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Primordial element (algebra)

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denote the set of all indices for which the expression of
545:{\displaystyle I(v)=\left\{i\in I:a_{i}(v)\neq 0\right\}} 1304: 794: 752: 715: 683: 654: 621: 598: 578: 558: 479: 452: 404: 384: 316: 290: 267: 242: 193: 171: 148: 1221: 1177: 1114: 1066: 1008: 893: 46:. Unsourced material may be challenged and removed. 803: 780: 736: 701: 669: 633: 607: 584: 564: 544: 465: 438: 390: 370: 302: 276: 248: 227: 179: 154: 645:if it has both of the following two properties: 1324: 853: 572:has a nonzero coefficient. Given a subspace 371:{\displaystyle v=\sum _{i\in I}a_{i}(v)e_{i}} 8: 439:{\displaystyle \left(a_{i}\right)_{i\in I}} 284:By the definition of a basis, every vector 228:{\displaystyle \left(e_{i}\right)_{i\in I}} 1331: 1317: 860: 846: 838: 793: 757: 751: 714: 682: 653: 620: 597: 577: 557: 516: 478: 457: 451: 424: 414: 403: 383: 362: 343: 327: 315: 289: 266: 241: 213: 203: 192: 173: 172: 170: 147: 106:Learn how and when to remove this message 820: 1251:Comparison of linear algebra libraries 7: 1285: 1283: 831:, updated March 23, 2013, Ch IV, §2. 44:adding citations to reliable sources 1303:. You can help Knowledge (XXG) by 14: 1355:Vectors (mathematics and physics) 55:"Primordial element" algebra 1287: 1264: 1263: 1241:Basic Linear Algebra Subprograms 999: 20: 1139:Seven-dimensional cross product 829:Class field theory course notes 31:needs additional citations for 769: 763: 693: 687: 664: 658: 528: 522: 489: 483: 355: 349: 1: 737:{\displaystyle 0\neq w\in W,} 310:can be expressed uniquely as 981:Eigenvalues and eigenvectors 446:where all but finitely many 180:{\displaystyle \mathbb {F} } 398:-indexed family of scalars 1376: 1282: 781:{\displaystyle a_{i}(p)=1} 677:is minimal among the sets 126:is a particular kind of a 1259: 997: 875: 162:be a vector space over a 1299:-related article is a 966:Row and column vectors 805: 782: 738: 703: 671: 635: 634:{\displaystyle p\in W} 609: 586: 566: 546: 467: 440: 392: 372: 304: 303:{\displaystyle v\in V} 278: 250: 229: 181: 156: 971:Row and column spaces 916:Scalar multiplication 806: 783: 739: 704: 702:{\displaystyle I(w),} 672: 636: 610: 587: 567: 547: 468: 466:{\displaystyle a_{i}} 441: 393: 373: 305: 279: 251: 230: 182: 157: 1360:Linear algebra stubs 1106:Gram–Schmidt process 1058:Gaussian elimination 792: 750: 713: 681: 670:{\displaystyle I(p)} 652: 619: 596: 576: 556: 477: 450: 402: 382: 314: 288: 265: 240: 191: 169: 146: 40:improve this article 1236:Numerical stability 1116:Multilinear algebra 1091:Inner product space 941:Linear independence 946:Linear combination 804:{\displaystyle i.} 801: 778: 734: 699: 667: 631: 608:{\displaystyle V,} 605: 582: 562: 542: 463: 436: 388: 368: 338: 300: 277:{\displaystyle V.