3444:
3708:
31:
3067:
2351:
528:
2536:
Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of
2246:
2734:
of the character can be viewed as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of
401:
2531:
2703:
647:
588:
1861:
850:
801:
1241:
2489:
2180:
1007:
699:
2088:
2445:
1321:
1182:
1122:
396:
88:
For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say
2604:
2491:
is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in
2409:
752:
232:
2346:{\displaystyle \operatorname {tr} \ \operatorname {id} _{\mathbb {R} ^{2}}=\operatorname {tr} \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)=1+1=2.}
2732:
1505:
2002:
359:
2970:
2630:
1540:
902:
1268:
1473:
1151:
1907:
1665:
1424:
1712:
1583:
1344:
725:
2758:
2377:
2110:
2026:
1973:
1953:
1933:
1881:
1772:
1752:
1732:
1689:
1623:
1603:
1560:
1444:
1394:
1366:
1288:
1069:
1049:
1029:
954:
930:
873:
327:
307:
287:
267:
193:
173:
134:
106:
3747:
2998:
3302:
523:{\displaystyle \left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}},{\begin{pmatrix}0\\0\\1\end{pmatrix}}\right\}}
3693:
2356:
Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an
2954:
2878:
2498:
1777:
2844:
2942:
3101:
2635:
3051:
2915:
2946:
3683:
3645:
3581:
593:
537:
2991:
806:
757:
1187:
3423:
3295:
3528:
3378:
3086:
2495:, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension (
2632:
is the dimension of the representation, as a representation sends the identity in the group to the identity matrix:
3433:
3327:
2771:
3673:
3322:
2984:
2456:
3548:
1629:
3665:
2199:
3737:
3711:
3418:
3288:
3229:
3224:
3204:
2565:
2115:
3742:
3475:
3408:
3398:
3214:
3209:
3189:
2236:
2230:
959:
58:
1376:
map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If
3635:
3490:
3485:
3480:
3413:
3358:
3219:
3199:
3194:
2807:
2801:
2561:
652:
2031:
2418:
2217:(1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of
1297:
1158:
1074:
372:
3500:
3465:
3452:
3343:
2357:
1011:
To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if
3678:
3558:
3533:
3383:
3096:
3091:
2736:
2574:
2569:
702:
66:
2382:
735:
198:
3732:
3388:
3270:
3111:
3066:
2195:
17:
3586:
3543:
3470:
3363:
3106:
2950:
2911:
2874:
2783:
2708:
2240:
1478:
1978:
332:
3591:
3495:
3348:
3036:
2868:
2763:
2609:
2568:
of a representation is the trace of the representation, hence a scalar-valued function on a
2546:
1510:
878:
1246:
3650:
3443:
3403:
3393:
3081:
3026:
2870:
Tensor
Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics
2832:
2789:
2206:
1668:
1449:
1291:
1127:
933:
1886:
1642:
1399:
1694:
1565:
1326:
707:
3655:
3640:
3576:
3311:
3163:
3148:
2743:
2362:
2214:
2210:
2095:
2011:
1958:
1938:
1918:
1866:
1757:
1737:
1717:
1674:
1608:
1588:
1545:
1429:
1379:
1351:
1273:
1054:
1034:
1014:
939:
915:
858:
730:
531:
312:
292:
272:
237:
178:
158:
119:
91:
2908:
Moonshine beyond the
Monster: The Bridge Connecting Algebra, Modular Forms and Physics
3726:
3688:
3611:
3571:
3538:
3518:
3153:
2934:
2767:
2542:
3621:
3510:
3460:
3353:
3173:
3138:
3031:
2795:
2550:
47:
2971:
MIT Linear
Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang
3601:
3566:
3523:
3368:
3258:
3041:
2740:
2538:
2353:
This appears to be a circular definition, but it allows useful generalizations.
2293:
2218:
1912:
74:
54:
39:
30:
3630:
3373:
3253:
3133:
2786: – Ratio providing a statistical index of complexity variation with scale
1633:
1369:
137:
3428:
3234:
3143:
3056:
3007:
2560:
of operators as a kind of "twisted" dimension. This occurs significantly in
2492:
1373:
82:
2235:
The dimension of a vector space may alternatively be characterized as the
3596:
3158:
3121:
3046:
149:
2533:), so in these cases the normalizing constant corresponds to dimension.
2202:
both have several properties similar to the dimension of vector spaces.
3168:
2526:{\displaystyle \epsilon :=\textstyle {\frac {1}{n}}\operatorname {tr} }
2191:
2810: – Topologically invariant definition of the dimension of a space
3606:
2449:
2194:, and in the latter there is a well-defined notion of dimension. The
3125:
29:
3280:
3284:
2980:
2698:{\displaystyle \chi (1_{G})=\operatorname {tr} \ I_{V}=\dim V.}
1915:
of the base field and the cardinality of the space itself. If
1911:
Some formulae relate the dimension of a vector space with the
1585:
These functions can be added and multiplied with elements of
2976:
2770:, and replacing the dimension with the character gives the
2798: – Maximum size of an independent set of the matroid
1774:-vector space. The dimensions are related by the formula
642:{\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{n})=n,}
583:{\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{3})=3.}
2766:
of an infinite-dimensional graded representation of the
2812:
Pages displaying short descriptions of redirect targets
2556:
A subtler generalization is to consider the trace of a
1863:
In particular, every complex vector space of dimension
2508:
1856:{\displaystyle \dim _{K}(V)=\dim _{K}(F)\dim _{F}(V).}
904:
the vector space consisting only of its zero element.
