
Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general,
11723:
10942:
10500:
7661:
7139:{\displaystyle A\Lambda ={\begin{bmatrix}1&-3\\1&2\end{bmatrix}}{\begin{bmatrix}\lambda _{1}\\\lambda _{2}\end{bmatrix}}.}
107:
11814:
8505:{\displaystyle a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+\cdots +a_{n}\mathbf {e} _{n}=\left(a_{1},a_{2},\ldots ,a_{n}\right),}
10833:
79:
10891:
9417:
2598:
2538:
1844:
1772:
11733:
11499:
86:
60:
7185:
3612:
695:{\displaystyle \mathbf {v} _{1}={\frac {-a_{2}}{a_{1}}}\mathbf {v} _{2}+\cdots +{\frac {-a_{k}}{a_{1}}}\mathbf {v} _{k},}
6816:
2841:
10414:
9826:
10379:
126:
93:
17:
6695:
1952:
1891:
are necessarily linearly dependent (and consequently, they are not linearly independent). To see why, suppose that
10881:
10466:
9640:
each. The original vectors are affinely independent if and only if the augmented vectors are linearly independent.
9348:
If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous
3685:
3655:
3555:
3497:
2773:
737:
9507:
11534:
10843:
10779:
3343:
238:
of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be
3720:
3464:
3146:
3047:
75:
11081:
10347:
8188:
3753:
3379:
3313:
3183:
1729:
the Earth's surface). The person might add, "The place is 5 miles northeast of here." This last statement is
344:
64:
5269:
5163:
4092:
10453:
5803:
4145:
4043:
10242:
5756:
5222:
5116:
5069:
1676:
11298:
10935:
10621:
10493:
11373:
10726:
10576:
10448:
10443:
10196:
9756:
7516:
equations; any solution of the full list of equations must also be true of the reduced list. In fact, if
7851:
2588:
As a consequence, the zero vector can not possibly belong to any collection of vectors that is linearly
11529:
11051:
10631:
10525:
9349:
8837:
8346:{\displaystyle a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+\cdots +a_{n}\mathbf {e} _{n}=\mathbf {0} .}
7812:
5861:
977:{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{n}\mathbf {v} _{n}=\mathbf {0} ,}
505:{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} ,}
182:
8776:
6920:
6863:
2812:
2723:
2694:
146:
11887:
11633:
11504:
11418:
10871:
10520:
1914:
9715:
8551:
3888:(while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false).
11738:
11628:
11336:
11016:
10863:
10746:
1023:
247:
4010:
3988:
3964:
3942:
3916:
3894:
3869:
3847:
3825:
3803:
3409:
3291:
3104:
3082:
3005:
2983:
2954:
2932:
2910:
2888:
1822:
1114:
1065:
518:
11892:
11773:
11702:
11584:
11444:
11041:
10928:
10909:
10838:
10616:
10486:
2505:
543:
100:
53:
3585:
1647:
1618:
1586:
1557:
1528:
1499:
1470:
1441:
1410:
1381:
1350:
1321:
1292:
1257:
1228:
11643:
11226:
11031:
10673:
10606:
10596:
10460:
9353:
8977:
8622:
1187:
5720:
2138:
1992:
1084:
as a linear combination of its vectors is the trivial representation in which all the scalars
11589:
11326:
11176:
11171:
11006:
10981:
10976:
10688:
10683:
10678:
10611:
10556:
8518:
7370:
3527:
2112:
1745:, that is, one of the three vectors is unnecessary to define a specific location on a plane.
990:
708:
10363:
9145:
9016:
8675:
7778:
7292:
1111:
are zero. Even more concisely, a sequence of vectors is linearly independent if and only if
11783:
11141:
10971:
10951:
10698:
10663:
10650:
10541:
10389:
9357:
9233:
9118:
8648:
7260:
6958:
6789:
2025:
339:
223:
9066:
8951:
3435:
3213:
774:
8:
11778:
11356:
11161:
11151:
10876:
10756:
10731:
9617:
9092:
8925:
7752:
2668:
9894:
8902:
11855:
11809:
11799:
11753:
11748:
11677:
11613:
11479:
11216:
11211:
11146:
11136:
11001:
10586:
10367:
10170:
9802:
9691:
9671:
9651:
9487:
9467:
9397:
9384:
9046:
8753:
8733:
8705:
8628:
8600:
7886:
7732:
7638:
7539:
7499:
7479:
7459:
7439:
7411:
7350:
7330:
7162:
6900:
3783:
3126:
3027:
2752:
2648:
2092:
2072:
2052:
1894:
1738:
1087:
800:
235:
11866:
11850:
11653:
11648:
11638:
11618:
11579:
11574:
11403:
11398:
11383:
11378:
11369:
11364:
11311:
11206:
11156:
11101:
11071:
11066:
11046:
11036:
10996:
10784:
10741:
10668:
10561:
10420:
10410:
10375:
10343:
2049:
be equal any other non-zero scalar will also work) and then let all other scalars be
799:
Thus, a set of vectors is linearly dependent if and only if one of them is zero or a
219:
4192:
then the condition for linear dependence seeks a set of non-zero scalars, such that
11861:
11829:
11758:
11697:
11692:
11672:
11608:
11514:
11484:
11469:
11449:
11388:
11341:
11316:
11306:
11277:
11196:
11191:
11166:
11096:
11076:
10986:
10966:
10789:
10693:
10546:
1284:
11454:
11559:
11494:
11474:
11459:
11439:
11423:
11321:
11252:
11242:
11201:
11086:
11056:
10848:
10641:
10601:
10591:
10385:
9313:{\displaystyle a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}=\mathbf {0} .}
9185:
9568:
9522:
11819:
11763:
11743:
11728:
11687:
11564:
11524:
11489:
11413:
11352:
11331:
11272:
11262:
11247:
11181:
11126:
11116:
11111:
11021:
10853:
10774:
10509:
9323:
If such a linear dependence exists with at least a nonzero component, then the
7904:
6950:
2974:
1168:
11881:
11824:
11682:
11623:
11554:
11544:
11539:
11464:
11393:
11267:
11257:
11186:
11106:
11091:
11026:
10886:
10809:
10769:
10736:
10716:
10424:
4525:
this matrix equation by subtracting the first row from the second to obtain,
4522:
8974:
In order to do this, we subtract the first equation from the second, giving
7625:{\displaystyle A_{\langle i_{1},\dots ,i_{m}\rangle }\Lambda =\mathbf {0} .}
11707:
11664:
11569:
11282:
11221:
11131:
11011:
10819:
10708:
10658:
10551:
9374:
8618:
2885:
This example considers the special case where there are exactly two vector
1133:
can be represented as a linear combination of its vectors in a unique way.
328:
215:
8171:{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\ldots ,\mathbf {e} _{n}}
1218:
11549:
11519:
11287:
11121:
10991:
10799:
10764:
10721:
10566:
7254:
6954:
3310:
is), which shows that (1) is true in this particular case. Similarly, if
1183:
864:{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}}
320:{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{k}}
9356:
of the vector space of linear dependencies can therefore be computed by
11600:
11061:
10828:
10571:
10189:
1752:
linearly independent vectors are required to describe all locations in
1191:
31:
11834:
11408:
10626:
3281:{\displaystyle c\mathbf {v} =0\mathbf {v} =\mathbf {0} =\mathbf {u} }
9115:
too. Therefore, according to the definition of linear independence,
2880:
2226:{\displaystyle a_{j}\mathbf {v} _{j}=0\mathbf {v} _{j}=\mathbf {0} }
42:
11768:
10794:
7635:
Furthermore, the reverse is true. That is, we can test whether the
2296:{\displaystyle a_{1}\mathbf {v} _{1}+\cdots +a_{k}\mathbf {v} _{k}}
1160:
is linearly independent. Conversely, an infinite set of vectors is
7025:{\displaystyle A={\begin{bmatrix}1&-3\\1&2\end{bmatrix}}.}
10920:
10478:
10302:
1376:
are dependent because all three are contained in the same plane.
10804:
10374:, Annals of Discrete Mathematics, vol. 29, North-Holland,
9383:
if at least one of the vectors in the set can be defined as an
1157:
7159:
for some nonzero vector Λ. This depends on the determinant of
540:
This implies that at least one of the scalars is nonzero, say
10328:(Trans. R. A. Silverman), Dover Publications, New York, 1977.
9207:
7719:{\displaystyle \det A_{\langle i_{1},\dots ,i_{m}\rangle }=0}
10338:
Friedberg, Stephen; Insel, Arnold; Spence, Lawrence (2003).
1190:
for that vector space. For example, the vector space of all
139:
10409:(Second ed.). Mineola, New York: Dover Publications.
9327:
vectors are linearly dependent. Linear dependencies among
1741:
of the other two vectors, and it makes the set of vectors
875:
if it is not linearly dependent, that is, if the equation
175:
9457:{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}}
9352:, with the coordinates of the vectors as coefficients. A
6961:
formed by taking the vectors as its columns is non-zero.
3582:; in this case, it is possible to multiply both sides by
3406:(for instance, if they are both equal to the zero vector
9643:
7436:. As we saw previously, this is equivalent to a list of
6892:
2638:{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}
2578:{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}
1884:{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}
1812:{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}
1769:
If one or more vectors from a given sequence of vectors
9920:
9829:
9510:
7918:
7098:
7059:
6985:
6642:
6586:
6547:
6477:
6420:
6334:
6268:
6211:
6116:
6044:
5980:
5919:
5680:
5637:
5601:
5549:
5506:
5467:
5415:
5383:
5344:
5219:
Now consider the linear dependence of the two vectors
5004:
4948:
4905:
4869:
4817:
4760:
4698:
4646:
4589:
4540:
4485:
4428:
4379:
4327:
4298:
4256:
4217:
3615:
3588:
1644:(null vector, whose components are equal to zero) and
1436:
are dependent because they are parallel to each other.
1090:
10305: – Abstraction of linear independence of vectors
10245:
10199:
10173:
9897:
9805:
9759:
9718:
9694:
9674:
9654:
9620:
9490:
9470:
9420:
9400:
9245:
9148:
9121:
9095:
9069:
9049:
9019:
8980:
8954:
8928:
8905:
8840:
8779:
8756:
8736:
8708:
8678:
8651:
8631:
8603:
8554:
8521:
8365:
8253:
8191:
8119:
7916:
7889:
7854:
7815:
7781:
7755:
7735:
7664:
7641:
7565:
7556:
rows, then the equation must be true for those rows.
7542:
7502:
7482:
7462:
7442:
7414:
7373:
7353:
7333:
7295:
7263:
7188:
7165:
7044:
6973:
6923:
6903:
6866:
6819:
6792:
6698:
6541:
6328:
6110:
5898:
5864:
5806:
5759:
5723:
5595:
5461:
5328:
5272:
5225:
5166:
5119:
5072:
4863:
4692:
4534:
4373:
4201:
4148:
4095:
4046:
4013:
3991:
3967:
3945:
3919:
3897:
3872:
3850:
3828:
3806:
3786:
3756:
3723:
3688:
3658:
3645:{\textstyle \mathbf {v} ={\frac {1}{c}}\mathbf {u} .}
3558:
3530:
3500:
3467:
3438:
3412:
3382:
3346:
3316:
3294:
3242:
3216:
3186:
3149:
3129:
3107:
3085:
3050:
3030:
3008:
2986:
2957:
2935:
2929:
from some real or complex vector space. The vectors
2913:
2891:
2844:
2815:
2776:
2755:
2726:
2697:
2671:
2651:
2601:
2541:
2508:
2312:
2239:
2174:
2141:
2115:
2095:
2075:
2055:
2028:
1995:
1955:
1917:
1897:
1847:
1825:
1775:
1679:
1650:
1621:
1589:
1560:
1531:
1502:
1473:
1444:
1413:
1384:
1353:
1324:
1295:
1260:
1231:
1117:
1068:
1026:
993:
884:
812:
777:
740:
711:
582:
546:
521:
412:
347:
268:
185:
149:
10337:
7243:{\displaystyle \det A=1\cdot 2-1\cdot (-3)=5\neq 0.}
7035:
We may write a linear combination of the columns as
2595:
Now consider the special case where the sequence of
1733:, but it is not necessary to find the location.
8592:
7775:, this requires only one determinant, as above. If
3123:(explicitly, this means that there exists a scalar
3024:(explicitly, this means that there exists a scalar
1182:A set of vectors which is linearly independent and
67:. Unsourced material may be challenged and removed.
10286:
10231:
10179:
10159:
9906:
9883:
9811:
9791:
9745:
9700:
9680:
9660:
9632:
9606:
9496:
9476:
9456:
9406:
9312:
9164:
9134:
9107:
9081:
9055:
9035:
9005:
8966:
8940:
8914:
8884:
8820:
8762:
8742:
8714:
8694:
8664:
8637:
8609:
8581:
8540:
8504:
8345:
8236:
8170:
8102:
7895:
7875:
7833:
7793:
7767:
7741:
7718:
7655:vectors are linearly dependent by testing whether
7647:
7624:
7548:
7508:
7488:
7468:
7448:
7420:
7388:
7359:
7339:
7316:
7281:
7242:
7171:
7138:
7024:
6964:In this case, the matrix formed by the vectors is
6938:
6909:
6881:
6853:{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},}
6852:
6805:
6775:
6682:This equation is easily solved to define non-zero
6671:
6517:
6308:
6101:are linearly dependent, form the matrix equation,
6090:
5882:
5842:
5792:
5745:
5706:
5575:
5441:
5311:
5258:
5205:
5152:
5105:
5046:
4843:
4672:
4511:
4353:
4184:
4134:
4082:
4021:
3999:
3975:
3953:
3927:
3905:
3880:
3858:
3836:
3814:
3792:
3772:
3742:
3704:
3674:
3644:
3601:
3574:
3542:
3516:
3486:
3450:
3420:
3398:
3368:
3332:
3302:
3280:
3228:
3202:
3168:
3135:
3115:
3093:
3069:
3036:
3016:
2994:
2965:
2943:
2921:
2899:
2871:{\displaystyle \mathbf {v} _{1}\neq \mathbf {0} .}
2870:
2830:
2799:
2761:
2741:
2712:
2683:
2657:
2637:
2577:
2527:
2491:
2295:
2225:
2160:
2127:
2101:
2081:
2061:
2041:
2014:
1981:
1941:
1903:
1883:
1833:
1811:
1712:
1665:
1636:
1604:
1575:
1546:
1517:
1488:
1459:
1428:
1399:
1368:
1339:
1310:
1275:
1246:
1125:
1103:
1076:
1053:
1012:
976:
863:
789:
763:
726:
694:
573:, and the above equation is able to be written as
565:
529:
504:
395:
319:
246:. These concepts are central to the definition of
203:
164:
10066:
9952:
9884:{\textstyle M_{i}\cap \sum _{k\neq i}M_{k}=\{0\}}
8831:Take the first derivative of the above equation:
3288:(this equality holds no matter what the value of
2881:Linear dependence and independence of two vectors
2502:Because not all scalars are zero (in particular,
1759:
11879:
10239:if these subspaces are linearly independent and
7665:
7189:
5213:Thus, the three vectors are linearly dependent.
9175:
7804:
5586:The same row reduction presented above yields,
30:For linear dependence of random variables, see
6897:An alternative method relies on the fact that
6813:can be chosen arbitrarily. Thus, the vectors
6776:{\displaystyle a_{1}=-3a_{3}/2,a_{2}=a_{3}/2,}
5858:In order to determine if the three vectors in
4854:Rearranging this equation allows us to obtain
1982:{\displaystyle \mathbf {v} _{i}=\mathbf {0} .}
10936:
10494:
10404:
10362:
9607:{\textstyle \left(\left,\ldots ,\left\right)}
3705:{\displaystyle \mathbf {v} \neq \mathbf {0} }
3675:{\displaystyle \mathbf {u} \neq \mathbf {0} }
3575:{\displaystyle \mathbf {v} \neq \mathbf {0} }
3517:{\displaystyle \mathbf {u} \neq \mathbf {0} }
2800:{\displaystyle \mathbf {v} _{1}=\mathbf {0} }
764:{\displaystyle \mathbf {v} _{1}=\mathbf {0} }
27:Vectors whose linear combinations are nonzero
10139:
10091:
9878:
9872:
9737:
9731:
9387:of the others. Otherwise, the set is called
7705:
7673:
7603:
7571:
1936:
1918:
3822:must be zero. Moreover, if exactly one of
3369:{\displaystyle \mathbf {v} =0\mathbf {u} .}
2069:(explicitly, this means that for any index
11510:Fundamental (linear differential equation)
10943:
10929:
10501:
10487:
10405:Bachman, George; Narici, Lawrence (2000).