} 274: 246: 225: 177: 152: 124:primordial element 1312: 1311: 1277: 1276: 1144:Geometric algebra 1101:Kronecker product 936:Linear projection 921:Vector projection 615:a nonzero vector 585:{\displaystyle W} 565:{\displaystyle v} 391:{\displaystyle I} 323: 249:{\displaystyle I} 155:{\displaystyle V} 116: 115: 108: 90: 1367: 1333: 1326: 1319: 1291: 1284: 1267: 1266: 1149:Exterior algebra 1086:Hadamard product 1003: 991:Linear equations 862: 855: 848: 839: 832: 825: 810: 808: 807: 802: 787: 785: 784: 779: 762: 761: 743: 741: 740: 735: 708: 706: 705: 700: 676: 674: 673: 668: 640: 638: 637: 632: 614: 612: 611: 606: 591: 589: 588: 583: 571: 569: 568: 563: 551: 549: 548: 543: 541: 537: 521: 520: 472: 470: 469: 464: 462: 461: 445: 443: 442: 437: 435: 434: 423: 419: 418: 397: 395: 394: 389: 377: 375: 374: 369: 367: 366: 348: 347: 337: 309: 307: 306: 301: 283: 281: 280: 275: 255: 253: 252: 247: 234: 232: 231: 226: 224: 223: 212: 208: 207: 186: 184: 183: 178: 176: 161: 159: 158: 153: 111: 104: 100: 97: 91: 89: 48: 24: 16: 1375: 1374: 1370: 1369: 1368: 1366: 1365: 1364: 1340: 1339: 1338: 1337: 1280: 1278: 1273: 1255: 1217: 1173: 1110: 1062: 1004: 995: 961:Change of basis 951:Multilinear map 889: 871: 866: 836: 835: 826: 822: 817: 790: 789: 788:for some index 753: 748: 747: 711: 710: 679: 678: 650: 649: 617: 616: 594: 593: 574: 573: 554: 553: 512: 499: 495: 475: 474: 473:are zero. Let 453: 448: 447: 410: 406: 405: 400: 399: 380: 379: 358: 339: 312: 311: 286: 285: 263: 262: 261:of vectors for 238: 237: 199: 195: 194: 189: 188: 167: 166: 144: 143: 140: 112: 101: 95: 92: 49: 47: 37: 25: 12: 11: 5: 1373: 1371: 1363: 1362: 1357: 1352: 1342: 1341: 1336: 1335: 1328: 1321: 1313: 1310: 1309: 1297:linear algebra 1292: 1275: 1274: 1272: 1271: 1260: 1257: 1256: 1254: 1253: 1248: 1243: 1238: 1233: 1231:Floating-point 1227: 1225: 1219: 1218: 1216: 1215: 1213:Tensor product 1210: 1205: 1200: 1198:Function space 1195: 1190: 1184: 1182: 1175: 1174: 1172: 1171: 1166: 1161: 1156: 1151: 1146: 1141: 1136: 1134:Triple product 1131: 1126: 1120: 1118: 1112: 1111: 1109: 1108: 1103: 1098: 1093: 1088: 1083: 1078: 1072: 1070: 1064: 1063: 1061: 1060: 1055: 1050: 1048:Transformation 1045: 1040: 1038:Multiplication 1035: 1030: 1025: 1020: 1014: 1012: 1006: 1005: 998: 996: 994: 993: 988: 983: 978: 973: 968: 963: 958: 953: 948: 943: 938: 933: 928: 923: 918: 913: 908: 903: 897: 895: 894:Basic concepts 891: 890: 888: 887: 882: 876: 873: 872: 869:Linear algebra 867: 865: 864: 857: 850: 842: 834: 833: 819: 818: 816: 813: 812: 811: 800: 797: 777: 774: 771: 768: 765: 760: 756: 745: 733: 730: 727: 724: 721: 718: 698: 695: 692: 689: 686: 666: 663: 660: 657: 644: 641:is said to be 630: 627: 624: 604: 601: 581: 561: 540: 536: 533: 530: 527: 524: 519: 515: 511: 508: 505: 502: 498: 494: 491: 488: 485: 482: 460: 456: 433: 430: 427: 422: 417: 413: 409: 387: 365: 361: 357: 354: 351: 346: 342: 336: 333: 330: 326: 322: 319: 299: 296: 293: 273: 270: 245: 222: 219: 216: 211: 206: 202: 198: 175: 151: 139: 136: 114: 113: 28: 26: 19: 13: 10: 9: 6: 4: 3: 2: 1372: 1361: 1358: 1356: 1353: 1351: 1350:Vector spaces 1348: 1347: 1345: 1334: 1329: 1327: 1322: 1320: 1315: 1314: 1308: 1306: 1302: 1298: 1293: 1290: 1286: 1281: 1270: 1262: 1261: 1258: 1252: 1249: 1247: 1246:Sparse matrix 1244: 1242: 1239: 1237: 1234: 1232: 1229: 1228: 1226: 1224: 1220: 1214: 1211: 1209: 1206: 1204: 1201: 1199: 1196: 1194: 1191: 1189: 1186: 1185: 1183: 1181:constructions 1180: 1176: 1170: 1169:Outermorphism 1167: 1165: 1162: 1160: 1157: 1155: 1152: 1150: 1147: 1145: 1142: 1140: 1137: 1135: 1132: 1130: 1129:Cross product 