487:
451:
415:
2746:
2711:
2638:
2612:
2577:
2501:
2459:
2421:
2385:
2365:
2249:
2190:
A vector space can be seen as a particular case of a
2118:
2098:
2034:
2014:
1981:
1961:
1941:
1921:
1889:
1869:
1780:
1760:
1740:
1720:
1697:
1677:
1645:
1628:
An important result about dimensions is given by the
1611:
1591:
1568:
1548:
1513:
1481:
1452:
1432:
1402:
1382:
1354:
1329:
1300:
1276:
1249:
1190:
1161:
1130:
1077:
1057:
1037:
1017:
962:
942:
918:
881:
861:
845:{\displaystyle \dim _{\mathbb {C} }(\mathbb {C} )=1.}
809:
760:
738:
710:
655:
596:
540:
404:
375:
335:
315:
295:
275:
240:
201:
181:
161:
122:
94:
796:{\displaystyle \dim _{\mathbb {R} }(\mathbb {C} )=2}
3664:
3620:
3557:
3509:
3451:
3336:
3246:
3182:
3120:
3074:
3014:
1236:{\displaystyle \left\{e_{1},\ldots ,e_{n}\right\},}
2752:
2726:
2697:
2624:
2598:
2525:
2483:
2439:
2403:
2371:
2345:
2174:
2104:
2082:
2020:
1996:
1967:
1947:
1927:
1901:
1875:
1855:
1766:
1746:
1726:
1706:
1683:
1659:
1617:
1597:
1577:
1554:
1534:
1499:
1467:
1438:
1418:
1388:
1360:
1338:
1315:
1282:
1262:
1235:
1176:
1145:
1116:
1063:
1043:
1023:
1001:
948:
924:
896:
867:
844:
795:
754:are both a real and complex vector space; we have
746:
719:
693:
641:
582:
522:
390:
353:
321:
301:
281:
261:
226:
187:
167:
128:
100:
27:Number of vectors in any basis of the vector space
2804: – Dimension of the column space of a matrix
2135:
1348:Any two finite dimensional vector spaces over
3296:
2992:
8:
1446:can be constructed as follows: take the set
888:
882:
852:So the dimension depends on the base field.
2792: – In mathematics, dimension of a ring
2484:{\displaystyle \epsilon \circ \eta :K\to K}
1396:is some set, a vector space with dimension
3303:
3289:
3281:
2999:
2985:
2977:
2814:, also called Lebesgue covering dimension
2745:
2710:
2674:
2649:
2637:
2611:
2576:
2509:
2500:
2458:
2420:
2384:
2364:
2291:
2270:
2266:
2265:
2263:
2248:
2149:
2141:
2127:
2119:
2117:
2097:
2065:
2060:
2051:
2043:
2035:
2033:
2013:
1980:
1960:
1940:
1920:
1888:
1868:
1832:
1810:
1785:
1779:
1759:
1739:
1719:
1696:
1676:
1649:
1644:
1610:
1590:
1567:
1547:
1512:
1480:
1451:
1431:
1411:
1403:
1401:
1381:
1353:
1328:
1307:
1303:
1302:
1299:
1275:
1254:
1248:
1219:
1200:
1189:
1168:
1164:
1163:
1160:
1129:
1076:
1056:
1036:
1031:is a finite-dimensional vector space and
1016:
961:
941:
917:
880:
860:
829:
828:
816:
815:
814:
808:
780:
779:
767:
766:
765:
759:
740:
739:
737:
709:
676:
660:
654:
621:
617:
616:
603:
602:
601:
595:
565:
561:
560:
547:
546:
545:
539:
482:
446:
410:
403:
382:
378:
377:
374:
334:
314:
294:
274:
239:
206:
200:
180:
160:
121:
93:
2859:
2824:
2774:for each element of the Monster group.