3743:{\displaystyle \mathbf {u} =c\mathbf {v} }
3487:{\displaystyle \mathbf {u} =c\mathbf {v} }
3169:{\displaystyle \mathbf {v} =c\mathbf {u} }
3070:{\displaystyle \mathbf {u} =c\mathbf {v} }
8237:{\displaystyle a_{1},a_{2},\ldots ,a_{n}}
7863:
7818:
6926:
5867:
3773:{\displaystyle \mathbf {u} =\mathbf {0} }
3399:{\displaystyle \mathbf {u} =\mathbf {v} }
3333:{\displaystyle \mathbf {v} =\mathbf {0} }
3203:{\displaystyle \mathbf {u} =\mathbf {0} }
1198:over the reals has the (infinite) subset
396:{\displaystyle a_{1},a_{2},\dots ,a_{k},}
188:
179:Linearly dependent vectors in a plane in
152:
127:Learn how and when to remove this message
10342:. Pearson, 4th Edition. pp. 48–49.
7843:
7428:entries, and we are again interested in
5312:{\displaystyle \mathbf {v} _{2}=(-3,2),}
5206:{\displaystyle \mathbf {v} _{2}=(-3,2).}
4135:{\displaystyle \mathbf {v} _{2}=(-3,2),}
2977:at least one of the following is true:
174:
138:
11815:Matrix representation of conic sections
7883:and consider the following elements in
5843:{\displaystyle \mathbf {v} _{2}=(-3,2)}
4185:{\displaystyle \mathbf {v} _{3}=(2,4),}
4083:{\displaystyle \mathbf {v} _{1}=(1,1),}
2838:is linearly independent if and only if
2769:) is linearly dependent if and only if
14:
11880:
10892:Comparison of linear algebra libraries
10400:
10398:
10287:{\displaystyle M_{1}+\cdots +M_{d}=X.}
9368:
5793:{\displaystyle \mathbf {v} _{1}=(1,1)}
5259:{\displaystyle \mathbf {v} _{1}=(1,1)}
5153:{\displaystyle \mathbf {v} _{1}=(1,1)}
5106:{\displaystyle \mathbf {v} _{3}=(2,4)}
1723:
1713:{\displaystyle {\vec {o}}=0{\vec {k}}}
10924:
10482:
9644:Linearly independent vector subspaces
7408:matrix and Λ is a column vector with
6893:Alternative method using determinants
1213:
6319:Row reduce this equation to obtain,
65:adding citations to reliable sources
36:
10473:Introduction to Linear Independence
10395:
10232:{\displaystyle M_{1},\ldots ,M_{d}}
9792:{\displaystyle M_{1},\ldots ,M_{d}}
24:
10950:
10508:
9363:
7876:{\displaystyle V=\mathbb {R} ^{n}}
7608:
7048:
1764:
1583:are independent of each other and
1217:
25:
11904:
10436:
10356:
9567:
9521:
8885:{\displaystyle ae^{t}+2be^{2t}=0}
7834:{\displaystyle \mathbb {R} ^{2}.}
5883:{\displaystyle \mathbb {R} ^{4},}
204:{\displaystyle \mathbb {R} ^{3}.}
11849:
10905:
10904:
10882:Basic Linear Algebra Subprograms
10640:
10467:Tutorial and interactive program
9580:
9534:
9444:
9423:
9303:
9289:
9258:
8821:{\displaystyle ae^{t}+be^{2t}=0}
8593:Linear independence of functions
8434:
8403:
8378:
8336:
8322:
8291:
8266:
8158:
8137:
8122:
8043:
7979:
7923:
7615:
6939:{\displaystyle \mathbb {R} ^{n}}
6882:{\displaystyle \mathbf {v} _{3}}
6869:
6837:
6822:
6026:
5962:
5901:
5853:
5809:
5762:
5275:
5228:
5169:
5122:
5075:
4151:
4098:
4049:
4032:
4015:
3993:
3969:
3947:
3921:
3899:
3874:
3852:
3830:
3808:
3766:
3758:
3736:
3725:
3698:
3690:
3668:
3660:
3635:
3617:
3568:
3560:
3510:
3502:
3480:
3469:
3414:
3392:
3384:
3359:
3348:
3326:
3318:
3296:
3274:
3266:
3258:
3247:
3196:
3188:
3162:
3151:
3109:
3087:
3063:
3052:
3010:
2988:
2959:
2937:
2915:
2893:
2861:
2847:
2831:{\displaystyle \mathbf {v} _{1}}
2818:
2793:
2779:
2742:{\displaystyle \mathbf {v} _{1}}
2729:
2720:is any vector then the sequence
2713:{\displaystyle \mathbf {v} _{1}}
2700:
2625:
2604:
2565:
2544:
2535:), this proves that the vectors
2482:
2474:
2450:
2431:
2417:
2403:
2384:
2370:
2356:
2325:
2283:
2252:
2219:
2205:
2187:
1972:
1958:
1911:is an index (i.e. an element of
1871:
1850:
1827:
1799:
1778:
1147:
1119:
1070:
967:
953:
922:
897:
851:
830:
815:
757:
743:
679:
629:
585:
523:
495:
481:
450:
425:
307:
286:
271:
165:{\displaystyle \mathbb {R} ^{3}}
143:Linearly independent vectors in
41:
11717:Used in science and engineering
10780:Seven-dimensional cross product
9379:A set of vectors is said to be
8770:are two real numbers such that
3432:(1) and (2) are true (by using
2749:(which is a sequence of length
1942:{\displaystyle \{1,\ldots ,k\}}
1283:are independent and define the
52:needs additional citations for
10960:Explicitly constrained entries
10331:
10316:
9746:{\displaystyle M\cap N=\{0\}.}
9484:each, and consider the set of
8582:{\displaystyle i=1,\ldots ,n.}
8090:
8060:
8026:
7996:
7970:
7940:
7456:equations. Consider the first
7311:
7296:
7276:
7264:
7225:
7216:
5837:
5822:
5787:
5775:
5303:
5288:
5253:
5241:
5197:
5182:
5147:
5135:
5100:
5088:
4176:
4164:
4126:
4111:
4074:
4062:
2809:alternatively, the collection
1760:Evaluating linear independence
1704:
1686:
1657:
1628:
1596:
1567:
1538:
1509:
1480:
1451:
1420:
1391:
1360:
1331:
1302:
1267:
1238:
1152:An infinite set of vectors is
234:if there exists no nontrivial
13:
1:
11734:Fundamental (computer vision)
10309:
9753:More generally, a collection
7149:We are interested in whether
5753:which means that the vectors
1054:{\displaystyle i=1,\dots ,n.}
257:
254:dimension of a vector space.
10622:Eigenvalues and eigenvectors
10461:Linearly Dependent Functions
9176:Space of linear dependencies
7805:More vectors than dimensions
4040:Consider the set of vectors
4022:{\displaystyle \mathbf {u} }
4007:is not a scalar multiple of
4000:{\displaystyle \mathbf {v} }
3976:{\displaystyle \mathbf {v} }
3961:is not a scalar multiple of
3954:{\displaystyle \mathbf {u} }
3928:{\displaystyle \mathbf {v} }
3906:{\displaystyle \mathbf {u} }
3881:{\displaystyle \mathbf {0} }
3859:{\displaystyle \mathbf {v} }
3837:{\displaystyle \mathbf {u} }
3815:{\displaystyle \mathbf {v} }
3421:{\displaystyle \mathbf {0} }
3303:{\displaystyle \mathbf {v} }
3116:{\displaystyle \mathbf {u} }
3094:{\displaystyle \mathbf {v} }
3017:{\displaystyle \mathbf {v} }
2995:{\displaystyle \mathbf {u} }
2966:{\displaystyle \mathbf {v} }
2944:{\displaystyle \mathbf {u} }
2922:{\displaystyle \mathbf {v} }
2900:{\displaystyle \mathbf {u} }
1834:{\displaystyle \mathbf {0} }
1126:{\displaystyle \mathbf {0} }
1077:{\displaystyle \mathbf {0} }
530:{\displaystyle \mathbf {0} }
7:
11500:Duplication and elimination
11299:eigenvalues or eigenvectors
10449:Encyclopedia of Mathematics
10296:
8244:are real numbers such that
7327:Otherwise, suppose we have
5113:can be defined in terms of
3602:{\textstyle {\frac {1}{c}}}
2528:{\displaystyle a_{i}\neq 0}
1186:some vector space, forms a
566:{\displaystyle a_{1}\neq 0}
10:
11909:
11433:With specific applications
11062:Discrete Fourier Transform
9372:
9350:system of linear equations
9172:are linearly independent.
8722:are linearly independent.
8178:are linearly independent.
7729:for all possible lists of
7324:are linearly independent.
5850:are linearly independent.
5057:which shows that non-zero
1666:{\displaystyle {\vec {k}}}
1637:{\displaystyle {\vec {o}}}
1605:{\displaystyle {\vec {k}}}
1576:{\displaystyle {\vec {v}}}
1547:{\displaystyle {\vec {u}}}
1518:{\displaystyle {\vec {k}}}
1489:{\displaystyle {\vec {v}}}
1460:{\displaystyle {\vec {u}}}
1429:{\displaystyle {\vec {j}}}
1400:{\displaystyle {\vec {u}}}
1369:{\displaystyle {\vec {w}}}
1340:{\displaystyle {\vec {v}}}
1311:{\displaystyle {\vec {u}}}
1276:{\displaystyle {\vec {v}}}
1247:{\displaystyle {\vec {u}}}
29:
11843:
11792:
11724:Cabibbo–Kobayashi–Maskawa
11716:
11662:
11598:
11432:
11351:Satisfying conditions on
11350:
11296:
11235:
10959:
10900:
10862:
10818:
10755:
10707:
10649:
10638:
10534:
10516:
9006:{\displaystyle be^{2t}=0}
7257:is non-zero, the vectors
3939:dependent if and only if
3340:then (2) is true because
1156:if every nonempty finite
987:can only be satisfied by
537:denotes the zero vector.
8725:
6889:are linearly dependent.
6528:Rearrange to solve for v
5746:{\displaystyle a_{i}=0,}
3101:is a scalar multiple of
3002:is a scalar multiple of
2585:are linearly dependent.
2161:{\displaystyle a_{j}:=0}
2022:(alternatively, letting
2015:{\displaystyle a_{i}:=1}
1525:are independent because
403:not all zero, such that
11082:Generalized permutation
10469:on Linear Independence.
8541:{\displaystyle a_{i}=0}
7389:{\displaystyle m<n.}
3543:{\displaystyle c\neq 0}
2973:are linearly dependent
2128:{\displaystyle j\neq i}
1013:{\displaystyle a_{i}=0}
727:{\displaystyle k>1,}
11856:Mathematics portal
10607:Row and column vectors
10288:
10233:
10181:
10161:
9908:
9885:
9813:
9793:
9747:
9702:
9682:
9662:
9634:
9608:
9498:
9478:
9458:
9408:
9314:
9166:
9165:{\displaystyle e^{2t}}
9136:
9109:
9083:
9057:
9037:
9036:{\displaystyle e^{2t}}
9007:
8968:
8942:
8916:
8886:
8822:
8764:
8744:
8716:
8696:
8695:{\displaystyle e^{2t}}
8666:
8639:
8621:of all differentiable
8611:
8583:
8542:
8506:
8347:
8238:
8172:
8104:
7897:
7877:
7835:
7795:
7794:{\displaystyle m>n}
7769:
7743:
7720:
7649:
7626:
7550:
7510:
7490:
7470:
7450:
7422:
7390:
7361:
7341:
7318:
7317:{\displaystyle (-3,2)}
7283:
7244:
7173:
7140:
7026:
6940:
6911:
6883:
6854:
6807:
6777:
6673:
6519:
6310:
6092:
5884:
5844:
5794:
5747:
5708:
5577:
5443:
5313:
5260:
5207:
5154:
5107:
5048:
4845:
4674:
4513:
4355:
4186:
4136:
4084:
4023:
4001:
3977:
3955:
3929:
3907:
3882:
3860:
3838:
3816:
3794:
3774:
3744:
3706:
3676:
3646:
3603:
3576:
3544:
3518:
3488:
3452:
3422:
3400:
3370:
3334:
3304:
3282:
3230:
3204:
3170:
3137:
3117:
3095:
3071:
3038:
3018:
2996:
2967:
2945:
2923:
2901:
2872:
2832:
2801:
2763:
2743:
2714:
2685:
2659:
2639:
2579:
2529:
2493:
2297:
2227:
2162:
2129:
2103:
2083:
2063:
2043:
2016:
1983:
1943:
1905:
1885:
1835:
1813:
1714:
1667:
1638:
1606:
1577:
1548:
1519:
1490:
1461:
1430:
1401:
1370:
1341:
1312:
1277:
1248:
1222:
1127:
1105:
1078:
1055:
1014:
978:
865:
806:A sequence of vectors
791:
765:
728:
696:
567:
531:
506:
397:
321:
262:A sequence of vectors
211:
205:
172:
166:
10612:Row and column spaces
10557:Scalar multiplication
10444:"Linear independence"
10289:
10234:
10182:
10162:
9909:
9886:
9814:
9794:
9748:
9703:
9683:
9663:
9648:Two vector subspaces
9635:
9609:
9499:
9479:
9459:
9409:
9345:form a vector space.
9315:
9236:components such that
9167:
9137:
9135:{\displaystyle e^{t}}
9110:
9084:
9058:
9043:is not zero for some
9038:
9008:
8969:
8943:
8922:We need to show that
8917:
8887:
8823:
8765:
8745:
8717:
8697:
8667:
8665:{\displaystyle e^{t}}
8645:. Then the functions
8640:
8612:
8584:
8543:
8507:
8348:
8239:
8173:
8105:
7898:
7878:
7844:Natural basis vectors
7836:
7796:
7770:
7744:
7721:
7650:
7627:
7551:
7511:
7491:
7471:
7451:
7423:
7391:
7362:
7342:
7319:
7284:
7282:{\displaystyle (1,1)}
7245:
7174:
7141:
7027:
6941:
6912:
6884:
6855:
6808:
6806:{\displaystyle a_{3}}
6778:
6674:
6520:
6311:
6093:
5885:
5845:
5795:
5748:
5709:
5578:
5444:
5314:
5261:
5208:
5155:
5108:
5049:
4846:
4675:
4514:
4356:
4187:
4137:
4085:
4024:
4002:
3978:
3956:
3930:
3908:
3883:
3861:
3839:
3817:
3795:
3780:then at least one of
3775:
3745:
3707:
3677:
3647:
3604:
3577:
3545:
3519:
3489:
3453:
3423:
3401:
3371:
3335:
3305:
3283:
3231:
3205:
3171:
3138:
3118:
3096:
3072:
3039:
3019:
2997:
2968:
2946:
2924:
2902:
2873:
2833:
2802:
2764:
2744:
2715:
2686:
2665:(i.e. the case where
2660:
2640:
2580:
2530:
2494:
2298:
2228:
2168:so that consequently
2163:
2130:
2104:
2084:
2064:
2044:
2042:{\displaystyle a_{i}}
2017:
1984:
1944:
1906:
1886:
1836:
1814:
1715:
1668:
1639:
1607:
1578:
1549:
1520:
1491:
1462:
1431:
1402:
1371:
1342:
1313:
1278:
1249:
1221:
1128:
1106:
1079:
1056:
1015:
979:
866:
792:
766:
729:
697:
568:
532:
507:
398:
322:
206:
178:
167:
142:
76:"Linear independence"
10747:Gram–Schmidt process
10699:Gaussian elimination
10463:at WolframMathWorld.