1127: 1125: 1122: 1121: 1119: 1117: 1113: 1107: 1104: 1102: 1099: 1097: 1096:Outer product 1094: 1092: 1089: 1087: 1084: 1082: 1079: 1077: 1076:Orthogonality 1074: 1073: 1071: 1069: 1065: 1059: 1056: 1054: 1053:Cramer's rule 1051: 1049: 1046: 1044: 1041: 1039: 1036: 1034: 1031: 1029: 1026: 1024: 1023:Decomposition 1021: 1019: 1016: 1015: 1013: 1011: 1007: 1002: 992: 989: 987: 984: 982: 979: 977: 974: 972: 969: 967: 964: 962: 959: 957: 954: 952: 949: 947: 944: 942: 939: 937: 934: 932: 929: 927: 924: 922: 919: 917: 914: 912: 909: 907: 904: 902: 899: 898: 896: 892: 886: 883: 881: 878: 877: 874: 870: 863: 858: 856: 851: 849: 844: 843: 840: 830: 824: 821: 814: 798: 795: 775: 772: 766: 758: 754: 746: 731: 728: 725: 722: 719: 716: 696: 690: 684: 661: 655: 648: 647: 646: 642: 628: 625: 622: 602: 599: 579: 559: 538: 534: 531: 525: 517: 513: 509: 506: 503: 500: 496: 492: 486: 480: 458: 454: 431: 428: 425: 420: 415: 411: 407: 385: 363: 359: 352: 344: 340: 334: 331: 328: 324: 320: 317: 297: 294: 291: 271: 268: 260: 257: 243: 220: 217: 214: 209: 204: 200: 196: 165: 149: 137: 135: 133: 129: 125: 121: 110: 107: 99: 88: 85: 81: 78: 74: 71: 67: 64: 60: 57: –  56: 52: 51:Find sources: 45: 41: 35: 34: 29:This article 27: 23: 18: 17: 1305:expanding it 1294: 1279: 1179:Vector space 911:Vector space 823: 141: 132:vector space 123: 117: 102: 96:January 2013 93: 83: 76: 69: 62: 50: 38:Please help 33:verification 30: 1159:Multivector 1124:Determinant 1081:Dot product 926:Linear span 827:Milne, J., 1344:Categories 1193:Direct sum 1028:Invertible 931:Linear map 815:References 643:primordial 138:Definition 66:newspapers 1223:Numerical 986:Transpose 726:∈ 720:≠ 626:∈ 532:≠ 504:∈ 429:∈ 378:for some 332:∈ 325:∑ 295:∈ 218:∈ 1269:Category 1208:Subspace 1203:Quotient 1154:Bivector 1068:Bilinear 1010:Matrices 885:Glossary 256:-indexed 187:and let 880:Outline 120:algebra 80:scholar 1164:Tensor 976:Kernel 906:Vector 901:Scalar 709:where 235:be an 128:vector 82:  75:  68:  61:  53:  1295:This 1033:Minor 1018:Block 956:Basis 259:basis 164:field 130:in a 87:JSTOR 73:books 1301:stub 1188:Dual 1043:Rank 142:Let 122:, a 59:news 744:and 592:of 134:. 118:In 42:by 1346:: 1332:e 1325:t 1318:v 1307:. 861:e 854:t 847:v 799:. 796:i 776:1 773:= 770:) 767:p 764:( 759:i 755:a 732:, 729:W 723:w 717:0 697:, 694:) 691:w 688:( 685:I 665:) 662:p 659:( 656:I 629:W 623:p 603:, 600:V 580:W 560:v 539:} 535:0 529:) 526:v 523:( 518:i 514:a 510:: 507:I 501:i 497:{ 493:= 490:) 487:v 484:( 481:I 459:i 455:a 432:I 426:i 421:) 416:i 412:a 408:( 386:I 364:i 360:e 356:) 353:v 350:( 345:i 341:a 335:I 329:i 321:= 318:v 298:V 292:v 272:. 269:V 244:I 221:I 215:i 210:) 205:i 201:e 197:( 174:F 150:V 109:) 103:( 98:) 94:( 84:· 77:· 70:· 63:· 36:.

Index


verification
improve this article
adding citations to reliable sources
"Primordial element" algebra
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scholar
JSTOR
Learn how and when to remove this message
algebra
vector
vector space
field
I {\displaystyle I} -indexed
basis
Class field theory course notes
v
t
e
Linear algebra
Outline
Glossary
Scalar
Vector
Vector space
Scalar multiplication
Vector projection
Linear span

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