3694:Comparison of linear algebra libraries
2411:(the inclusion of scalars, called the
81:to distinguish it from other types of
34:A diagram of dimensions 1, 2, 3, and 4
2893:
2175:{\displaystyle |V|=\max(|F|,\dim V).}
1691:is in particular a vector space over
855:The only vector space with dimension
7:
2447:(corresponding to trace, called the
1883:is a real vector space of dimension
1002:{\displaystyle \dim(W)\leq \dim(V).}
2845:dimension theorem for vector spaces
57:(i.e., the number of vectors) of a
2943:Undergraduate Texts in Mathematics
694:{\displaystyle \dim _{F}(F^{n})=n}
155:The dimension of the vector space
25:
3748:Vectors (mathematics and physics)
2292:
2083:{\displaystyle |V|=|F|^{\dim V}.}
3707:
3706:
3684:Basic Linear Algebra Subprograms
3442:
3065:
2440:{\displaystyle \epsilon :A\to K}
1316:{\displaystyle \mathbb {R} ^{n}}
1290:-th column of the corresponding
1177:{\displaystyle \mathbb {R} ^{n}}
1117:{\displaystyle \dim(W)=\dim(V),}
391:{\displaystyle \mathbb {R} ^{3}}
3582:Seven-dimensional cross product
1935:is a vector space over a field
18:Finite-dimensional vector space
2910:, Cambridge University Press,
2721:
2715:
2655:
2642:
2587:
2475:
2431:
2395:
2166:
2150:
2142:
2138:
2128:
2120:
2061:
2052:
2044:
2036:
1847:
1841:
1825:
1819:
1800:
1794:
1523:
1517:
1491:
1462:
1456:
1412:
1404:
1108:
1102:
1090:
1084:
993:
987:
975:
969:
833:
825:
784:
776:
682:
669:
627:
612:
571:
556:
348:
342:
329:can be inferred from context,
253:
241:
221:
215:
1:
2599:{\displaystyle \chi :G\to K,}
3424:Eigenvalues and eigenvectors
2606:whose value on the identity
2404:{\displaystyle \eta :K\to A}
1368:with the same dimension are
747:{\displaystyle \mathbb {C} }
227:{\displaystyle \dim _{F}(V)}
3764:
2228:
1542:for all but finitely many
3702:
3440:
3318:
3267:
3063:
2939:Linear Algebra Done Right
649:and even more generally,
69:. It is sometimes called
2867:Itzkov, Mikhail (2009).
2727:{\displaystyle \chi (g)}
2200:rank of an abelian group
1955:and if the dimension of
1500:{\displaystyle f:B\to F}
1051:is a linear subspace of
2873:. Springer. p. 4.
1997:{\displaystyle \dim V,}
1184:has the standard basis
354:{\displaystyle \dim(V)}
3409:Row and column vectors
2906:Gannon, Terry (2006),
2754:
2728:
2699:
2626:
2625:{\displaystyle 1\in G}
2600:
2527:
2485:
2441:
2405:
2373:
2347:
2231:Trace (linear algebra)
2176:
2106:
2084:
2022:
1998:
1969:
1949:
1929:
1903:
1877:
1857:
1768:
1748:
1728:
1708:
1685:
1661:
1619:
1605:to obtain the desired
1599:
1579:
1556:
1536:
1535:{\displaystyle f(b)=0}
1501:
1469:
1440:
1420:
1390:
1362:
1340:
1317:
1284:
1264:
1237:
1178:
1147:
1118:
1065:
1045:
1025:
1003:
950:
926:
898:
897:{\displaystyle \{0\},}
869:
846:
797:
748:
721:
695:
643:
584:
524:
392:
361:is typically written.
355:
323:
303:
283:
263:
228:
189:
169:
130:
102:
35:
3414:Row and column spaces
3359:Scalar multiplication
2973:at MIT OpenCourseWare
2808:Topological dimension
2802:Rank (linear algebra)
2772:McKay–Thompson series
2755:
2729:
2700:
2627:
2601:
2562:representation theory
2528:
2486:
2442:
2406:
2374:
2348:
2177:
2107:
2085:
2023:
1999:
1970:
1950:
1930:
1904:
1878:
1858:
1769:
1749:
1729:
1709:
1686:
1662:
1620:
1600:
1580:
1557:
1537:
1502:
1470:
1441:
1421:
1391:
1363:
1341:
1318:
1285:
1265:
1263:{\displaystyle e_{i}}
1238:
1179:
1148:
1119:
1066:
1046:
1026:
1004:
951:
927:
899:
870:
847:
798:
749:
722:
696:
644:
585:
525:
393:
356:
324:
304:
284:
264:
229:
190:
170:
131:
103:
33:
3549:Gram–Schmidt process
3501:Gaussian elimination
3183:Dimensions by number
2744:
2709:
2636:
2610:
2575:
2545:, or more generally
2499:
2457:
2419:
2383:
2363:
2247:
2116:
2096:
2032:
2012:
1979:
1959:
1939:
1919:
1887:
1867:
1778:
1758:
1738:
1718:
1695:
1675:
1643:
1630:rank–nullity theorem
1609:
1589:
1566:
1546:
1511:
1479:
1468:{\displaystyle F(B)}
1450:
1430:
1400:
1380:
1352:
1327:
1298:
1274:
1247:
1188:
1159:
1146:{\displaystyle W=V.}
1128:
1075:
1055:
1035:
1015:
960:
940:
916:
879:
859:
807:
758:
736:
708:
653:
594:
538:
402:
373:
333:
313:
293:
273:
238:
199:
179:
159:
148:if its dimension is
144:infinite-dimensional
120:
116:if the dimension of
92:
3679:Numerical stability
3559:Multilinear algebra
3534:Inner product space