10243:
10197:
10171:
9918:
9895:
9827:
9821:linearly independent
9803:
9757:
9716:
9710:linearly independent
9692:
9672:
9652:
9618:
9508:
9488:
9468:
9418:
9398:
9389:affinely independent
9358:Gaussian elimination
9243:
9146:
9119:
9093:
9082:{\displaystyle b=0.}
9067:
9047:
9017:
8978:
8967:{\displaystyle b=0.}
8952:
8926:
8903:
8838:
8777:
8754:
8734:
8706:
8676:
8649:
8629:
8601:
8552:
8519:
8363:
8251:
8189:
8117:
7914:
7887:
7852:
7813:
7779:
7753:
7733:
7662:
7639:
7563:
7540:
7500:
7480:
7460:
7440:
7412:
7371:
7351:
7331:
7293:
7261:
7186:
7163:
7042:
6971:
6921:
6901:
6864:
6817:
6790:
6696:
6539:
6326:
6108:
5896:
5862:
5804:
5757:
5721:
5593:
5459:
5326:
5270:
5223:
5164:
5117:
5070:
4861:
4690:
4532:
4371:
4199:
4146:
4093:
4044:
4011:
3989:
3965:
3943:
3917:
3895:
3870:
3848:
3826:
3804:
3784:
3754:
3721:
3686:
3656:
3613:
3586:
3556:
3528:
3524:is only possible if
3498:
3465:
3451:{\displaystyle c:=1}
3436:
3410:
3380:
3344:
3314:
3292:
3240:
3229:{\displaystyle c:=0}
3214:
3184:
3147:
3127:
3105:
3083:
3048:
3028:
3006:
2984:
2955:
2933:
2911:
2889:
2842:
2813:
2774:
2753:
2724:
2695:
2669:
2649:
2599:
2539:
2506:
2310:
2237:
2172:
2139:
2113:
2093:
2073:
2053:
2026:
1993:
1953:
1915:
1895:
1845:
1823:
1773:
1756:-dimensional space.
1677:
1673:are dependent since
1648:
1619:
1587:
1558:
1529:
1500:
1471:
1442:
1411:
1382:
1351:
1322:
1293:
1258:
1229:
1173:linearly independent
1154:linearly independent
1139:linearly independent
1115:
1088:
1066:
1024:
991:
882:
873:linearly independent
810:
790:{\displaystyle k=1.}
775:
738:
709:
580:
544:
519:
410:
345:
266:
230:linearly independent
183:
147:
61:improve this article
11805:Linear independence
11052:Diagonally dominant
10877:Numerical stability
10757:Multilinear algebra
10732:Inner product space
10582:Linear independence
10407:Functional Analysis
10058: for all
9633:{\displaystyle n+1}
9369:Affine independence
9108:{\displaystyle a=0}
8941:{\displaystyle a=0}
8625:of a real variable
7768:{\displaystyle m=n}
3652:This shows that if
2684:{\displaystyle k=1}
1819:is the zero vector
1724:Geographic location
11810:Matrix exponential
11800:Jordan normal form
11634:Fisher information
11505:Euclidean distance
11419:Totally unimodular
10587:Linear combination
10284:
10229:
10177:
10157:
10143:
9936:
9907:{\displaystyle i,}
9904:
9881:
9858:
9809:
9789:
9743:
9698:
9688:of a vector space
9678:
9658:
9630:
9604:
9593:
9592:
9547:
9546:
9504:augmented vectors
9494:
9474:
9454:
9404:
9394:Consider a set of
9385:affine combination
9381:affinely dependent
9310:
9162:
9132:
9105:
9079:
9053:
9033:
9003:
8964:
8938:
8915:{\displaystyle t.}
8912:
8882:
8818:
8760:
8740:
8712:
8692:
8662:
8635:
8607:
8579:
8538:
8502:
8343:
8234:
8183:
8168:
8100:
8098:
7893:
7873:
7831:
7791:
7765:
7739:
7716:
7645:
7622:
7546:
7506:
7486:
7466:
7446:
7418:
7386:
7367:coordinates, with
7357:
7337:
7314:
7279:
7240:
7169:
7136:
7127:
7087:
7022:
7013:
6936:
6907:
6879:
6850:
6803:
6773:
6669:
6660:
6615:
6575:
6515:
6506:
6463:
6409:
6306:
6297:
6254:
6200:
6088:
6079:
6015:
5951:
5880:
5840:
5790:
5743:
5704:
5695:
5666:
5626:
5573:
5564:
5535:
5495:
5439:
5430:
5401:
5359:
5309:
5256:
5203:
5150:
5103:
5044:
5035:
4977:
4934:
4894:
4841:
4832:
4803:
4749:
4670:
4661:
4632:
4578:
4509:
4500:
4471:
4417:
4351:
4342:
4313:
4274:
4232:
4182:
4132:
4080:
4019:
3997:
3973:
3951:
3925:
3903:
3878:
3856:
3834:
3812:
3790:
3770:
3740:
3702:
3672:
3642:
3599:
3572:
3540:
3514:
3484:
3448:
3418:
3396:
3366:
3330:
3300:
3278:
3226:
3200:
3166:
3133:
3113:
3091:
3067:
3034:
3014:
2992:
2963:
2941:
2919:
2897:
2868:
2828:
2797:
2759:
2739:
2710:
2681:
2655:
2635:
2575:
2525:
2489:
2293:
2223:
2158:
2125:
2099:
2079:
2059:
2039:
2012:
1979:
1939:
1901:
1881:
1831:
1809:
1743:linearly dependent
1739:linear combination
1710:
1663:
1634:
1602:
1573:
1544:
1515:
1486:
1457:
1426:
1397:
1366:
1337:
1308:
1273:
1244:
1223:
1214:Geometric examples
1177:linearly dependent
1162:linearly dependent
1123:
1104:{\textstyle a_{i}}
1101:
1074:
1051:
1010:
974:
861:
801:linear combination
787:
761:
724:
692:
563:
527:
502:
393:
336:linearly dependent
317:
242:linearly dependent
236:linear combination
212:
201:
173:
162:
11875:
11874:
11867:Category:Matrices
11739:Fuzzy associative
11629:Doubly stochastic
11337:Positive-definite
11017:Block tridiagonal
10918:
10917:
10785:Geometric algebra
10742:Kronecker product
10577:Linear projection
10562:Vector projection
10180:{\displaystyle X}
10167:The vector space
10080:
10059:
9921:
9843:
9812:{\displaystyle X}
9701:{\displaystyle X}
9681:{\displaystyle N}
9661:{\displaystyle M}
9497:{\displaystyle m}
9477:{\displaystyle n}
9407:{\displaystyle m}
9182:linear dependency
9056:{\displaystyle t}
8763:{\displaystyle b}
8743:{\displaystyle a}
8715:{\displaystyle V}
8638:{\displaystyle t}
8610:{\displaystyle V}
8181:
7896:{\displaystyle V}
7742:{\displaystyle m}
7648:{\displaystyle m}
7549:{\displaystyle m}
7509:{\displaystyle m}
7489:{\displaystyle A}
7469:{\displaystyle m}
7449:{\displaystyle n}
7421:{\displaystyle m}
7360:{\displaystyle n}
7340:{\displaystyle m}
7172:{\displaystyle A}
6910:{\displaystyle n}
3793:{\displaystyle c}
3632:
3597:
3136:{\displaystyle c}
3037:{\displaystyle c}
2762:{\displaystyle 1}
2658:{\displaystyle 1}
2102:{\displaystyle i}
2082:{\displaystyle j}
2062:{\displaystyle 0}
1904:{\displaystyle i}
1707:
1689:
1660:
1631:
1599:
1570:
1541:
1512:
1483:
1454:
1423:
1394:
1363:
1334:
1305:
1270:
1241:
675:
625:
338:, if there exist
214:In the theory of
137:
136:
129:
111:
18:Linear dependency
16:(Redirected from
11900:
11888:Abstract algebra
11862:List of matrices
11854:
11853:
11830:Row echelon form
11774:State transition
11703:Seidel adjacency
11585:Totally positive
11445:Alternating sign
11042:Complex Hadamard
10945:
10938:
10931:
10922:
10921:
10908:
10907:
10790:Exterior algebra
10727:Hadamard product
10644:
10632:Linear equations
10503:
10496:
10489:
10480:
10479:
10457:
10430:
10428:
10402:
10393:
10392:
10360:
10354:
10353:
10335:
10329:
10320:
10293:
10291:
10290:
10285:
10274:
10273:
10255:
10254:
10238:
10236:
10235:
10230:
10228:
10227:
10209:
10208:
10187:is said to be a
10186:
10184:
10183:
10178:
10166:
10164:
10163:
10158:
10153:
10152:
10142:
10070:
10069:
10060:
10057:
10055:
10054:
10042:
10041:
10029:
10028:
10010:
10009:
9991:
9990:
9966:
9965:
9956:
9955:
9946:
9945:
9935:
9913:
9911:
9910:
9905:
9891:for every index
9890:
9888:
9887:
9882:
9868:
9867:
9857:
9839:
9838:
9818:
9816:
9815:
9810:
9799:of subspaces of
9798:
9796:
9795:
9790:
9788:
9787:
9769:
9768:
9752:
9750:
9749:
9744:
9707:
9705:
9704:
9699:
9687:
9685:
9684:
9679:
9667:
9665:
9664:
9659:
9639:
9637:
9636:
9631:
9613:
9611:
9610:
9605:
9603:
9599:
9598:
9594:
9589:
9588:
9583:
9552:
9548:
9543:
9542:
9537:
9503:
9501:
9500:
9495:
9483:
9481:
9480:
9475:
9463:
9461:
9460:
9455:
9453:
9452:
9447:
9432:
9431:
9426:
9413:
9411:
9410:
9405:
9344:
9326:
9319:
9317:
9316:
9311:
9306:
9298:
9297:
9292:
9286:
9285:
9267:
9266:
9261:
9255:
9254:
9232:
9228:
9205:
9171:
9169:
9168:
9163:
9161:
9160:
9141:
9139:
9138:
9133:
9131:
9130:
9114:
9112:
9111:
9106:
9089:It follows that
9088:
9086:
9085:
9080:
9062:
9060:
9059:
9054:
9042:
9040:
9039:
9034:
9032:
9031:
9012:
9010:
9009:
9004:
8996:
8995:
8973:
8971:
8970:
8965:
8947:
8945:
8944:
8939:
8921:
8919:
8918:
8913:
8891:
8889:
8888:
8883:
8875:
8874:
8853:
8852:
8827:
8825:
8824:
8819:
8811:
8810:
8792:
8791:
8769:
8767:
8766:
8761:
8749:
8747:
8746:
8741:
8721:
8719:
8718:
8713:
8701:
8699:
8698:
8693:
8691:
8690:
8671:
8669:
8668:
8663:
8661:
8660:
8644:
8642:
8641:
8636:
8616:
8614:
8613:
8608:
8588:
8586:
8585:
8580:
8547:
8545:
8544:
8539:
8531:
8530:
8511:
8509:
8508:
8503:
8498:
8494:
8493:
8492:
8474:
8473:
8461:
8460:
8443:
8442:
8437:
8431:
8430:
8412:
8411:
8406:
8400:
8399:
8387:
8386:
8381:
8375:
8374:
8352:
8350:
8349:
8344:
8339:
8331:
8330:
8325:
8319:
8318:
8300:
8299:
8294:
8288:
8287:
8275:
8274:
8269:
8263:
8262:
8243:
8241:
8240:
8235:
8233:
8232:
8214:
8213:
8201:
8200:
8177:
8175:
8174:
8169:
8167:
8166:
8161:
8146:
8145:
8140:
8131:
8130:
8125:
8109:
8107:
8106:
8101:
8099:
8052:
8051:
8046:
8032:
7988:
7987:
7982:
7932:
7931:
7926:
7902:
7900:
7899:
7894:
7882:
7880:
7879:
7874:
7872:
7871:
7866:
7840:
7838:
7837:
7832:
7827:
7826:
7821:
7800:
7798:
7797:
7792:
7774:
7772:
7771:
7766:
7748:
7746:
7745:
7740:
7725:
7723:
7722:
7717:
7709:
7708:
7704:
7703:
7685:
7684:
7654:
7652:
7651:
7646:
7631:
7629:
7628:
7623:
7618:
7607:
7606:
7602:
7601:
7583:
7582:
7555:
7553:
7552:
7547:
7535:
7515:
7513:
7512:
7507:
7495:
7493:
7492:
7487:
7475:
7473:
7472:
7467:
7455:
7453:
7452:
7447:
7427:
7425:
7424:
7419:
7395:
7393:
7392:
7387:
7366:
7364:
7363:
7358:
7346:
7344:
7343:
7338:
7323:
7321:
7320:
7315:
7288:
7286:
7285:
7280:
7249:
7247:
7246:
7241:
7178:
7176:
7175:
7170:
7158:
7145:
7143:
7142:
7137:
7132:
7131:
7124:
7123:
7110:
7109:
7092:
7091:
7031:
7029:
7028:
7023:
7018:
7017:
6945:
6943:
6942:
6937:
6935:
6934:
6929:
6916:
6914:
6913:
6908:
6888:
6886:
6885:
6880:
6878:
6877:
6872:
6859:
6857:
6856:
6851:
6846:
6845:
6840:
6831:
6830:
6825:
6812:
6810:
6809:
6804:
6802:
6801:
6782:
6780:
6779:
6774:
6766:
6761:
6760:
6748:
6747:
6732:
6727:
6726:
6708:
6707:
6678:
6676:
6675:
6670:
6665:
6664:
6636:
6635:
6620:
6619:
6612:
6611:
6598:
6597:
6580:
6579:
6524:
6522:
6521:
6516:
6511:
6510:
6468:
6467:
6460:
6459:
6446:
6445:
6432:
6431:
6414:
6413:
6315:
6313:
6312:
6307:
6302:
6301:
6259:
6258:
6251:
6250:
6237:
6236:
6223:
6222:
6205:
6204:
6097:
6095:
6094:
6089:
6084:
6083:
6035:
6034:
6029:
6020:
6019:
5971:
5970:
5965:
5956:
5955:
5910:
5909:
5904:
5889:
5887:
5886:
5881:
5876:
5875:
5870:
5849:
5847:
5846:
5841:
5818:
5817:
5812:
5799:
5797:
5796:
5791:
5771:
5770:
5765:
5752:
5750:
5749:
5744:
5733:
5732:
5717:This shows that
5713:
5711:
5710:
5705:
5700:
5699:
5671:
5670:
5663:
5662:
5649:
5648:
5631:
5630:
5582:
5580:
5579:
5574:
5569:
5568:
5540:
5539:
5532:
5531:
5518:
5517:
5500:
5499:
5448:
5446:
5445:
5440:
5435:
5434:
5406:
5405:
5377:
5376:
5364:
5363:
5338:
5337:
5318:
5316:
5315:
5310:
5284:
5283:
5278:
5265:
5263:
5262:
5257:
5237:
5236:
5231:
5212:
5210:
5209:
5204:
5178:
5177:
5172:
5159:
5157:
5156:
5151:
5131:
5130:
5125:
5112:
5110:
5109:
5104:
5084:
5083:
5078:
5066:exist such that
5053:
5051:
5050:
5045:
5040:
5039:
5029:
5014:
4998:
4997:
4982:
4981:
4974:
4973:
4960:
4959:
4939:
4938:
4931:
4930:
4917:
4916:
4899:
4898:
4850:
4848:
4847:
4842:
4837:
4836:
4808:
4807:
4800:
4799:
4786:
4785:
4772:
4771:
4754:
4753:
4743:
4718:
4679:
4677:
4676:
4671:
4666:
4665:
4637:
4636:
4629:
4628:
4615:
4614:
4601:
4600:
4583:
4582:
4518:
4516:
4515:
4510:
4505:
4504:
4476:
4475:
4468:
4467:
4454:
4453:
4440:
4439:
4422:
4421:
4360:
4358:
4357:
4352:
4347:
4346:
4318:
4317:
4292:
4291:
4279:
4278:
4250:
4249:
4237:
4236:
4211:
4210:
4191:
4189:
4188:
4183:
4160:
4159:
4154:
4141:
4139:
4138:
4133:
4107:
4106:
4101:
4089:
4087:
4086:
4081:
4058:
4057:
4052:
4028:
4026:
4025:
4020:
4018:
4006:
4004:
4003:
3998:
3996:
3982:
3980:
3979:
3974:
3972:
3960:
3958:
3957:
3952:
3950:
3934:
3932:
3931:
3926:
3924:
3912:
3910:
3909:
3904:
3902:
3887:
3885:
3884:
3879:
3877:
3865:
3863:
3862:
3857:
3855:
3843:
3841:
3840:
3835:
3833:
3821:
3819:
3818:
3813:
3811:
3799:
3797:
3796:
3791:
3779:
3777:
3776:
3771:
3769:
3761:
3749:
3747:
3746:
3741:
3739:
3728:
3717:dependent). If
3711:
3709:
3708:
3703:
3701:
3693:
3681:
3679:
3678:
3673:
3671:
3663:
3651:
3649:
3648:
3643:
3638:
3633:
3625:
3620:
3608:
3606:
3605:
3600:
3598:
3590:
3581:
3579:
3578:
3573:
3571:
3563:
3549:
3547:
3546:
3541:
3523:
3521:
3520:
3515:
3513:
3505:
3493:
3491:
3490:
3485:
3483:
3472:
3457:
3455:
3454:
3449:
3427:
3425:
3424:
3419:
3417:
3405:
3403:
3402:
3397:
3395:
3387:
3375:
3373:
3372:
3367:
3362:
3351:
3339:
3337:
3336:
3331:
3329:
3321:
3309:
3307:
3306:
3301:
3299:
3287:
3285:
3284:
3279:
3277:
3269:
3261:
3250:
3235:
3233:
3232:
3227:
3210:then by setting
3209:
3207:
3206:
3201:
3199:
3191:
3175:
3173:
3172:
3167:
3165:
3154:
3142:
3140:
3139:
3134:
3122:
3120:
3119:
3114:
3112:
3100:
3098:
3097:
3092:
3090:
3076:
3074:
3073:
3068:
3066:
3055:
3043:
3041:
3040:
3035:
3023:
3021:
3020:
3015:
3013:
3001:
2999:
2998:
2993:
2991:
2972:
2970:
2969:
2964:
2962:
2950:
2948:
2947:
2942:
2940:
2928:
2926:
2925:
2920:
2918:
2906:
2904:
2903:
2898:
2896:
2877:
2875:
2874:
2869:
2864:
2856:
2855:
2850:
2837:
2835:
2834:
2829:
2827:
2826:
2821:
2808:
2806:
2804:
2803:
2798:
2796:
2788:
2787:
2782:
2768:
2766:
2765:
2760:
2748:
2746:
2745:
2740:
2738:
2737:
2732:
2719:
2717:
2716:
2711:
2709:
2708:
2703:
2690:
2688:
2687:
2682:
2664:
2662:
2661:
2656:
2644:
2642:
2641:
2636:
2634:
2633:
2628:
2613:
2612:
2607:
2584:
2582:
2581:
2576:
2574:
2573:
2568:
2553:
2552:
2547:
2534:
2532:
2531:
2526:
2518:
2517:
2498:
2496:
2495:
2490:
2485:
2477:
2472:
2471:
2459:
2458:
2453:
2447:
2446:
2434:
2420:
2412:
2411:
2406:
2400:
2399:
2387:
2373:
2365:
2364:
2359:
2353:
2352:
2334:
2333:
2328:
2322:
2321:
2302:
2300:
2299:
2294:
2292:
2291:
2286:
2280:
2279:
2261:
2260:
2255:
2249:
2248:
2233:). Simplifying
2232:
2230:
2229:
2224:
2222:
2214:
2213:
2208:
2196:
2195:
2190:
2184:
2183:
2167:
2165:
2164:
2159:
2151:
2150:
2134:
2132:
2131:
2126:
2108:
2106:
2105:
2100:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2048:
2046:
2045:
2040:
2038:
2037:
2021:
2019:
2018:
2013:
2005:
2004:
1988:
1986:
1985:
1980:
1975:
1967:
1966:
1961:
1948:
1946:
1945:
1940:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1880:
1879:
1874:
1859:
1858:
1853:
1841:then the vector
1840:
1838:
1837:
1832:
1830:
1818:
1816:
1815:
1810:
1808:
1807:
1802:
1787:
1786:
1781:
1755:
1751:
1719:
1717:
1716:
1711:
1709:
1708:
1700:
1691:
1690:
1682:
1672:
1670:
1669:
1664:
1662:
1661:
1653:
1643:
1641:
1640:
1635:
1633:
1632:
1624:
1611:
1609:
1608:
1603:
1601:
1600:
1592:
1582:
1580:
1579:
1574:
1572:
1571:
1563:
1553:
1551:
1550:
1545:
1543:
1542:
1534:
1524:
1522:
1521:
1516:
1514:
1513:
1505:
1495:
1493:
1492:
1487:
1485:
1484:
1476:
1466:
1464:
1463:
1458:
1456:
1455:
1447:
1435:
1433:
1432:
1427:
1425:
1424:
1416:
1406:
1404:
1403:
1398:
1396:
1395:
1387:
1375:
1373:
1372:
1367:
1365:
1364:
1356:
1346:
1344:
1343:
1338:
1336:
1335:
1327:
1317:
1315:
1314:
1309:
1307:
1306:
1298:
1282:
1280:
1279:
1274:
1272:
1271:
1263:
1253:
1251:
1250:
1245:
1243:
1242:
1234:
1209:
1197:
1132:
1130:
1129:
1124:
1122:
1110:
1108:
1107:
1102:
1100:
1099:
1083:
1081:
1080:
1075:
1073:
1060:
1058:
1057:
1052:
1019:
1017:
1016:
1011:
1003:
1002:
983:
981:
980:
975:
970:
962:
961:
956:
950:
949:
931:
930:
925:
919:
918:
906:
905:
900:
894:
893:
870:
868:
867:
862:
860:
859:
854:
839:
838:
833:
824:
823:
818:
796:
794:
793:
788:
770:
768:
767:
762:
760:
752:
751:
746:
733:
731:
730:
725:
701:
699:
698:
693:
688:
687:
682:
676:
674:
673:
664:
663:
662:
649:
638:
637:
632:
626:
624:
623:
614:
613:
612:
599:
594:
593:
588:
572:
570:
569:
564:
556:
555:
536:
534:
533:
528:
526:
511:
509:
508:
503:
498:
490:
489:
484:
478:
477:
459:
458:
453:
447:
446:
434:
433:
428:
422:
421:
402:
400:
399:
394:
389:
388:
370:
369:
357:
356:
333:
326:
324:
323:
318:
316:
315:
310:
295:
294:
289:
280:
279:
274:
244:
243:
232:
231:
210:
208:
207:
202:
197:
196:
191:
171:
169:
168:
163:
161:
160:
155:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
11908:
11907:
11903:
11902:
11901:
11899:
11898:
11897:
11878:
11877:
11876:
11871:
11848:
11839:
11788:
11712:
11658:
11594:
11428:
11346:
11292:
11231:
11032:Centrosymmetric
10955:
10949:
10919:
10914:
10896:
10858:
10814:
10751:
10703:
10645:
10636:
10602:Change of basis
10592:Multilinear map
10530:
10512:
10507:
10475:at KhanAcademy.
10442:
10439:
10434:
10433:
10417:
10403:
10396:
10382:
10372:Matching Theory
10361:
10357:
10350:
10336:
10332:
10321:
10317:
10312:
10299:
10269:
10265:
10250:
10246:
10244:
10241:
10240:
10223:
10219:
10204:
10200:
10198:
10195:
10194:
10172:
10169:
10168:
10148:
10144:
10084:
10065:
10064:
10056:
10050:
10046:
10037:
10033:
10024:
10020:
9999:
9995:
9980:
9976:
9961:
9957:
9951:
9950:
9941:
9937:
9925:
9919:
9916:
9915:
9896:
9893:
9892:
9863:
9859:
9847:
9834:
9830:
9828:
9825:
9824:
9819:are said to be
9804:
9801:
9800:
9783:
9779:
9764:
9760:
9758:
9755:
9754:
9717:
9714:
9713:
9708:are said to be
9693:
9690:
9689:
9673:
9670:
9669:
9653:
9650:
9649:
9646:
9619:
9616:
9615:
9591:
9590:
9584:
9579:
9578:
9575:
9574:
9566:
9562:
9545:
9544:
9538:
9533:
9532:
9529:
9528:
9520:
9516:
9515:
9511:
9509:
9506:
9505:
9489:
9486:
9485:
9469:
9466:
9465:
9448:
9443:
9442:
9427:
9422:
9421:
9419:
9416:
9415:
9399:
9396:
9395:
9377:
9371:
9366:
9364:Generalizations
9343:
9334:
9328:
9324:
9302:
9293:
9288:
9287:
9281:
9277:
9262:
9257:
9256:
9250:
9246:
9244:
9241:
9240:
9230:
9226:
9217:
9210:
9204:
9195:
9189:
9186:linear relation
9178:
9153:
9149:
9147:
9144:
9143:
9126:
9122:
9120:
9117:
9116:
9094:
9091:
9090:
9068:
9065:
9064:
9048:
9045:
9044:
9024:
9020:
9018:
9015:
9014:
8988:
8984:
8979:
8976:
8975:
8953:
8950:
8949:
8927:
8924:
8923:
8904:
8901:
8900:
8867:
8863:
8848:
8844:
8839:
8836:
8835:
8803:
8799:
8787:
8783:
8778:
8775:
8774:
8755:
8752:
8751:
8735:
8732:
8731:
8728:
8707:
8704:
8703:
8683:
8679:
8677:
8674:
8673:
8656:
8652:
8650:
8647:
8646:
8630:
8627:
8626:
8602:
8599:
8598:
8595:
8590:
8553:
8550:
8549:
8526:
8522:
8520:
8517:
8516:
8488:
8484:
8469:
8465:
8456:
8452:
8451:
8447:
8438:
8433:
8432:
8426:
8422:
8407:
8402:
8401:
8395:
8391:
8382:
8377:
8376:
8370:
8366:
8364:
8361:
8360:
8335:
8326:
8321:
8320:
8314:
8310:
8295:
8290:
8289:
8283:
8279:
8270:
8265:
8264:
8258:
8254:
8252:
8249:
8248:
8228:
8224:
8209:
8205:
8196:
8192:
8190:
8187:
8186:
8162:
8157:
8156:
8141:
8136:
8135:
8126:
8121:
8120:
8118:
8115:
8114:
8097:
8096:
8058:
8053:
8047:
8042:
8041:
8038:
8037:
8030:
8029:
7994:
7989:
7983:
7978:
7977:
7974:
7973:
7938:
7933:
7927:
7922:
7921:
7917:
7915:
7912:
7911:
7903:, known as the
7888:
7885:
7884:
7867:
7862:
7861:
7853:
7850:
7849:
7846:
7822:
7817:
7816:
7814:
7811:
7810:
7807:
7780:
7777:
7776:
7754:
7751:
7750:
7749:rows. (In case
7734:
7731:
7730:
7699:
7695:
7680:
7676:
7672:
7668:
7663:
7660:
7659:
7640:
7637:
7636:
7614:
7597:
7593:
7578:
7574:
7570:
7566:
7564:
7561:
7560:
7541:
7538:
7537:
7536:is any list of
7533:
7524:
7517:
7501:
7498:
7497:
7481:
7478:
7477:
7461:
7458:
7457:
7441:
7438:
7437:
7413:
7410:
7409:
7372:
7369:
7368:
7352:
7349:
7348:
7332:
7329:
7328:
7294:
7291:
7290:
7262:
7259:
7258:
7187:
7184:
7183:
7164:
7161:
7160:
7150:
7126:
7125:
7119:
7115:
7112:
7111:
7105:
7101:
7094:
7093:
7086:
7085:
7080:
7074:
7073:
7065:
7055:
7054:
7043:
7040:
7039:
7012:
7011:
7006:
7000:
6999:
6991:
6981:
6980:
6972:
6969:
6968:
6930:
6925:
6924:
6922:
6919:
6918:
6902:
6899:
6898:
6895:
6873:
6868:
6867:
6865:
6862:
6861:
6841:
6836:
6835:
6826:
6821:
6820:
6818:
6815:
6814:
6797:
6793:
6791:
6788:
6787:
6762:
6756:
6752:
6743:
6739:
6728:
6722:
6718:
6703:
6699:
6697:
6694:
6693:
6688:
6659:
6658:
6652:
6651:
6638:
6637:
6631:
6627:
6614:
6613:
6607:
6603:
6600:
6599:
6593:
6589:
6582:
6581:
6574:
6573:
6565:
6559:
6558:
6553:
6543:
6542:
6540:
6537:
6536:
6531:
6505:
6504:
6498:
6497:
6491:
6490:
6484:
6483:
6473:
6472:
6462:
6461:
6455:
6451:
6448:
6447:
6441:
6437:
6434:
6433:
6427:
6423:
6416:
6415:
6408:
6407:
6402:
6397:
6391:
6390:
6385:
6380:
6374:
6373:
6368:
6360:
6354:
6353:
6345:
6340:
6330:
6329:
6327:
6324:
6323:
6296:
6295:
6289:
6288:
6282:
6281:
6275:
6274:
6264:
6263:
6253:
6252:
6246:
6242:
6239:
6238:
6232:
6228:
6225:
6224:
6218:
6214:
6207:
6206:
6199:
6198:
6190:
6182:
6173:
6172:
6167:
6159:
6153:
6152:
6147:
6142:
6136:
6135:
6127:
6122:
6112:
6111:
6109:
6106:
6105:
6078:
6077:
6068:
6067:
6061:
6060:
6054:
6053:
6040:
6039:
6030:
6025:
6024:
6014:
6013:
6004:
6003:
5994:
5993:
5987:
5986:
5976:
5975:
5966:
5961:
5960:
5950:
5949:
5940:
5939:
5933:
5932:
5926:
5925:
5915:
5914:
5905:
5900:
5899:
5897:
5894:
5893:
5871:
5866:
5865:
5863:
5860:
5859:
5856:
5813:
5808:
5807:
5805:
5802:
5801:
5766:
5761:
5760:
5758:
5755:
5754:
5728:
5724:
5722:
5719:
5718:
5694:
5693:
5687:
5686:
5676:
5675:
5665:
5664:
5658:
5654:
5651:
5650:
5644:
5640:
5633:
5632:
5625:
5624:
5619:
5613:
5612:
5607:
5597:
5596:
5594:
5591:
5590:
5563:
5562:
5556:
5555:
5545:
5544:
5534:
5533:
5527:
5523:
5520:
5519:
5513:
5509:
5502:
5501:
5494:
5493:
5488:
5482:
5481:
5473:
5463:
5462:
5460:
5457:
5456:
5429:
5428:
5422:
5421:
5411:
5410:
5400:
5399:
5393:
5392:
5379:
5378:
5372:
5368:
5358:
5357:
5351:
5350:
5340:
5339:
5333:
5329:
5327:
5324:
5323:
5279:
5274:
5273:
5271:
5268:
5267:
5232:
5227:
5226:
5224:
5221:
5220:
5173:
5168:
5167:
5165:
5162:
5161:
5126:
5121:
5120:
5118:
5115:
5114:
5079:
5074:
5073:
5071:
5068:
5067:
5065:
5034:
5033:
5025:
5019:
5018:
5010:
5000:
4999:
4993:
4989:
4976:
4975:
4969:
4965:
4962:
4961:
4955:
4951:
4944:
4943:
4933:
4932:
4926:
4922:
4919:
4918:
4912:
4908:
4901:
4900:
4893:
4892:
4887:
4881:
4880:
4875:
4865:
4864:
4862:
4859:
4858:
4831:
4830:
4824:
4823:
4813:
4812:
4802:
4801:
4795:
4791:
4788:
4787:
4781:
4777:
4774:
4773:
4767:
4763:
4756:
4755:
4748:
4747:
4739:
4734:
4729:
4723:
4722:
4714:
4709:
4704:
4694:
4693:
4691:
4688:
4687:
4660:
4659:
4653:
4652:
4642:
4641:
4631:
4630:
4624:
4620:
4617:
4616:
4610:
4606:
4603:
4602:
4596:
4592:
4585:
4584:
4577:
4576:
4571:
4566:
4560:
4559:
4554:
4546:
4536:
4535:
4533:
4530:
4529:
4499:
4498:
4492:
4491:
4481:
4480:
4470:
4469:
4463:
4459:
4456:
4455:
4449:
4445:
4442:
4441:
4435:
4431:
4424:
4423:
4416:
4415:
4410:
4405:
4399:
4398:
4393:
4385:
4375:
4374:
4372:
4369:
4368:
4341:
4340:
4334:
4333:
4323:
4322:
4312:
4311:
4305:
4304:
4294:
4293:
4287:
4283:
4273:
4272:
4266:
4265:
4252:
4251:
4245:
4241:
4231:
4230:
4224:
4223:
4213:
4212:
4206:
4202:
4200:
4197:
4196:
4155:
4150:
4149:
4147:
4144:
4143:
4102:
4097:
4096:
4094:
4091:
4090:
4053:
4048:
4047:
4045:
4042:
4041:
4035:
4014:
4012:
4009:
4008:
3992:
3990:
3987:
3986:
3968:
3966:
3963:
3962:
3946:
3944:
3941:
3940:
3920:
3918:
3915:
3914:
3898:
3896:
3893:
3892:
3873:
3871:
3868:
3867:
3851:
3849:
3846:
3845:
3829:
3827:
3824:
3823:
3807:
3805:
3802:
3801:
3785:
3782:
3781:
3765:
3757:
3755:
3752:
3751:
3735:
3724:
3722:
3719:
3718:
3697:
3689:
3687:
3684:
3683:
3667:
3659:
3657:
3654:
3653:
3634:
3624:
3616:
3614:
3611:
3610:
3589:
3587:
3584:
3583:
3567:
3559:
3557:
3554:
3553:
3529:
3526:
3525:
3509:
3501:
3499:
3496:
3495:
3479:
3468:
3466:
3463:
3462:
3437:
3434:
3433:
3413:
3411:
3408:
3407:
3391:
3383:
3381:
3378:
3377:
3358:
3347:
3345:
3342:
3341:
3325:
3317:
3315:
3312:
3311:
3295:
3293:
3290:
3289:
3273:
3265:
3257:
3246:
3241:
3238:
3237:
3215:
3212:
3211:
3195:
3187:
3185:
3182:
3181:
3161:
3150:
3148:
3145:
3144:
3128:
3125:
3124:
3108:
3106:
3103:
3102:
3086:
3084:
3081:
3080:
3062:
3051:
3049:
3046:
3045:
3029:
3026:
3025:
3009:
3007:
3004:
3003:
2987:
2985:
2982:
2981:
2958:
2956:
2953:
2952:
2936:
2934:
2931:
2930:
2914:
2912:
2909:
2908:
2892:
2890:
2887:
2886:
2883:
2860:
2851:
2846:
2845:
2843:
2840:
2839:
2822:
2817:
2816:
2814:
2811:
2810:
2792:
2783:
2778:
2777:
2775:
2772:
2771:
2770:
2754:
2751:
2750:
2733:
2728:
2727:
2725:
2722:
2721:
2704:
2699:
2698:
2696:
2693:
2692:
2670:
2667:
2666:
2650:
2647:
2646:
2629:
2624:
2623:
2608:
2603:
2602:
2600:
2597:
2596:
2569:
2564:
2563:
2548:
2543:
2542:
2540:
2537:
2536:
2513:
2509:
2507:
2504:
2503:
2481:
2473:
2467:
2463:
2454:
2449:
2448:
2442:
2438:
2430:
2416:
2407:
2402:
2401:
2395:
2391:
2383:
2369:
2360:
2355:
2354:
2348:
2344:
2329:
2324:
2323:
2317:
2313:
2311:
2308:
2307:
2287:
2282:
2281:
2275:
2271:
2256:
2251:
2250:
2244:
2240:
2238:
2235:
2234:
2218:
2209:
2204:
2203:
2191:
2186:
2185:
2179:
2175:
2173:
2170:
2169:
2146:
2142:
2140:
2137:
2136:
2114:
2111:
2110:
2094:
2091:
2090:
2074:
2071:
2070:
2054:
2051:
2050:
2033:
2029:
2027:
2024:
2023:
2000:
1996:
1994:
1991:
1990:
1971:
1962:
1957:
1956:
1954:
1951:
1950:
1916:
1913:
1912:
1896:
1893:
1892:
1875:
1870:
1869:
1854:
1849:
1848:
1846:
1843:
1842:
1826:
1824:
1821:
1820:
1803:
1798:
1797:
1782:
1777:
1776:
1774:
1771:
1770:
1767:
1765:The zero vector
1762:
1753:
1749:
1726:
1699:
1698:
1681:
1680:
1678:
1675:
1674:
1652:
1651:
1649:
1646:
1645:
1623:
1622:
1620:
1617:
1616:
1591:
1590:
1588:
1585:
1584:
1562:
1561:
1559:
1556:
1555:
1533:
1532:
1530:
1527:
1526:
1504:
1503:
1501:
1498:
1497:
1475:
1474:
1472:
1469:
1468:
1446:
1445:
1443:
1440:
1439:
1415:
1414:
1412:
1409:
1408:
1386:
1385:
1383:
1380:
1379:
1355:
1354:
1352:
1349:
1348:
1326:
1325:
1323:
1320:
1319:
1297:
1296:
1294:
1291:
1290:
1262:
1261:
1259:
1256:
1255:
1233:
1232:
1230:
1227:
1226:
1216:
1199:
1195:
1150:
1118:
1116:
1113:
1112:
1095:
1091:
1089:
1086:
1085:
1069:
1067:
1064:
1063:
1025:
1022:
1021:
998:
994:
992:
989:
988:
966:
957:
952:
951:
945:
941:
926:
921:
920:
914:
910:
901:
896:
895:
889:
885:
883:
880:
879:
855:
850:
849:
834:
829:
828:
819:
814:
813:
811:
808:
807:
803:of the others.