3384:Linear independence
2831:if one assumes the
2737:monstrous moonshine
2453:). The composition
1902:{\displaystyle 2n.}
1714:Furthermore, every
1660:{\displaystyle F/K}
1419:{\displaystyle |B|}
269:read "dimension of
79:algebraic dimension
3389:Linear combination
3112:Degrees of freedom
3015:Dimensional spaces
2750:
2724:
2695:
2622:
2596:
2523:
2522:
2481:
2437:
2401:
2369:
2343:
2319:
2318:
2196:length of a module
2172:
2102:
2080:
2018:
1994:
1965:
1945:
1925:
1899:
1873:
1853:
1764:
1744:
1724:
1707:{\displaystyle K.}
1704:
1681:
1657:
1615:
1595:
1578:{\displaystyle B.}
1575:
1552:
1532:
1497:
1465:
1436:
1416:
1386:
1358:
1339:{\displaystyle n.}
1336:
1313:
1280:
1260:
1233:
1174:
1143:
1114:
1061:
1041:
1021:
999:
946:
922:
894:
865:
842:
793:
744:
720:{\displaystyle F.}
717:
691:
639:
580:
520:
509:
473:
437:
388:
351:
319:
299:
279:
259:
224:
195:can be written as
185:
165:
126:
112:finite-dimensional
98:
36:
3720:
3719:
3587:Geometric algebra
3544:Kronecker product
3379:Linear projection
3364:Vector projection
3278:
3277:
3087:Lebesgue covering
3052:Algebraic variety
2956:978-3-319-11079-0
2880:978-3-540-93906-1
2784:Fractal dimension
2753:{\displaystyle j}
2705:The other values
2669:
2547:nuclear operators
2517:
2372:{\displaystyle A}
2258:
2243:. For instance,
2241:identity operator
2209:of a commutative
2112:is infinite then
2105:{\displaystyle V}
2021:{\displaystyle V}
1968:{\displaystyle V}
1948:{\displaystyle F}
1928:{\displaystyle V}
1876:{\displaystyle n}
1767:{\displaystyle K}
1747:{\displaystyle V}
1727:{\displaystyle F}
1684:{\displaystyle F}
1618:{\displaystyle F}
1598:{\displaystyle F}
1555:{\displaystyle b}
1475:of all functions
1439:{\displaystyle F}
1389:{\displaystyle B}
1361:{\displaystyle F}
1283:{\displaystyle i}
1064:{\displaystyle V}
1044:{\displaystyle W}
1024:{\displaystyle V}
949:{\displaystyle V}
925:{\displaystyle W}
868:{\displaystyle 0}
369:The vector space
322:{\displaystyle F}
302:{\displaystyle F}
282:{\displaystyle V}
262:{\displaystyle ,}
188:{\displaystyle F}
168:{\displaystyle V}
129:{\displaystyle V}
101:{\displaystyle V}
16:(Redirected from
3755:
3710:
3709:
3592:Exterior algebra
3529:Hadamard product
3446:
3434:Linear equations
3305:
3298:
3291:
3282:
3075:Other dimensions
3069:
3037:Projective space
3001:
2994:
2987:
2978:
2960:
2945:(3rd ed.).
2921:
2920:
2903:
2897:
2891:
2885:
2884:
2864:
2847:
2841:
2835:
2829:
2813:
2764:graded dimension
2759:
2757:
2756:
2751:
2733:
2731:
2730:
2725:
2704:
2702:
2701:
2696:
2679:
2678:
2667:
2654:
2653:
2631:
2629:
2628:
2623:
2605:
2603:
2602:
2597:
2541:operators" on a
2532:
2530:
2529:
2524:
2518:
2510:
2490:
2488:
2487:
2482:
2446:
2444:
2443:
2438:
2410:
2408:
2407:
2402:
2378:
2376:
2375:
2370:
2352:
2350:
2349:
2344:
2324:
2320:
2277:
2276:
2275:
2274:
2269:
2256:
2181:
2179:
2178:
2173:
2153:
2145:
2131:
2123:
2111:
2109:
2108:
2103:
2089:
2087:
2086:
2081:
2076:
2075:
2064:
2055:
2047:
2039:
2027:
2025:
2024:
2019:
2003:
2001:
2000:
1995:
1974:
1972:
1971:
1966:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1908:
1906:
1905:
1900:
1882:
1880:
1879:
1874:
1862:
1860:
1859:
1854:
1837:
1836:
1815:
1814:
1790:
1789:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1733:
1731:
1730:
1725:
1713:
1711:
1710:
1705:
1690:
1688:
1687:
1682:
1666:
1664:
1663:
1658:
1653:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1561:
1559:
1558:
1553:
1541:
1539:
1538:
1533:
1506:
1504:
1503:
1498:
1474:
1472:
1471:
1466:
1445:
1443:
1442:
1437:
1425:
1423:
1422:
1417:
1415:
1407:
1395:
1393:
1392:
1387:
1367:
1365:
1364:
1359:
1345:
1343:
1342:
1337:
1322:
1320:
1319:
1314:
1312:
1311:
1306:
1289:
1287:
1286:
1281:
1269:
1267:
1266:
1261:
1259:
1258:
1242:
1240:
1239:
1234:
1229:
1225:
1224:
1223:
1205:
1204:
1183:
1181:
1180:
1175:
1173:
1172:
1167:
1152:
1150:
1149:
1144:
1123:
1121:
1120:
1115:
1070:
1068:
1067:
1062:
1050:
1048:
1047:
1042:
1030:
1028:
1027:
1022:
1008:
1006:
1005:
1000:
955:
953:
952:
947:
931:
929:
928:
923:
903:
901:
900:
895:
874:
872:
871:
866:
851:
849:
848:
843:
832:
821:
820:
819:
802:
800:
799:
794:
783:
772:
771:
770:
753:
751:
750:
745:
743:
726:
724:
723:
718:
700:
698:
697:
692:
681:
680:
665:
664:
648:
646:
645:
640:
626:
625:
620:
608:
607:
606:
590:More generally,
589:
587:
586:
581:
570:
569:
564:
552:
551:
550:
534:, and therefore
529:
527:
526:
521:
519:
515:
514:
513:
478:
477:
442:
441:
397:
395:
394:
389:
387:
386:
381:
360:
358:
357:
352:
328:
326:
325:
320:
308:
306:
305:
300:
288:
286:
285:
280:
268:
266:
265:
260:
233:
231:
230:
225:
211:
210:
194:
192:
191:
186:
174:
172:
171:
166:
146:
145:
135:
133:
132:
127:
114:
113:
107:
105:
104:
99:
21:
3763:
3762:
3758:
3757:
3756:
3754:
3753:
3752:
3723:
3722:
3721:
3716:
3698:
3660:
3616:
3553:
3505:
3447:
3438:
3404:Change of basis
3394:Multilinear map
3332:
3314:
3309:
3279:
3274:
3263:
3242:
3178:
3116:
3070:
3061:
3027:Euclidean space
3010:
3005:
2967:
2957:
2933:
2930:
2925:
2924:
2918:
2905:
2904:
2900:
2892:
2888:
2881:
2866:
2865:
2861:
2856:
2851:
2850:
2842:
2838:
2833:axiom of choice
2830:
2826:
2821:
2811:
2790:Krull dimension
2780:
2742:
2741:
2707:
2706:
2670:
2645:
2634:
2633:
2608:
2607:
2573:
2572:
2497:
2496:
2455:
2454:
2417:
2416:
2381:
2380:
2361:
2360:
2317:
2316:
2311:
2305:
2304:
2299:
2287:
2264:
2259:
2245:
2244:
2233:
2227:
2207:Krull dimension
2188:
2186:Generalizations
2114:
2113:
2094:
2093:
2059:
2030:
2029:
2028:is finite then
2010:
2009:
1977:
1976:
1957:
1956:
1937:
1936:
1917:
1916:
1885:
1884:
1865:
1864:
1828:
1806:
1781:
1776:
1775:
1756:
1755:
1736:
1735:
1716:
1715:
1693:
1692:
1673:
1672:
1669:field extension
1641:
1640:
1625:-vector space.
1607:
1606:
1587:
1586:
1564:
1563:
1544:
1543:
1509:
1508:
1477:
1476:
1448:
1447:
1428:
1427:
1398:
1397:
1378:
1377:
1350:
1349:
1325:
1324:
1301:
1296:
1295:
1292:identity matrix
1272:
1271:
1250:
1245:
1244:
1215:
1196:
1195:
1191:
1186:
1185:
1162:
1157:
1156:
1126:
1125:
1073:
1072:
1053:
1052:
1033:
1032:
1013:
1012:
958:
957:
938:
937:
934:linear subspace
914:
913:
910:
877:
876:
857:
856:
810:
805:
804:
761:
756:
755:
734:
733:
731:complex numbers
706:
705:
672:
656:
651:
650:
615:
597:
592:
591:
559:
541:
536:
535:
508:
507:
501:
500:
494:
493:
483:
472:
471:
465:
464:
458:
457:
447:
436:
435:
429:
428:
422:
421:
411:
409:
405:
400:
399:
376:
371:
370:
367:
331:
330:
311:
310:
291:
290:
271:
270:
236:
235:
202:
197:
196:
177:
176:
175:over the field
157:
156:
143:
142:
118:
117:
111:
110:
90:
89:
71:Hamel dimension
28:
23:
22:
15:
12:
11:
5:
3761:
3759:
3751:
3750:
3745:
3740:
3738:Linear algebra
3735:
3725:
3724:
3718:
3717:
3715:
3714:
3703:
3700:
3699:
3697:
3696:
3691:
3686:
3681:
3676:
3674:Floating-point
3670:
3668:
3662:
3661:
3659:
3658:
3656:Tensor product
3653:
3648:
3643:
3641:Function space
3638:
3633:
3627:
3625:
3618:
3617:
3615:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3577:Triple product
3574:
3569:
3563:
3561:
3555:
3554:
3552:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3515:
3513:
3507:
3506:
3504:
3503:
3498:
3493:
3491:Transformation
3488:
3483:
3481:Multiplication
3478:
3473:
3468:
3463:
3457:
3455:
3449:
3448:
3441:
3439:
3437:
3436:
3431:
3426:
3421:
3416:
3411:
3406:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3361:
3356:
3351:
3346:
3340:
3338:
3337:Basic concepts
3334:
3333:
3331:
3330:
3325:
3319:
3316:
3315:
3312:Linear algebra
3310:
3308:
3307:
3300:
3293:
3285:
3276:
3275:
3268:
3265:
3264:
3262:
3261:
3256:
3250:
3248:
3244:
3243:
3241:
3240:
3232:
3227:
3222:
3217:
3212:
3207:
3202:
3197:
3192:
3186:
3184:
3180:
3179:
3177:
3176:
3171:
3166:
3164:Cross-polytope
3161:
3156:
3151:
3149:Hyperrectangle
3146:
3141:
3136:
3130:
3128:
3118:
3117:
3115:
3114:
3109:
3104:
3099:
3094:
3089:
3084:
3078:
3076:
3072:
3071:
3064:
3062:
3060:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3018:
3016:
3012:
3011:
3006:
3004:
3003:
2996:
2989:
2981:
2975:
2974:
2966:
2965:External links
2963:
2962:
2961:
2955:
2935:Axler, Sheldon
2929:
2926:
2923:
2922:
2916:
2898:
2886:
2879:
2858:
2857:
2855:
2852:
2849:
2848:
2836:
2823:
2822:
2820:
2817:
2816:
2815:
2805:
2799:
2793:
2787:
2779:
2776:
2749:
2723:
2720:
2717:
2714:
2694:
2691:
2688:
2685:
2682:
2677:
2673:
2666:
2663:
2660:
2657:
2652:
2648:
2644:
2641:
2621:
2618:
2615:
2595:
2592:
2589:
2586:
2583:
2580:
2521:
2516:
2513:
2507:
2504:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2436:
2433:
2430:
2427:
2424:
2400:
2397:
2394:
2391:
2388:
2368:
2342:
2339:
2336:
2333:
2330:
2327:
2323:
2315:
2312:
2310:
2307:
2306:
2303:
2300:
2298:
2295:
2294:
2290:
2286:
2283:
2280:
2273:
2268:
2262:
2255:
2252:
2226:
2223:
2215:Wolfgang Krull
2213:, named after
2187:
2184:
2183:
2182:
2171:
2168:
2165:
2162:
2159:
2156:
2152:
2148:
2144:
2140:
2137:
2134:
2130:
2126:
2122:
2101:
2090:
2079:
2074:
2071:
2068:
2063:
2058:
2054:
2050:
2046:
2042:
2038:
2017:
1993:
1990:
1987:
1984:
1975:is denoted by
1964:
1944:
1924:
1898:
1895:
1892:
1872:
1852:
1849:
1846:
1843:
1840:
1835:
1831:
1827:
1824:
1821:
1818:
1813:
1809:
1805:
1802:
1799:
1796:
1793:
1788:
1784:
1763:
1743:
1734:-vector space
1723:
1703:
1700:
1680:
1656:
1652:
1648:
1614:
1594:
1574:
1571:
1551:
1531:
1528:
1525:
1522:
1519:
1516:
1496:
1493:
1490:
1487:
1484:
1464:
1461:
1458:
1455:
1435:
1414:
1410:
1406:
1385:
1357:
1335:
1332:
1323:has dimension
1310:
1305:
1279:
1257:
1253:
1232:
1228:
1222:
1218:
1214:
1211:
1208:
1203:
1199:
1194:
1171:
1166:
1142:
1139:
1136:
1133:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1060:
1040:
1020:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
945:
921:
909:
906:
893:
890:
887:
884:
864:
841:
838:
835:
831:
827:
824:
818:
813:
792:
789:
786:
782:
778:
775:
769:
764:
742:
716:
713:
690:
687:
684:
679:
675:
671:
668:
663:
659:
638:
635:
632:
629:
624:
619:
614:
611:
605:
600:
579:
576:
573:
568:
563:
558:
555:
549:
544:
532:standard basis
518:
512:
506:
503:
502:
499:
496:
495:
492:
489:
488:
486:
481:
476:
470:
467:
466:
463:
460:
459:
456:
453:
452:
450:
445:
440:
434:
431:
430:
427:
424:
423:
420:
417:
416:
414:
408:
385:
380:
366:
363:
350:
347:
344:
341:
338:
318:
298:
278:
258:
255:
252:
249:
246:
243:
223:
220:
217:
214:
209:
205:
184:
164:
125:
97:
65:over its base
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3760:
3749:
3746:
3744:
3743:Vector spaces
3741:
3739:
3736:
3734:
3731:
3730:
3728:
3713:
3705:
3704:
3701:
3695:
3692:
3690:
3689:Sparse matrix
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3671:
3669:
3667:
3663:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3628:
3626:
3624:constructions
3623:
3619:
3613:
3612:Outermorphism
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3572:Cross product
3570:
3568:
3565:
3564:
3562:
3560:
3556:
3550:
3547:
3545:
3542:
3540:
3539:Outer product
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3519:Orthogonality
3517:
3516:
3514:
3512:
3508:
3502:
3499:
3497:
3496:Cramer's rule
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3466:Decomposition
3464:
3462:
3459:
3458:
3456:
3454:
3450:
3445:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3341:
3339:
3335:
3329:
3326:
3324:
3321:
3320:
3317:
3313:
3306:
3301:
3299:
3294:
3292:
3287:
3286:
3283:
3273:
3272:
3266:
3260:
3257:
3255:
3252:
3251:
3249:
3245:
3239:
3237:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3185:
3181:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3154:Demihypercube
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3131:
3129:
3127:
3123:
3119:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3079:
3077:
3073:
3068:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3019:
3017:
3013:
3009:
3002:
2997:
2995:
2990:
2988:
2983:
2982:
2979:
2972:
2969:
2968:
2964:
2958:
2952:
2948:
2944:
2940:
2936:
2932:
2931:
2927:
2919:
2917:0-521-83531-3
2913:
2909:
2902:
2899:
2895:
2890:
2887:
2882:
2876:
2872:
2871:
2863:
2860:
2853:
2846:
2840:
2837:
2834:
2828:
2825:
2818:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2781:
2777:
2775:
2773:
2769:
2768:monster group
2765:
2761:
2747:
2738:
2718:
2712:
2692:
2689:
2686:
2683:
2680:
2675:
2671:
2664:
2661:
2658:
2650:
2646:
2639:
2619:
2616:
2613:
2593:
2590:
2584:
2581:
2578:
2571:
2567:
2563:
2559:
2554:
2552:
2548:
2544:
2543:Hilbert space
2540:
2534:
2519:
2514:
2511:
2505:
2502:
2494:
2478:
2472:
2469:
2466:
2463:
2460:
2452:
2451:
2434:
2428:
2425:
2422:
2414:
2398:
2392:
2389:
2386:
2366:
2359:
2354:
2340:
2337:
2334:
2331:
2328:
2325:
2321:
2313:
2308:
2301:
2296:
2288:
2284:
2281:
2278:
2271:
2260:
2253:
2250:
2242:
2238:
2232:
2224:
2222:
2221:in the ring.
2220:
2216:
2212:
2208:
2203:
2201:
2197:
2193:
2185:
2169:
2163:
2160:
2157:
2154:
2146:
2132:
2124:
2099:
2091:
2077:
2072:
2069:
2066:
2056:
2048:
2040:
2015:
2007:
2006:
2005:
1991:
1988:
1985:
1982:
1962:
1942:
1922:
1914:
1909:
1896:
1893:
1890:
1870:
1850:
1844:
1838:
1833:
1829:
1822:
1816:
1811:
1807:
1803:
1797:
1791:
1786:
1782:
1761:
1741:
1721:
1701:
1698:
1678:
1670:
1654:
1650:
1646:
1637:
1635:
1631:
1626:
1612:
1592:
1572:
1569:
1549:
1529:
1526:
1520:
1514:
1494:
1488:
1485:
1482:
1459:
1453:
1433:
1408:
1383:
1375:
1371:
1355:
1346:
1333:
1330:
1308:
1294:. Therefore,
1293:
1277:
1255:
1251:
1230:
1226:
1220:
1216:
1212:
1209:
1206:
1201:
1197:
1192:
1169:
1153:
1140:
1137:
1134:
1131:
1111:
1105:
1099:
1096:
1093:
1087:
1081:
1078:
1058:
1038:
1018:
1009:
996:
990:
984:
981:
978:
972:
966:
963:
943:
935:
919:
907:
905:
891:
885:
862:
853:
839:
836:
822:
811:
790:
787:
773:
762:
732:
727:
714:
711:
704:
688:
685:
677:
673:
666:
661:
657:
636:
633:
630:
622:
609:
598:
577:
574:
566:
553:
542:
533:
516:
510:
504:
497:
490:
484:
479:
474:
468:
461:
454:
448:
443:
438:
432:
425:
418:
412:
406:
383:
364:
362:
345:
339:
336:
316:
296:
276:
256:
250:
247:
244:
218:
212:
207:
203:
182:
162:
153:
151:
147:
139:
123:
115:
95:
86:
84:
80:
76:
72:
68:
64:
60:
56:
52:
49:
45:
41:
32:
19:
3622:Vector space
3354:Vector space
3269:
3235:
3174:Hyperpyramid
3139:Hypersurface
3032:Affine space
3022:Vector space
3021:
2938:
2907:
2901:
2896:p. 44, §2.36
2894:Axler (2015)
2889:
2869:
2862:
2839:
2827:
2796:Matroid rank
2564:, where the
2557:
2555:
2551:Banach space
2535:
2448:
2415:) and a map
2412:
2355:
2234:
2219:prime ideals
2204:
2189:
1910:
1638:
1627:
1347:
1154:
1010:
911:
854:
728:
368:
154:
141:
109:
87:
78:
70:
62:
50:
48:vector space
43:
37:
3602:Multivector
3567:Determinant
3524:Dot product
3369:Linear span
3259:Codimension
3238:-dimensions
3159:Hypersphere
3042:Free module
2539:trace class
1913:cardinality
1634:linear maps
75:Georg Hamel
55:cardinality
40:mathematics
3727:Categories
3636:Direct sum
3471:Invertible
3374:Linear map
3254:Hyperspace
3134:Hyperplane
2854:References
2760:-invariant
2493:bialgebras
2379:with maps
2229:See also:
1754:is also a
1507:such that
1370:isomorphic
1155:The space
908:Properties
3733:Dimension
3666:Numerical
3429:Transpose
3144:Hypercube
3122:Polytopes
3102:Minkowski
3097:Hausdorff
3092:Inductive
3057:Spacetime
3008:Dimension
2713:χ
2687:
2665:
2640:χ
2617:∈
2588:→
2579:χ
2566:character
2503:ϵ
2476:→
2467:η
2464:∘
2461:ϵ
2432:→
2423:ϵ
2396:→
2387:η
2285:
2254:
2161:
2070:
1986:
1839:
1817:
1792:
1492:→
1374:bijective
1210:…
1100:
1082:
985:
979:≤
967:
823:
774:
667:
610:
554:
340:
213:
83:dimension
44:dimension
3712:Category
3651:Subspace
3646:Quotient
3597:Bivector
3511:Bilinear
3453:Matrices
3328:Glossary
3271:Category
3247:See also
3047:Manifold
2947:Springer
2937:(2015).