776:
773:
772:
756:
747:
742:
741:
739:
736:
735:
710:
707:
706:
683:
678:
677:
669:
665:
658:
654:
650:
648:
633:
628:
627:
619:
615:
608:
604:
600:
598:
589:
584:
583:
581:
578:
577:
551:
547:
545:
542:
541:
522:
520:
517:
516:
494:
485:
480:
479:
473:
469:
454:
449:
448:
442:
438:
429:
424:
423:
417:
413:
411:
408:
407:
384:
380:
365:
361:
352:
348:
346:
343:
342:
331:
311:
306:
305:
290:
285:
284:
275:
270:
269:
267:
264:
263:
260:
241:
240:
229:
228:
192:
187:
186:
184:
181:
180:
156:
151:
150:
148:
145:
144:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
11906:
11896:
11895:
11893:Linear algebra
11890:
11873:
11872:
11870:
11869:
11864:
11859:
11844:
11841:
11840:
11838:
11837:
11832:
11827:
11822:
11820:Perfect matrix
11817:
11812:
11807:
11802:
11796:
11794:
11790:
11789:
11787:
11786:
11781:
11776:
11771:
11766:
11761:
11756:
11751:
11746:
11741:
11736:
11731:
11726:
11720:
11718:
11714:
11713:
11711:
11710:
11705:
11700:
11695:
11690:
11685:
11680:
11675:
11669:
11667:
11660:
11659:
11657:
11656:
11651:
11646:
11641:
11636:
11631:
11626:
11621:
11616:
11611:
11605:
11603:
11596:
11595:
11593:
11592:
11590:Transformation
11587:
11582:
11577:
11572:
11567:
11562:
11557:
11552:
11547:
11542:
11537:
11532:
11527:
11522:
11517:
11512:
11507:
11502:
11497:
11492:
11487:
11482:
11477:
11472:
11467:
11462:
11457:
11452:
11447:
11442:
11436:
11434:
11430:
11429:
11427:
11426:
11421:
11416:
11411:
11406:
11401:
11396:
11391:
11386:
11381:
11376:
11367:
11361:
11359:
11348:
11347:
11345:
11344:
11339:
11334:
11329:
11327:Diagonalizable
11324:
11319:
11314:
11309:
11303:
11301:
11297:Conditions on
11294:
11293:
11291:
11290:
11285:
11280:
11275:
11270:
11265:
11260:
11255:
11250:
11245:
11239:
11237:
11233:
11232:
11230:
11229:
11224:
11219:
11214:
11209:
11204:
11199:
11194:
11189:
11184:
11179:
11177:Skew-symmetric
11174:
11172:Skew-Hermitian
11169:
11164:
11159:
11154:
11149:
11144:
11139:
11134:
11129:
11124:
11119:
11114:
11109:
11104:
11099:
11094:
11089:
11084:
11079:
11074:
11069:
11064:
11059:
11054:
11049:
11044:
11039:
11034:
11029:
11024:
11019:
11014:
11009:
11007:Block-diagonal
11004:
10999:
10994:
10989:
10984:
10982:Anti-symmetric
10979:
10977:Anti-Hermitian
10974:
10969:
10963:
10961:
10957:
10956:
10948:
10947:
10940:
10933:
10925:
10916:
10915:
10913:
10912:
10901:
10898:
10897:
10895:
10894:
10889:
10884:
10879:
10874:
10872:Floating-point
10868:
10866:
10860:
10859:
10857:
10856:
10854:Tensor product
10851:
10846:
10841:
10839:Function space
10836:
10831:
10825:
10823:
10816:
10815:
10813:
10812:
10807:
10802:
10797:
10792:
10787:
10782:
10777:
10775:Triple product
10772:
10767:
10761:
10759:
10753:
10752:
10750:
10749:
10744:
10739:
10734:
10729:
10724:
10719:
10713:
10711:
10705:
10704:
10702:
10701:
10696:
10691:
10689:Transformation
10686:
10681:
10679:Multiplication
10676:
10671:
10666:
10661:
10655:
10653:
10647:
10646:
10639:
10637:
10635:
10634:
10629:
10624:
10619:
10614:
10609:
10604:
10599:
10594:
10589:
10584:
10579:
10574:
10569:
10564:
10559:
10554:
10549:
10544:
10538:
10536:
10535:Basic concepts
10532:
10531:
10529:
10528:
10523:
10517:
10514:
10513:
10510:Linear algebra
10506:
10505:
10498:
10491:
10483:
10477:
10476:
10470:
10464:
10458:
10438:
10437:External links
10435:
10432:
10431:
10416:978-0486402512
10415:
10394:
10380:
10368:Plummer, M. D.
10364:Lovász, László
10355:
10348:
10340:Linear Algebra
10330:
10325:Linear Algebra
10322:G. E. Shilov,
10314:
10313:
10311:
10308:
10307:
10306:
10298:
10295:
10283:
10280:
10277:
10272:
10268:
10264:
10261:
10258:
10253:
10249:
10226:
10222:
10218:
10215:
10212:
10207:
10203:
10192:
10176:
10156:
10151:
10147:
10141:
10138:
10135:
10132:
10129:
10126:
10123:
10120:
10117:
10114:
10111:
10108:
10105:
10102:
10099:
10096:
10093:
10090:
10087:
10083:
10079:
10076:
10073:
10068:
10063:
10053:
10049:
10045:
10040:
10036:
10032:
10027:
10023:
10019:
10016:
10013:
10008:
10005:
10002:
9998:
9994:
9989:
9986:
9983:
9979:
9975:
9972:
9969:
9964:
9960:
9954:
9949:
9944:
9940:
9934:
9931:
9928:
9924:
9903:
9900:
9880:
9877:
9874:
9871:
9866:
9862:
9856:
9853:
9850:
9846:
9842:
9837:
9833:
9822:
9808:
9786:
9782:
9778:
9775:
9772:
9767:
9763:
9742:
9739:
9736:
9733:
9730:
9727:
9724:
9721:
9711:
9697:
9677:
9657:
9645:
9642:
9629:
9626:
9623:
9602:
9597:
9587:
9582:
9577:
9576:
9573:
9570:
9569:
9565:
9561:
9558:
9555:
9551:
9541:
9536:
9531:
9530:
9527:
9524:
9523:
9519:
9514:
9493:
9473:
9451:
9446:
9441:
9438:
9435:
9430:
9425:
9403:
9370:
9367:
9365:
9362:
9339:
9332:
9321:
9320:
9309:
9305:
9301:
9296:
9291:
9284:
9280:
9276:
9273:
9270:
9265:
9260:
9253:
9249:
9222:
9215:
9200:
9193:
9188:among vectors
9177:
9174:
9159:
9156:
9152:
9129:
9125:
9104:
9101:
9098:
9078:
9075:
9072:
9052:
9030:
9027:
9023:
9002:
8999:
8994:
8991:
8987:
8983:
8963:
8960:
8957:
8937:
8934:
8931:
8911:
8908:
8898:
8893:
8892:
8881:
8878:
8873:
8870:
8866:
8862:
8859:
8856:
8851:
8847:
8843:
8829:
8828:
8817:
8814:
8809:
8806:
8802:
8798:
8795:
8790:
8786:
8782:
8759:
8739:
8727:
8724:
8711:
8689:
8686:
8682:
8659:
8655:
8634:
8606:
8594:
8591:
8578:
8575:
8572:
8569:
8566:
8563:
8560:
8557:
8537:
8534:
8529:
8525:
8513:
8512:
8501:
8497:
8491:
8487:
8483:
8480:
8477:
8472:
8468:
8464:
8459:
8455:
8450:
8446:
8441:
8436:
8429:
8425:
8421:
8418:
8415:
8410:
8405:
8398:
8394:
8390:
8385:
8380:
8373:
8369:
8354:
8353:
8342:
8338:
8334:
8329:
8324:
8317:
8313:
8309:
8306:
8303:
8298:
8293:
8286:
8282:
8278:
8273:
8268:
8261:
8257:
8231:
8227:
8223:
8220:
8217:
8212:
8208:
8204:
8199:
8195:
8180:
8165:
8160:
8155:
8152:
8149:
8144:
8139:
8134:
8129:
8124:
8111:
8110:
8095:
8092:
8089:
8086:
8083:
8080:
8077:
8074:
8071:
8068:
8065:
8062:
8059:
8057:
8054:
8050:
8045:
8040:
8039:
8036:
8033:
8031:
8028:
8025:
8022:
8019:
8016:
8013:
8010:
8007:
8004:
8001:
7998:
7995:
7993:
7990:
7986:
7981:
7976:
7975:
7972:
7969:
7966:
7963:
7960:
7957:
7954:
7951:
7948:
7945:
7942:
7939:
7937:
7934:
7930:
7925:
7920:
7919:
7892:
7870:
7865:
7860:
7857:
7845:
7842:
7830:
7825:
7820:
7806:
7803:
7790:
7787:
7784:
7764:
7761:
7758:
7738:
7727:
7726:
7715:
7712:
7707:
7702:
7698:
7694:
7691:
7688:
7683:
7679:
7675:
7671:
7667:
7644:
7633:
7632:
7621:
7617:
7613:
7610:
7605:
7600:
7596:
7592:
7589:
7586:
7581:
7577:
7573:
7569:
7545:
7529:
7522:
7505:
7485:
7465:
7445:
7417:
7385:
7382:
7379:
7376:
7356:
7336:
7313:
7310:
7307:
7304:
7301:
7298:
7278:
7275:
7272:
7269:
7266:
7251:
7250:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7168:
7147:
7146:
7135:
7130:
7122:
7118:
7114:
7113:
7108:
7104:
7100:
7099:
7097:
7090:
7084:
7081:
7079:
7076:
7075:
7072:
7069:
7066:
7064:
7061:
7060:
7058:
7053:
7050:
7047:
7033:
7032:
7021:
7016:
7010:
7007:
7005:
7002:
7001:
6998:
6995:
6992:
6990:
6987:
6986:
6984:
6979:
6976:
6951:if and only if
6933:
6928:
6906:
6894:
6891:
6876:
6871:
6849:
6844:
6839:
6834:
6829:
6824:
6800:
6796:
6784:
6783:
6772:
6769:
6765:
6759:
6755:
6751:
6746:
6742:
6738:
6735:
6731:
6725:
6721:
6717:
6714:
6711:
6706:
6702:
6686:
6680:
6679:
6668:
6663:
6657:
6654:
6653:
6650:
6647:
6644:
6643:
6641:
6634:
6630:
6626:
6623:
6618:
6610:
6606:
6602:
6601:
6596:
6592:
6588:
6587:
6585:
6578:
6572:
6569:
6566:
6564:
6561:
6560:
6557:
6554:
6552:
6549:
6548:
6546:
6529:
6526:
6525:
6514:
6509:
6503:
6500:
6499:
6496:
6493:
6492:
6489:
6486:
6485:
6482:
6479:
6478:
6476:
6471:
6466:
6458:
6454:
6450:
6449:
6444:
6440:
6436:
6435:
6430:
6426:
6422:
6421:
6419:
6412:
6406:
6403:
6401:
6398:
6396:
6393:
6392:
6389:
6386:
6384:
6381:
6379:
6376:
6375:
6372:
6369:
6367:
6364:
6361:
6359:
6356:
6355:
6352:
6349:
6346:
6344:
6341:
6339:
6336:
6335:
6333:
6317:
6316:
6305:
6300:
6294:
6291:
6290:
6287:
6284:
6283:
6280:
6277:
6276:
6273:
6270:
6269:
6267:
6262:
6257:
6249:
6245:
6241:
6240:
6235:
6231:
6227:
6226:
6221:
6217:
6213:
6212:
6210:
6203:
6197:
6194:
6191:
6189:
6186:
6183:
6181:
6178:
6175:
6174:
6171:
6168:
6166:
6163:
6160:
6158:
6155:
6154:
6151:
6148:
6146:
6143:
6141:
6138:
6137:
6134:
6131:
6128:
6126:
6123:
6121:
6118:
6117:
6115:
6099:
6098:
6087:
6082:
6076:
6073:
6070:
6069:
6066:
6063:
6062:
6059:
6056:
6055:
6052:
6049:
6046:
6045:
6043:
6038:
6033:
6028:
6023:
6018:
6012:
6009:
6006:
6005:
6002:
5999:
5996:
5995:
5992:
5989:
5988:
5985:
5982:
5981:
5979:
5974:
5969:
5964:
5959:
5954:
5948:
5945:
5942:
5941:
5938:
5935:
5934:
5931:
5928:
5927:
5924:
5921:
5920:
5918:
5913:
5908:
5903:
5879:
5874:
5869:
5855:
5852:
5839:
5836:
5833:
5830:
5827:
5824:
5821:
5816:
5811:
5789:
5786:
5783:
5780:
5777:
5774:
5769:
5764:
5742:
5739:
5736:
5731:
5727:
5715:
5714:
5703:
5698:
5692:
5689:
5688:
5685:
5682:
5681:
5679:
5674:
5669:
5661:
5657:
5653:
5652:
5647:
5643:
5639:
5638:
5636:
5629:
5623:
5620:
5618:
5615:
5614:
5611:
5608:
5606:
5603:
5602:
5600:
5584:
5583:
5572:
5567:
5561:
5558:
5557:
5554:
5551:
5550:
5548:
5543:
5538:
5530:
5526:
5522:
5521:
5516:
5512:
5508:
5507:
5505:
5498:
5492:
5489:
5487:
5484:
5483:
5480:
5477:
5474:
5472:
5469:
5468:
5466:
5450:
5449:
5438:
5433:
5427:
5424:
5423:
5420:
5417:
5416:
5414:
5409:
5404:
5398:
5395:
5394:
5391:
5388:
5385:
5384:
5382:
5375:
5371:
5367:
5362:
5356:
5353:
5352:
5349:
5346:
5345:
5343:
5336:
5332:
5308:
5305:
5302:
5299:
5296:
5293:
5290:
5287:
5282:
5277:
5255:
5252:
5249:
5246:
5243:
5240:
5235:
5230:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5176:
5171:
5149:
5146:
5143:
5140:
5137:
5134:
5129:
5124:
5102:
5099:
5096:
5093:
5090:
5087:
5082:
5077:
5061:
5055:
5054:
5043:
5038:
5032:
5028:
5024:
5021:
5020:
5017:
5013:
5009:
5006:
5005:
5003:
4996:
4992:
4988:
4985:
4980:
4972:
4968:
4964:
4963:
4958:
4954:
4950:
4949:
4947:
4942:
4937:
4929:
4925:
4921:
4920:
4915:
4911:
4907:
4906:
4904:
4897:
4891:
4888:
4886:
4883:
4882:
4879:
4876:
4874:
4871:
4870:
4868:
4852:
4851:
4840:
4835:
4829:
4826:
4825:
4822:
4819:
4818:
4816:
4811:
4806:
4798:
4794:
4790:
4789:
4784:
4780:
4776:
4775:
4770:
4766:
4762:
4761:
4759:
4752:
4746:
4742:
4738:
4735:
4733:
4730:
4728:
4725:
4724:
4721:
4717:
4713:
4710:
4708:
4705:
4703:
4700:
4699:
4697:
4681:
4680:
4669:
4664:
4658:
4655:
4654:
4651:
4648:
4647:
4645:
4640:
4635:
4627:
4623:
4619:
4618:
4613:
4609:
4605:
4604:
4599:
4595:
4591:
4590:
4588:
4581:
4575:
4572:
4570:
4567:
4565:
4562:
4561:
4558:
4555:
4553:
4550:
4547:
4545:
4542:
4541:
4539:
4520:
4519:
4508:
4503:
4497:
4494:
4493:
4490:
4487:
4486:
4484:
4479:
4474:
4466:
4462:
4458:
4457:
4452:
4448:
4444:
4443:
4438:
4434:
4430:
4429:
4427:
4420:
4414:
4411:
4409:
4406:
4404:
4401:
4400:
4397:
4394:
4392:
4389:
4386:
4384:
4381:
4380:
4378:
4362:
4361:
4350:
4345:
4339:
4336:
4335:
4332:
4329:
4328:
4326:
4321:
4316:
4310:
4307:
4306:
4303:
4300:
4299:
4297:
4290:
4286:
4282:
4277:
4271:
4268:
4267:
4264:
4261:
4258:
4257:
4255:
4248:
4244:
4240:
4235:
4229:
4226:
4225:
4222:
4219:
4218:
4216:
4209:
4205:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4158:
4153:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4105:
4100:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4056:
4051:
4038:Three vectors:
4034:
4031:
4017:
3995:
3971:
3949:
3923:
3901:
3876:
3854:
3832:
3810:
3789:
3768:
3764:
3760:
3738:
3734:
3731:
3727:
3700:
3696:
3692:
3670:
3666:
3662:
3641:
3637:
3631:
3628:
3623:
3619:
3596:
3593:
3570:
3566:
3562:
3539:
3536:
3533:
3512:
3508:
3504:
3482:
3478:
3475:
3471:
3447:
3444:
3441:
3416:
3394:
3390:
3386:
3365:
3361:
3357:
3354:
3350:
3328:
3324:
3320:
3298:
3276:
3272:
3268:
3264:
3260:
3256:
3253:
3249:
3245:
3225:
3222:
3219:
3198:
3194:
3190:
3178:
3177:
3164:
3160:
3157:
3153:
3132:
3111:
3089:
3078:
3065:
3061:
3058:
3054:
3033:
3012:
2990:
2975:if and only if
2961:
2939:
2917:
2895:
2882:
2879:
2867:
2863:
2859:
2854:
2849:
2825:
2820:
2795:
2791:
2786:
2781:
2758:
2736:
2731:
2707:
2702:
2680:
2677:
2674:
2654:
2632:
2627:
2622:
2619:
2616:
2611:
2606:
2572:
2567:
2562:
2559:
2556:
2551:
2546:
2524:
2521:
2516:
2512:
2500:
2499:
2488:
2484:
2480:
2476:
2470:
2466:
2462:
2457:
2452:
2445:
2441:
2437:
2433:
2429:
2426:
2423:
2419:
2415:
2410:
2405:
2398:
2394:
2390:
2386:
2382:
2379:
2376:
2372:
2368:
2363:
2358:
2351:
2347:
2343:
2340:
2337:
2332:
2327:
2320:
2316:
2290:
2285:
2278:
2274:
2270:
2267:
2264:
2259:
2254:
2247:
2243:
2221:
2217:
2212:
2207:
2202:
2199:
2194:
2189:
2182:
2178:
2157:
2154:
2149:
2145:
2124:
2121:
2118:
2098:
2078:
2058:
2036:
2032:
2011:
2008:
2003:
1999:
1978:
1974:
1970:
1965:
1960:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1900:
1878:
1873:
1868:
1865:
1862:
1857:
1852:
1829:
1806:
1801:
1796:
1793:
1790:
1785:
1780:
1766:
1763:
1761:
1758:
1725:
1722:
1721:
1720:
1706:
1703:
1697:
1694:
1688:
1685:
1659:
1656:
1630:
1627:
1613:
1598:
1595:
1569:
1566:
1540:
1537:
1511:
1508:
1482:
1479:
1453:
1450:
1437:
1422:
1419:
1393:
1390:
1377:
1362:
1359:
1333:
1330:
1304:
1301:
1288:
1269:
1266:
1240:
1237:
1215:
1212:
1171:of vectors is
1169:indexed family
1149:
1146:
1121:
1098:
1094:
1072:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1009:
1006:
1001:
997:
985:
984:
973:
969:
965:
960:
955:
948:
944:
940:
937:
934:
929:
924:
917:
913:
909:
904:
899:
892:
888:
871:is said to be
858:
853:
848:
845:
842:
837:
832:
827:
822:
817:
786:
783:
780:
759:
755:
750:
745:
723:
720:
717:
714:
703:
702:
691:
686:
681:
672:
668:
661:
657:
653:
647:
644:
641:
636:
631:
622:
618:
611:
607:
603:
597:
592:
587:
562:
559:
554:
550:
525:
513:
512:
501:
497:
493:
488:
483:
476:
472:
468:
465:
462:
457:
452:
445:
441:
437:
432:
427:
420:
416:
392:
387:
383:
379:
376:
373:
368:
364:
360:
355:
351:
334:is said to be
314:
309:
304:
301:
298:
293:
288:
283:
278:
273:
259:
256:
226:is said to be
200:
195:
190:
159:
154:
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
11905:
11894:
11891:
11889:
11886:
11885:
11883:
11868:
11865:
11863:
11860:
11858:
11857:
11852:
11846:
11845:
11842:
11836:
11833:
11831:
11828:
11826:
11825:Pseudoinverse
11823:
11821:
11818:
11816:
11813:
11811:
11808:
11806:
11803:
11801:
11798:
11797:
11795:
11793:Related terms
11791:
11785:
11784:Z (chemistry)
11782:
11780:
11777:
11775:
11772:
11770:
11767:
11765:
11762:
11760:
11757:
11755:
11752:
11750:
11747:
11745:
11742:
11740:
11737:
11735:
11732:
11730:
11727:
11725:
11722:
11721:
11719:
11715:
11709:
11706:
11704:
11701:
11699:
11696:
11694:
11691:
11689:
11686:
11684:
11681:
11679:
11676:
11674:
11671:
11670:
11668:
11666:
11661:
11655:
11652:
11650:
11647:
11645:
11642:
11640:
11637:
11635:
11632:
11630:
11627:
11625:
11622:
11620:
11617:
11615:
11612:
11610:
11607:
11606:
11604:
11602:
11597:
11591:
11588:
11586:
11583:
11581:
11578:
11576:
11573:
11571:
11568:
11566:
11563:
11561:
11558:
11556:
11553:
11551:
11548:
11546:
11543:
11541:
11538:
11536:
11533:
11531:
11528:
11526:
11523:
11521:
11518:
11516:
11513:
11511:
11508:
11506:
11503:
11501:
11498:
11496:
11493:
11491:
11488:
11486:
11483:
11481:
11478:
11476:
11473:
11471:
11468:
11466:
11463:
11461:
11458:
11456:
11453:
11451:
11448:
11446:
11443:
11441:
11438:
11437:
11435:
11431:
11425:
11422:
11420:
11417:
11415:
11412:
11410:
11407:
11405:
11402:
11400:
11397:
11395:
11392:
11390:
11387:
11385:
11382:
11380:
11377:
11375:
11371:
11368:
11366:
11363:
11362:
11360:
11358:
11354:
11349:
11343:
11340:
11338:
11335:
11333:
11330:
11328:
11325:
11323:
11320:
11318:
11315:
11313:
11310:
11308:
11305:
11304:
11302:
11300:
11295:
11289:
11286:
11284:
11281:
11279:
11276:
11274:
11271:
11269:
11266:
11264:
11261:
11259:
11256:
11254:
11251:
11249:
11246:
11244:
11241:
11240:
11238:
11234:
11228:
11225:
11223:
11220:
11218:
11215:
11213:
11210:
11208:
11205:
11203:
11200:
11198:
11195:
11193:
11190:
11188:
11185:
11183:
11180:
11178:
11175:
11173:
11170:
11168:
11165:
11163:
11160:
11158:
11155:
11153:
11150:
11148:
11145:
11143:
11142:Pentadiagonal
11140:
11138:
11135:
11133:
11130:
11128:
11125:
11123:
11120:
11118:
11115:
11113:
11110:
11108:
11105:
11103:
11100:
11098:
11095:
11093:
11090:
11088:
11085:
11083:
11080:
11078:
11075:
11073:
11070:
11068:
11065:
11063:
11060:
11058:
11055:
11053:
11050:
11048:
11045:
11043:
11040:
11038:
11035:
11033:
11030:
11028:
11025:
11023:
11020:
11018:
11015:
11013:
11010:
11008:
11005:
11003:
11000:
10998:
10995:
10993:
10990:
10988:
10985:
10983:
10980:
10978:
10975:
10973:
10972:Anti-diagonal
10970:
10968:
10965:
10964:
10962:
10958:
10953:
10946:
10941:
10939:
10934:
10932:
10927:
10926:
10923:
10911:
10903:
10902:
10899:
10893:
10890:
10888:
10887:Sparse matrix
10885:
10883:
10880:
10878:
10875:
10873:
10870:
10869:
10867:
10865:
10861:
10855:
10852:
10850:
10847:
10845:
10842:
10840:
10837:
10835:
10832:
10830:
10827:
10826:
10824:
10822:constructions
10821:
10817:
10811:
10810:Outermorphism
10808:
10806:
10803:
10801:
10798:
10796:
10793:
10791:
10788:
10786:
10783:
10781:
10778:
10776:
10773:
10771:
10770:Cross product
10768:
10766:
10763:
10762:
10760:
10758:
10754:
10748:
10745:
10743:
10740:
10738:
10737:Outer product
10735:
10733:
10730:
10728:
10725:
10723:
10720:
10718:
10717:Orthogonality
10715:
10714:
10712:
10710:
10706:
10700:
10697:
10695:
10694:Cramer's rule
10692:
10690:
10687:
10685:
10682:
10680:
10677:
10675:
10672:
10670:
10667:
10665:
10664:Decomposition
10662:
10660:
10657:
10656:
10654:
10652:
10648:
10643:
10633:
10630:
10628:
10625:
10623:
10620:
10618:
10615:
10613:
10610:
10608:
10605:
10603:
10600:
10598:
10595:
10593:
10590:
10588:
10585:
10583:
10580:
10578:
10575:
10573:
10570:
10568:
10565:
10563:
10560:
10558:
10555:
10553:
10550:
10548:
10545:
10543:
10540:
10539:
10537:
10533:
10527:
10524:
10522:
10519:
10518:
10515:
10511:
10504:
10499:
10497:
10492:
10490:
10485:
10484:
10481:
10474:
10471:
10468:
10465:
10462:
10459:
10455:
10451:
10450:
10445:
10441:
10440:
10426:
10422:
10418:
10412:
10408:
10401:
10399:
10391:
10387:
10383:
10381:0-444-87916-1
10377:
10373:
10369:
10365:
10359:
10351:
10345:
10341:
10334:
10327:
10326:
10319:
10315:
10304:
10301:
10300:
10294:
10281:
10278:
10275:
10270:
10266:
10262:
10259:
10256:
10251:
10247:
10224:
10220:
10216:
10213:
10210:
10205:
10201:
10191:
10188:
10174:
10154:
10149:
10145:
10136:
10133:
10130:
10127:
10124:
10121:
10118:
10115:
10112:
10109:
10106:
10103:
10100:
10097:
10094:
10088:
10085:
10081:
10077:
10074:
10071:
10061:
10051:
10047:
10043:
10038:
10034:
10030:
10025:
10021:
10017:
10014:
10011:
10006:
10003:
10000:
9996:
9992:
9987:
9984:
9981:
9977:
9973:
9970:
9967:
9962:
9958:
9947:
9942:
9938:
9932:
9929:
9926:
9922:
9901:
9898:
9875:
9869:
9864:
9860:
9854:
9851:
9848:
9844:
9840:
9835:
9831:
9820:
9806:
9784:
9780:
9776:
9773:
9770:
9765:
9761:
9740:
9734:
9728:
9725:
9722:
9719:
9709:
9695:
9675:
9655:
9641:
9627:
9624:
9621:
9600:
9595:
9585:
9571:
9563:
9559:
9556:
9553:
9549:
9539:
9525:
9517:
9512:
9491:
9471:
9449:
9439:
9436:
9433:
9428:
9401:
9392:
9390:
9386:
9382:
9376:
9361:
9359:
9355:
9351:
9346:
9342:
9338:
9331:
9307:
9299:
9294:
9282:
9278:
9274:
9271:
9268:
9263:
9251:
9247:
9239:
9238:
9237:
9235:
9225:
9221:
9214:
9209:
9203:
9199:
9192:
9187:
9183:
9173:
9157:
9154:
9150:
9127:
9123:
9102:
9099:
9096:
9076:
9073:
9070:
9050:
9028:
9025:
9021:
9000:
8997:
8992:
8989:
8985:
8981:
8961:
8958:
8955:
8935:
8932:
8929:
8909:
8906:
8896:
8879:
8876:
8871:
8868:
8864:
8860:
8857:
8854:
8849:
8845:
8841:
8834:
8833:
8832:
8815:
8812:
8807:
8804:
8800:
8796:
8793:
8788:
8784:
8780:
8773:
8772:
8771:
8757:
8737:
8723:
8709:
8687:
8684:
8680:
8657:
8653:
8632:
8624:
8620:
8604:
8589:
8576:
8573:
8570:
8567:
8564:
8561:
8558:
8555:
8535:
8532:
8527:
8523:
8499:
8495:
8489:
8485:
8481:
8478:
8475:
8470:
8466:
8462:
8457:
8453:
8448:
8444:
8439:
8427:
8423:
8419:
8416:
8413:
8408:
8396:
8392:
8388:
8383:
8371:
8367:
8359:
8358:
8357:
8340:
8332:
8327:
8315:
8311:
8307:
8304:
8301:
8296:
8284:
8280:
8276:
8271:
8259:
8255:
8247:
8246:
8245:
8229:
8225:
8221:
8218:
8215:
8210:
8206:
8202:
8197:
8193:
8185:Suppose that
8179:
8163:
8153:
8150:
8147:
8142:
8132:
8127:
8093:
8087:
8084:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8055:
8048:
8034:
8023:
8020:
8017:
8014:
8011:
8008:
8005:
8002:
7999:
7991:
7984:
7967:
7964:
7961:
7958:
7955:
7952:
7949:
7946:
7943:
7935:
7928:
7910:
7909:
7908:
7906:
7905:natural basis
7890:
7868:
7858:
7855:
7841:
7828:
7823:
7802:
7788:
7785:
7782:
7762:
7759:
7756:
7736:
7713:
7710:
7700:
7696:
7692:
7689:
7686:
7681:
7677:
7669:
7658:
7657:
7656:
7642:
7619:
7611:
7598:
7594:
7590:
7587:
7584:
7579:
7575:
7567:
7559:
7558:
7557:
7543:
7532:
7528:
7521:
7503:
7483:
7463:
7443:
7435:
7431:
7415:
7407:
7403:
7399:
7383:
7380:
7377:
7374:
7354:
7334:
7325:
7308:
7305:
7302:
7299:
7273:
7270:
7267:
7256:
7237:
7234:
7231:
7228:
7222:
7219:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7192:
7182:
7181:
7180:
7166:
7157:
7153:
7133:
7128:
7120:
7116:
7106:
7102:
7095:
7088:
7082:
7077:
7070:
7067:
7062:
7056:
7051:
7045:
7038:
7037:
7036:
7019:
7014:
7008:
7003:
6996:
6993:
6988:
6982:
6977:
6974:
6967:
6966:
6965:
6962:
6960:
6956:
6952:
6949:
6946:are linearly
6931:
6904:
6890:
6874:
6847:
6842:
6832:
6827:
6798:
6794:
6770:
6767:
6763:
6757:
6753:
6749:
6744:
6740:
6736:
6733:
6729:
6723:
6719:
6715:
6712:
6709:
6704:
6700:
6692:
6691:
6690:
6685:
6666:
6661:
6655:
6648:
6645:
6639:
6632:
6628:
6624:
6621:
6616:
6608:
6604:
6594:
6590:
6583:
6576:
6570:
6567:
6562:
6555:
6550:
6544:
6535:
6534:
6533:
6512:
6507:
6501:
6494:
6487:
6480:
6474:
6469:
6464:
6456:
6452:
6442:
6438:
6428:
6424:
6417:
6410:
6404:
6399:
6394:
6387:
6382:
6377:
6370:
6365:
6362:
6357:
6350:
6347:
6342:
6337:
6331:
6322:
6321:
6320:
6303:
6298:
6292:
6285:
6278:
6271:
6265:
6260:
6255:
6247:
6243:
6233:
6229:
6219:
6215:
6208:
6201:
6195:
6192:
6187:
6184:
6179:
6176:
6169:
6164:
6161:
6156:
6149:
6144:
6139:
6132:
6129:
6124:
6119:
6113:
6104:
6103:
6102:
6085:
6080:
6074:
6071:
6064:
6057:
6050:
6047:
6041:
6036:
6031:
6021:
6016:
6010:
6007:
6000:
5997:
5990:
5983:
5977:
5972:
5967:
5957:
5952:
5946:
5943:
5936:
5929:
5922:
5916:
5911:
5906:
5892:
5891:
5890:
5877:
5872:
5851:
5834:
5831:
5828:
5825:
5819:
5814:
5784:
5781:
5778:
5772:
5767:
5740:
5737:
5734:
5729:
5725:
5701:
5696:
5690:
5683:
5677:
5672:
5667:
5659:
5655:
5645:
5641:
5634:
5627:
5621:
5616:
5609:
5604:
5598:
5589:
5588:
5587:
5570:
5565:
5559:
5552:
5546:
5541:
5536:
5528:
5524:
5514:
5510:
5503:
5496:
5490:
5485:
5478:
5475:
5470:
5464:
5455:
5454:
5453:
5436:
5431:
5425:
5418:
5412:
5407:
5402:
5396:
5389:
5386:
5380:
5373:
5369:
5365:
5360:
5354:
5347:
5341:
5334:
5330:
5322:
5321:
5320:
5306:
5300:
5297:
5294:
5291:
5285:
5280:
5250:
5247:
5244:
5238:
5233:
5218:
5214:
5200:
5194:
5191:
5188:
5185:
5179:
5174:
5144:
5141:
5138:
5132:
5127:
5097:
5094:
5091:
5085:
5080:
5064:
5060:
5041:
5036:
5030:
5026:
5022:
5015:
5011:
5007:
5001:
4994:
4990:
4986:
4983:
4978:
4970:
4966:
4956:
4952:
4945:
4940:
4935:
4927:
4923:
4913:
4909:
4902:
4895:
4889:
4884:
4877:
4872:
4866:
4857:
4856:
4855:
4838:
4833:
4827:
4820:
4814:
4809:
4804:
4796:
4792:
4782:
4778:
4768:
4764:
4757:
4750:
4744:
4740:
4736:
4731:
4726:
4719:
4715:
4711:
4706:
4701:
4695:
4686:
4685:
4684:
4667:
4662:
4656:
4649:
4643:
4638:
4633:
4625:
4621:
4611:
4607:
4597:
4593:
4586:
4579:
4573:
4568:
4563:
4556:
4551:
4548:
4543:
4537:
4528:
4527:
4526:
4524:
4506:
4501:
4495:
4488:
4482:
4477:
4472:
4464:
4460:
4450:
4446:
4436:
4432:
4425:
4418:
4412:
4407:
4402:
4395:
4390:
4387:
4382:
4376:
4367:
4366:
4365:
4348:
4343:
4337:
4330:
4324:
4319:
4314:
4308:
4301:
4295:
4288:
4284:
4280:
4275:
4269:
4262:
4259:
4253:
4246:
4242:
4238:
4233:
4227:
4220:
4214:
4207:
4203:
4195:
4194:
4193:
4179:
4173:
4170:
4167:
4161:
4156:
4129:
4123:
4120:
4117:
4114:
4108:
4103:
4077:
4071:
4068:
4065:
4059:
4054:
4039:
4030:
3985:
3938:
3935:are linearly
3889:
3787:
3762:
3732:
3729:
3716:
3694:
3664:
3639:
3629:
3626:
3621:
3594:
3591:
3564:
3552:
3537:
3534:
3531:
3506:
3476:
3473:
3459:
3445:
3442:
3439:
3431:
3388:
3363:
3355:
3352:
3322:
3270:
3262:
3254:
3251:
3243:
3223:
3220:
3217:
3192:
3158:
3155:
3130:
3079:
3059:
3056:
3031:
2980:
2979:
2978:
2976:
2878:
2865:
2857:
2852:
2823:
2789:
2784:
2756:
2734:
2705:
2678:
2675:
2672:
2652:
2630:
2620:
2617:
2614:
2609:
2593:
2591:
2586:
2570:
2560:
2557:
2554:
2549:
2522:
2519:
2514:
2510:
2486:
2478:
2468:
2464:
2460:
2455:
2443:
2439:
2435:
2427:
2424:
2421:
2413:
2408:
2396:
2392:
2388:
2380:
2377:
2374:
2366:
2361:
2349:
2345:
2341:
2338:
2335:
2330:
2318:
2314:
2306:
2305:
2304:
2288:
2276:
2272:
2268:
2265:
2262:
2257:
2245:
2241:
2215:
2210:
2200:
2197:
2192:
2180:
2176:
2155:
2152:
2147:
2143:
2122:
2119:
2116:
2096:
2076:
2056:
2034:
2030:
2009:
2006:
2001:
1997:
1976:
1968:
1963:
1933:
1930:
1927:
1924:
1921:
1898:
1876:
1866:
1863:
1860:
1855:
1804:
1794:
1791:
1788:
1783:
1757:
1746:
1744:
1740:
1734:
1732:
1701:
1695:
1692:
1683:
1654:
1625:
1614:
1593:
1564:
1535:
1506:
1477:
1448:
1438:
1417:
1388:
1378:
1357:
1328:
1299:
1289:
1286:
1264:
1235:
1225:
1224:
1220:
1211:
1207:
1203:
1193:
1189:
1185:
1180:
1178:
1174:
1170:
1165:
1163:
1159:
1155:
1148:Infinite case
1145:
1142:
1140:
1134:
1096:
1092:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1007:
1004:
999:
995:
971:
963:
958:
946:
942:
938:
935:
932:
927:
915:
911:
907:
902:
890:
886:
878:
877:
876:
874:
856:
846:
843:
840:
835:
825:
820:
804:
802:
797:
784:
781:
778:
753:
748:
721:
718:
715:
712:
689:
684:
670:
666:
659:
655:
651:
645:
642:
639:
634:
620:
616:
609:
605:
601:
595:
590:
576:
575:
574:
560:
557:
552:
548:
538:
499:
491:
486:
474:
470:
466:
463:
460:
455:
443:
439:
435:
430:
418:
414:
406:
405:
404:
390:
385:
381:
377:
374:
371:
366:
362:
358:
353:
349:
341:
337:
330:
312:
302:
299:
296:
291:
281:
276:
255:
251:
249:
245:
237:
233:
225:
221:
217:
216:vector spaces
198:
193:
177:
157:
141:
131:
128:
120:
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:
11847:
11804:
11779:Substitution
11665:graph theory
11162:Quaternionic
11152:Persymmetric
10820:Vector space
10581:
10552:Vector space
10447:
10406:
10371:
10358:
10339:
10333:
10323:
10318:
9647:
9393:
9388:
9380:
9378:
9375:Affine space
9347:
9340:
9336:
9329:
9322:
9223:
9219:
9212:
9201:
9197:
9190:
9181:
9179:
8894:
8830:
8729:
8619:vector space
8596:
8514:
8355:
8184:
8112:
7847:
7808:
7728:
7634:
7530:
7526:
7519:
7496:, the first
7433:
7429:
7405:
7401:
7397:
7326:
7252:
7155:
7151:
7148:
7034:
6963:
6947:
6896:
6785:
6683:
6681:
6532:and obtain,
6527:
6318:
6100:
5857:
5854:Vectors in R
5716:
5585:
5451:
5319:and check,
5217:Two vectors:
5216:
5215:
5062:
5058:
5056:
4853:
4682:
4521:
4363:
4037:
4036:
4033:Vectors in R
3983:
3936:
3891:The vectors
3890:
3750:but instead
3714:
3609:to conclude
3550:
3460:
3429:
3179:
2884:
2594:
2589:
2587:
2501:
1949:) such that
1768:
1747:
1742:
1735:
1730:
1727:
1615:The vectors
1210:as a basis.
1205:
1201:
1181:
1176:
1172:
1166:
1161:
1153:
1151:
1143:
1138:
1135:
986:
872:
805:
798:
704:
539:
514:
335:
329:vector space
261:
252:
239:
227:
213:
123:
117:January 2019
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
11754:Hamiltonian
11678:Biadjacency
11614:Correlation
11530:Householder
11480:Commutation
11217:Vandermonde
11212:Tridiagonal
11147:Permutation
11137:Nonnegative
11122:Matrix unit
11002:Bisymmetric
10800:Multivector
10765:Determinant
10722:Dot product
10567:Linear span
7347:vectors of
7255:determinant
7179:, which is
6955:determinant
6948:independent
6917:vectors in
3458:for both).
2645:has length
2592:dependent.
2089:other than
1192:polynomials
11882:Categories
11654:Transition
11649:Stochastic
11619:Covariance
11601:statistics
11580:Symplectic
11575:Similarity
11404:Unimodular
11399:Orthogonal
11384:Involutory
11379:Invertible
11374:Projection
11370:Idempotent
11312:Convergent
11207:Triangular
11157:Polynomial
11102:Hessenberg
11072:Equivalent
11067:Elementary
11047:Copositive
11037:Conference
10997:Bidiagonal
10834:Direct sum
10669:Invertible
10572:Linear map
10349:0130084514
10310:References
10190:direct sum
9373:See also:
8899:values of
7253:Since the
4523:Row reduce
3143:such that
3044:such that
2109:(i.e. for
258:Definition
87:newspapers
32:Covariance
11835:Wronskian
11759:Irregular
11749:Gell-Mann
11698:Laplacian
11693:Incidence
11673:Adjacency
11644:Precision
11609:Centering
11515:Generator
11485:Confusion
11470:Circulant
11450:Augmented
11409:Unipotent
11389:Nilpotent
11365:Congruent
11342:Stieltjes
11317:Defective
11307:Companion
11278:Redheffer
11197:Symmetric
11192:Sylvester
11167:Signature
11097:Hermitian
11077:Frobenius
10987:Arrowhead
10967:Alternant
10864:Numerical
10627:Transpose
10454:EMS Press
10425:829157984
10260:⋯
10214:…
10131:…
10110:−
10101:…
10089:∈
10082:⋃
10078:
10044:∈
10015:⋯
9985:−
9971:⋯
9930:≠
9923:∑
9852:≠
9845:∑
9841:∩
9774:…
9723:∩
9557:…
9437:…
9272:⋯
8623:functions
8568:…
8479:…
8417:⋯
8305:⋯
8219:…
8151:…
8082:…
8035:⋮
8018:…
7962:…
7907:vectors:
7706:⟩
7690:…
7674:⟨
7609:Λ
7604:⟩
7588:…
7572:⟨
7432:Λ =
7300:−
7235:≠
7220:−
7214:⋅
7208:−
7202:⋅
7117:λ
7103:λ
7068:−
7049:Λ
6994:−
6713:−
6646:−
6625:−
6568:−
6363:−
6348:−
6193:−
6185:−
6177:−
6162:−
6130:−
6072:−
6048:−
6008:−
5998:−
5944:−
5826:−
5476:−
5387:−
5292:−
5186:−
4987:−
4549:−
4388:−
4260:−
4115:−
3695:≠
3665:≠
3565:≠
3535:≠
3507:≠
2858:≠
2618:…
2558:…
2520:≠
2425:⋯
2378:⋯
2339:⋯
2266:⋯
2120:≠
1989:Then let
1928:…
1864:…
1792:…
1705:→
1687:→
1658:→
1629:→
1597:→
1568:→
1539:→
1510:→
1481:→
1452:→
1421:→
1392:→
1361:→
1332:→
1303:→
1268:→
1239:→
1040:…
936:⋯
844:…
652:−
643:⋯
602:−
558:≠
464:⋯
375:…
300:…
248:dimension
11663:Used in
11599:Used in
11560:Rotation
11535:Jacobian
11495:Distance
11475:Cofactor
11460:Carleman
11440:Adjugate
11424:Weighing
11357:inverses
11353:products
11322:Definite
11253:Identity
11243:Exchange
11236:Constant
11202:Toeplitz
11087:Hadamard
11057:Diagonal
10910:Category
10849:Subspace
10844:Quotient
10795:Bivector
10709:Bilinear
10651:Matrices
10526:Glossary
10370:(1986),
10297:See also
9614:of size
9464:of size
9414:vectors
9013:. Since
8730:Suppose
8548:for all
7476:rows of
3236:we have
2303:gives:
11764:Overlap
11729:Density
11688:Edmonds
11565:Seifert
11525:Hessian
11490:Coxeter
11414:Unitary
11332:Hurwitz
11263:Of ones
11248:Hilbert
11182:Skyline
11127:Metzler
11117:Logical
11112:Integer
11022:Boolean
10954:classes
10521:Outline
10456:, 2001
10429:pp. 3–7
10390:0859549
10303:Matroid
9335:, ...,
9218:, ...,
9196:, ...,
8617:be the
6957:of the
3428:) then
2135:), let
340:scalars
327:from a
224:vectors
101:scholar
11683:Degree
11624:Design
11555:Random
11545:Payoff
11540:Moment
11465:Cartan
11455:Bézout
11394:Normal
11268:Pascal
11258:Lehmer
11187:Sparse
11107:Hollow
11092:Hankel
11027:Cauchy
10952:Matrix
10805:Tensor
10617:Kernel
10547:Vector
10542:Scalar
10423:
10413:
10388:
10378:
10346:
9914:where
9234:scalar
8356:Since
7400:is an
6959:matrix
6786:where
1208:, ...}
1158:subset
515:where
103:
96:
89:
82:
74:
11744:Gamma
11708:Tutte
11570:Shear
11283:Shift
11273:Pauli
11222:Walsh
11132:Moore
11012:Block
10674:Minor
10659:Block
10597:Basis
9354:basis
9229:with
9208:tuple
9206:is a
8726:Proof
8515:then
8182:Proof
8113:Then
7525:,...,
7396:Then
3494:then
1285:plane
1188:basis
1184:spans
218:, a
108:JSTOR
94:books
11550:Pick
11520:Gram
11288:Zero
10992:Band
10829:Dual
10684:Rank
10421:OCLC
10411:ISBN
10376:ISBN
10344:ISBN
10075:span
9668:and
9142:and
8948:and
8895:for
8750:and
8672:and
8597:Let
7848:Let
7786:>
7378:<
7289:and
7154:Λ =
6953:the
6860:and
5800:and
5266:and
5160:and
4142:and
3913:and
3844:and
3800:and
3682:and
3430:both
3077:) or
2951:and
2907:and
1731:true
1554:and
1496:and
1407:and
1347:and
1254:and
1200:{1,
1020:for
734:and
716:>
80:news
11639:Hat
11372:or
11355:or
10193:of
9823:if
9712:if
9184:or
8897:all
8702:in
7666:det
7190:det
5452:or
4364:or
3984:and
3866:is
3551:and
3461:If
3376:If
3180:If
1194:in
1167:An
771:if
705:if
222:of
220:set
63:by
11884::
10452:,
10446:,
10419:.