2778:See also
2198:and the
701:for any
365:Examples
309:". When
150:infinite
3323:Outline
3169:Simplex
3107:Fractal
2928:Sources
2762:is the
2358:algebra
2239:of the
2192:matroid
2092:If dim
2008:If dim
1671:, then
1270:is the
73:(after
53:is the
3607:Tensor
3419:Kernel
3349:Vector
3344:Scalar
3126:shapes
2953:
2914:
2877:
2739:: the
2668:
2558:family
2450:counit
2257:
2004:then:
1372:. Any
1243:where
234:or as
140:, and
138:finite
42:, the
3476:Minor
3461:Block
3399:Basis
3230:Eight
3225:Seven
3205:Three
3082:Krull
2819:Notes
2570:group
2549:on a
2237:trace
2225:Trace
1667:is a
1426:over
1124:then
1071:with
956:then
932:is a
703:field
530:as a
289:over
77:) or
67:field
59:basis
46:of a
3631:Dual
3486:Rank
3215:Five
3210:Four
3190:Zero
3124:and
2951:ISBN
2912:ISBN
2875:ISBN
2843:see
2413:unit
2211:ring
2205:The
1632:for
803:and
729:The
398:has
3220:Six
3200:Two
3195:One
2684:dim
2158:dim
2136:max
2067:dim
1983:dim
1830:dim
1808:dim
1783:dim
1639:If
1562:in
1097:dim
1079:dim
982:dim
964:dim
936:of
912:If
875:is
812:dim
763:dim
658:dim
599:dim
543:dim
337:dim
204:dim
136:is
108:is
61:of
38:In
3729::
2949:.
2941:.
2662:tr
2553:.
2520:tr
2506::=
2341:2.
2282:tr
2261:id
2251:tr
1636:.
840:1.
578:3.
152:.
85:.
3304:e
3297:t
3290:v
3236:n
3000:e
2993:t
2986:v
2959:.
2883:.
2748:j
2722:)
2719:g
2716:(
2693:.
2690:V
2681:=
2676:V
2672:I
2659:=
2656:)
2651:G
2647:1
2643:(
2620:G
2614:1
2594:,
2591:K
2585:G
2582::
2537:"
2515:n
2512:1
2479:K
2473:K
2470::
2435:K
2429:A
2426::
2399:A
2393:K
2390::
2367:A
2338:=
2335:1
2332:+
2329:1
2326:=
2322:)
2314:1
2309:0
2302:0
2297:1
2289:(
2279:=
2272:2
2267:R
2170:.
2167:)
2164:V
2155:,
2151:|
2147:F
2143:|
2139:(
2133:=
2129:|
2125:V
2121:|
2100:V
2078:.
2073:V
2062:|
2057:F
2053:|
2049:=
2045:|
2041:V
2037:|
2016:V
1992:,
1989:V
1963:V
1943:F
1923:V
1897:.
1894:n
1891:2
1871:n
1851:.
1848:)
1845:V
1842:(
1834:F
1826:)
1823:F
1820:(
1812:K
1804:=
1801:)
1798:V
1795:(
1787:K
1762:K
1742:V
1722:F
1702:.
1699:K
1679:F
1655:K
1651:/
1647:F
1613:F
1593:F
1573:.
1570:B
1550:b
1530:0
1527:=
1524:)
1521:b
1518:(
1515:f
1495:F
1489:B
1486::
1483:f
1463:)
1460:B
1457:(
1454:F
1434:F
1413:|
1409:B
1405:|
1384:B
1356:F
1334:.
1331:n
1309:n
1304:R
1278:i
1256:i
1252:e
1231:,
1227:}
1221:n
1217:e
1213:,
1207:,
1202:1
1198:e
1193:{
1170:n
1165:R
1141:.
1138:V
1135:=
1132:W
1112:,
1109:)
1106:V
1103:(
1094:=
1091:)
1088:W
1085:(
1059:V
1039:W
1019:V
997:.
994:)
991:V
988:(
976:)
973:W
970:(
944:V
920:W
892:,
889:}
886:0
883:{
863:0
837:=
834:)
830:C
826:(
817:C
791:2
788:=
785:)
781:C
777:(
768:R
741:C
715:.
712:F
689:n
686:=
683:)
678:n
674:F
670:(
662:F
637:,
634:n
631:=
628:)
623:n
618:R
613:(
604:R
575:=
572:)
567:3
562:R
557:(
548:R
517:}
511:)
505:1
498:0
491:0
485:(
480:,
475:)
469:0
462:1
455:0
449:(
444:,
439:)
433:0
426:0
419:1
413:(
407:{
384:3
379:R
349:)
346:V
343:(
317:F
297:F
277:V
257:,
254:]
251:F
248::
245:V
242:[
222:)
219:V
216:(
208:F
183:F
163:V
124:V
96:V
63:V
51:V
20:)
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