10397:^
10386:MR
10384:,
10366:;
9360:.
9180:A
9077:0.
9063:,
8962:0.
7238:0.
6689:,
6571:18
6366:18
6145:10
5991:10
5008:16
4712:16
4029:.
3937:in
3715:in
3443::=
3221::=
3176:).
2590:in
2153::=
2007::=
1467:,
1318:,
1287:P.
1204:,
1179:.
785:1.
250:.
11769:S
11227:Z
10944:e
10937:t
10930:v
10502:e
10495:t
10488:v
10427:.
10352:.
10282:.
10279:X
10276:=
10271:d
10267:M
10263:+
10257:+
10252:1
10248:M
10225:d
10221:M
10217:,
10211:,
10206:1
10202:M
10175:X
10155:.
10150:k
10146:M
10140:}
10137:d
10134:,
10128:,
10125:1
10122:+
10119:i
10116:,
10113:1
10107:i
10104:,
10098:,
10095:1
10092:{
10086:k
10072:=
10067:}
10062:k
10052:k
10048:M
10039:k
10035:m
10031::
10026:d
10022:m
10018:+
10012:+
10007:1
10004:+
10001:i
9997:m
9993:+
9988:1
9982:i
9978:m
9974:+
9968:+
9963:1
9959:m
9953:{
9948:=
9943:k
9939:M
9933:i
9927:k
9902:,
9899:i
9879:}
9876:0
9873:{
9870:=
9865:k
9861:M
9855:i
9849:k
9836:i
9832:M
9807:X
9785:d
9781:M
9777:,
9771:,
9766:1
9762:M
9741:.
9738:}
9735:0
9732:{
9729:=
9726:N
9720:M
9696:X
9676:N
9656:M
9628:1
9625:+
9622:n
9601:)
9596:]
9586:m
9581:v
9572:1
9564:[
9560:,
9554:,
9550:]
9540:1
9535:v
9526:1
9518:[
9513:(
9492:m
9472:n
9450:m
9445:v
9440:,
9434:,
9429:1
9424:v
9402:m
9341:n
9337:v
9333:1
9330:v
9325:n
9308:.
9304:0
9300:=
9295:n
9290:v
9283:n
9279:a
9275:+
9269:+
9264:1
9259:v
9252:1
9248:a
9231:n
9227:)
9224:n
9220:a
9216:1
9213:a
9211:(
9202:n
9198:v
9194:1
9191:v
9158:t
9155:2
9151:e
9128:t
9124:e
9103:0
9100:=
9097:a
9074:=
9071:b
9051:t
9029:t
9026:2
9022:e
9001:0
8998:=
8993:t
8990:2
8986:e
8982:b
8959:=
8956:b
8936:0
8933:=
8930:a
8910:.
8907:t
8880:0
8877:=
8872:t
8869:2
8865:e
8861:b
8858:2
8855:+
8850:t
8846:e
8842:a
8816:0
8813:=
8808:t
8805:2
8801:e
8797:b
8794:+
8789:t
8785:e
8781:a
8758:b
8738:a
8710:V
8688:t
8685:2
8681:e
8658:t
8654:e
8633:t
8605:V
8577:.
8574:n
8571:,
8565:,
8562:1
8559:=
8556:i
8536:0
8533:=
8528:i
8524:a
8500:,
8496:)
8490:n
8486:a
8482:,
8476:,
8471:2
8467:a
8463:,
8458:1
8454:a
8449:(
8445:=
8440:n
8435:e
8428:n
8424:a
8420:+
8414:+
8409:2
8404:e
8397:2
8393:a
8389:+
8384:1
8379:e
8372:1
8368:a
8341:.
8337:0
8333:=
8328:n
8323:e
8316:n
8312:a
8308:+
8302:+
8297:2
8292:e
8285:2
8281:a
8277:+
8272:1
8267:e
8260:1
8256:a
8230:n
8226:a
8222:,
8216:,
8211:2
8207:a
8203:,
8198:1
8194:a
8164:n
8159:e
8154:,
8148:,
8143:2
8138:e
8133:,
8128:1
8123:e
8094:.
8091:)
8088:1
8085:,
8079:,
8076:0
8073:,
8070:0
8067:,
8064:0
8061:(
8056:=
8049:n
8044:e
8027:)
8024:0
8021:,
8015:,
8012:0
8009:,
8006:1
8003:,
8000:0
7997:(
7992:=
7985:2
7980:e
7971:)
7968:0
7965:,
7959:,
7956:0
7953:,
7950:0
7947:,
7944:1
7941:(
7936:=
7929:1
7924:e
7891:V
7869:n
7864:R
7859:=
7856:V
7829:.
7824:2
7819:R
7789:n
7783:m
7763:n
7760:=
7757:m
7737:m
7714:0
7711:=
7701:m
7697:i
7693:,
7687:,
7682:1
7678:i
7670:A
7643:m
7620:.
7616:0
7612:=
7599:m
7595:i
7591:,
7585:,
7580:1
7576:i
7568:A
7544:m
7534:⟩
7531:m
7527:i
7523:1
7520:i
7518:⟨
7504:m
7484:A
7464:m
7444:n
7434:0
7430:A
7416:m
7406:m
7404:×
7402:n
7398:A
7384:.
7381:n
7375:m
7355:n
7335:m
7312:)
7309:2
7306:,
7303:3
7297:(
7277:)
7274:1
7271:,
7268:1
7265:(
7232:5
7229:=
7226:)
7223:3
7217:(
7211:1
7205:2
7199:1
7196:=
7193:A
7167:A
7156:0
7152:A
7134:.
7129:]
7121:2
7107:1
7096:[
7089:]
7083:2
7078:1
7071:3
7063:1
7057:[
7052:=
7046:A
7020:.
7015:]
7009:2
7004:1
6997:3
6989:1
6983:[
6978:=
6975:A
6932:n
6927:R
6905:n
6875:3
6870:v
6848:,
6843:2
6838:v
6833:,
6828:1
6823:v
6799:3
6795:a
6771:,
6768:2
6764:/
6758:3
6754:a
6750:=
6745:2
6741:a
6737:,
6734:2
6730:/
6724:3
6720:a
6716:3
6710:=
6705:1
6701:a
6687:i
6684:a
6667:.
6662:]
6656:9
6649:2
6640:[
6633:3
6629:a
6622:=
6617:]
6609:2
6605:a
6595:1
6591:a
6584:[
6577:]
6563:0
6556:7
6551:1
6545:[
6530:3
6513:.
6508:]
6502:0
6495:0
6488:0
6481:0
6475:[
6470:=
6465:]
6457:3
6453:a
6443:2
6439:a
6429:1
6425:a
6418:[
6411:]
6405:0
6400:0
6395:0
6388:0
6383:0
6378:0
6371:9
6358:0
6351:2
6343:7
6338:1
6332:[
6304:.
6299:]
6293:0
6286:0
6279:0
6272:0
6266:[
6261:=
6256:]
6248:3
6244:a
6234:2
6230:a
6220:1
6216:a
6209:[
6202:]
6196:4
6188:1
6180:3
6170:5
6165:4
6157:2
6150:1
6140:4
6133:2
6125:7
6120:1
6114:[
6086:.
6081:]
6075:4
6065:5
6058:1
6051:2
6042:[
6037:=
6032:3
6027:v
6022:,
6017:]
6011:1
6001:4
5984:7
5978:[
5973:=
5968:2
5963:v
5958:,
5953:]
5947:3
5937:2
5930:4
5923:1
5917:[
5912:=
5907:1
5902:v
5878:,
5873:4
5868:R
5838:)
5835:2
5832:,
5829:3
5823:(
5820:=
5815:2
5810:v
5788:)
5785:1
5782:,
5779:1
5776:(
5773:=
5768:1
5763:v
5741:,
5738:0
5735:=
5730:i
5726:a
5702:.
5697:]
5691:0
5684:0
5678:[
5673:=
5668:]
5660:2
5656:a
5646:1
5642:a
5635:[
5628:]
5622:1
5617:0
5610:0
5605:1
5599:[
5571:.
5566:]
5560:0
5553:0
5547:[
5542:=
5537:]
5529:2
5525:a
5515:1
5511:a
5504:[
5497:]
5491:2
5486:1
5479:3
5471:1
5465:[
5437:,
5432:]
5426:0
5419:0
5413:[
5408:=
5403:]
5397:2
5390:3
5381:[
5374:2
5370:a
5366:+
5361:]
5355:1
5348:1
5342:[
5335:1
5331:a
5307:,
5304:)
5301:2
5298:,
5295:3
5289:(
5286:=
5281:2
5276:v
5254:)
5251:1
5248:,
5245:1
5242:(
5239:=
5234:1
5229:v
5201:.
5198:)
5195:2
5192:,
5189:3
5183:(
5180:=
5175:2
5170:v
5148:)
5145:1
5142:,
5139:1
5136:(
5133:=
5128:1
5123:v
5101:)
5098:4
5095:,
5092:2
5089:(
5086:=
5081:3
5076:v
5063:i
5059:a
5042:.
5037:]
5031:5
5027:/
5023:2
5016:5
5012:/
5002:[
4995:3
4991:a
4984:=
4979:]
4971:2
4967:a
4957:1
4953:a
4946:[
4941:=
4936:]
4928:2
4924:a
4914:1
4910:a
4903:[
4896:]
4890:1
4885:0
4878:0
4873:1
4867:[
4839:.
4834:]
4828:0
4821:0
4815:[
4810:=
4805:]
4797:3
4793:a
4783:2
4779:a
4769:1
4765:a
4758:[
4751:]
4745:5
4741:/
4737:2
4732:1
4727:0
4720:5
4716:/
4707:0
4702:1
4696:[
4668:.
4663:]
4657:0
4650:0
4644:[
4639:=
4634:]
4626:3
4622:a
4612:2
4608:a
4598:1
4594:a
4587:[
4580:]
4574:2
4569:5
4564:0
4557:2
4552:3
4544:1
4538:[
4507:.
4502:]
4496:0
4489:0
4483:[
4478:=
4473:]
4465:3
4461:a
4451:2
4447:a
4437:1
4433:a
4426:[
4419:]
4413:4
4408:2
4403:1
4396:2
4391:3
4383:1
4377:[
4349:,
4344:]
4338:0
4331:0
4325:[
4320:=
4315:]
4309:4
4302:2
4296:[
4289:3
4285:a
4281:+
4276:]
4270:2
4263:3
4254:[
4247:2
4243:a
4239:+
4234:]
4228:1
4221:1
4215:[
4208:1
4204:a
4180:,
4177:)
4174:4
4171:,
4168:2
4165:(
4162:=
4157:3
4152:v
4130:,
4127:)
4124:2
4121:,
4118:3
4112:(
4109:=
4104:2
4099:v
4078:,
4075:)
4072:1
4069:,
4066:1
4063:(
4060:=
4055:1
4050:v
4016:u
3994:v
3970:v
3948:u
3922:v
3900:u
3875:0
3853:v
3831:u
3809:v
3788:c
3767:0
3763:=
3759:u
3737:v
3733:c
3730:=
3726:u
3699:0
3691:v
3669:0
3661:u
3640:.
3636:u
3630:c
3627:1
3622:=
3618:v
3595:c
3592:1
3569:0
3561:v
3538:0
3532:c
3511:0
3503:u
3481:v
3477:c
3474:=
3470:u
3446:1
3440:c
3415:0
3393:v
3389:=
3385:u
3364:.
3360:u
3356:0
3353:=
3349:v
3327:0
3323:=
3319:v
3297:v
3275:u
3271:=
3267:0
3263:=
3259:v
3255:0
3252:=
3248:v
3244:c
3224:0
3218:c
3197:0
3193:=
3189:u
3163:u
3159:c
3156:=
3152:v
3131:c
3110:u
3088:v
3064:v
3060:c
3057:=
3053:u
3032:c
3011:v
2989:u
2960:v
2938:u
2916:v
2894:u
2866:.
2862:0
2853:1
2848:v
2824:1
2819:v
2807:;
2794:0
2790:=
2785:1
2780:v
2757:1
2735:1
2730:v
2706:1
2701:v
2679:1
2676:=
2673:k
2653:1
2631:k
2626:v
2621:,
2615:,
2610:1
2605:v
2571:k
2566:v
2561:,
2555:,
2550:1
2545:v
2523:0
2515:i
2511:a
2487:.
2483:0
2479:=
2475:0
2469:i
2465:a
2461:=
2456:i
2451:v
2444:i
2440:a
2436:=
2432:0
2428:+
2422:+
2418:0
2414:+
2409:i
2404:v
2397:i
2393:a
2389:+
2385:0
2381:+
2375:+
2371:0
2367:=
2362:k
2357:v
2350:k
2346:a
2342:+
2336:+
2331:1
2326:v
2319:1
2315:a
2289:k
2284:v
2277:k
2273:a
2269:+
2263:+
2258:1
2253:v
2246:1
2242:a
2220:0
2216:=
2211:j
2206:v
2201:0
2198:=
2193:j
2188:v
2181:j
2177:a
2156:0
2148:j
2144:a
2123:i
2117:j
2097:i
2077:j
2057:0
2035:i
2031:a
2010:1
2002:i
1998:a
1977:.
1973:0
1969:=
1964:i
1959:v
1937:}
1934:k
1931:,
1925:,
1922:1
1919:{
1899:i
1877:k
1872:v
1867:,
1861:,
1856:1
1851:v
1828:0
1805:k
1800:v
1795:,
1789:,
1784:1
1779:v
1754:n
1750:n
1702:k
1696:0
1693:=
1684:o
1655:k
1626:o
1594:k
1565:v
1536:u
1507:k
1478:v
1449:u
1418:j
1389:u
1358:w
1329:v
1300:u
1265:v
1236:u
1206:x
1202:x
1196:x
1120:0
1097:i
1093:a
1071:0
1049:.
1046:n
1043:,
1037:,
1034:1
1031:=
1028:i
1008:0
1005:=
1000:i
996:a
972:,
968:0
964:=
959:n
954:v
947:n
943:a
939:+
933:+
928:2
923:v
916:2
912:a
908:+
903:1
898:v
891:1
887:a
857:n
852:v
847:,
841:,
836:2
831:v
826:,
821:1
816:v
782:=
779:k
758:0
754:=
749:1
744:v
722:,
719:1
713:k
690:,
685:k
680:v
671:1
667:a
660:k
656:a
646:+
640:+
635:2
630:v
621:1
617:a
610:2
606:a
596:=
591:1
586:v
561:0
553:1
549:a
524:0
500:,
496:0
492:=
487:k
482:v
475:k
471:a
467:+
461:+
456:2
451:v
444:2
440:a
436:+
431:1
426:v
419:1
415:a
391:,
386:k
382:a
378:,
372:,
367:2
363:a
359:,
354:1
350:a
332:V
313:k
308:v
303:,
297:,
292:2
287:v
282:,
277:1
272:v
199:.
194:3
189:R
158:3
153:R
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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