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Ein zentraler Satz besagt: ",[569,806,807],{},"Zeilenrang = Spaltenrang",", also ",[574,810,812,852],{"className":811},[577],[574,813,815],{"className":814},[581],[583,816,817],{"xmlns":585},[587,818,819,849],{},[590,820,821,823,825,827,829,831,833,835,837,839,847],{},[593,822,596],{"mathvariant":595},[598,824,600],{},[598,826,604],{"stretchy":603},[593,828,607],{},[598,830,610],{"stretchy":603},[598,832,677],{},[593,834,596],{"mathvariant":595},[598,836,600],{},[598,838,604],{"stretchy":603},[840,841,842,844],"msup",{},[593,843,607],{},[593,845,846],{},"T",[598,848,610],{"stretchy":603},[612,850,851],{"encoding":614},"\\operatorname{rg}(A) = \\operatorname{rg}(A^T)",[574,853,855,885],{"className":854,"ariaHidden":620},[619],[574,856,858,861,867,870,873,876,879,882],{"className":857},[624],[574,859],{"className":860,"style":629},[628],[574,862,864],{"className":863},[633],[574,865,596],{"className":866,"style":639},[637,638],[574,868,604],{"className":869},[643],[574,871,607],{"className":872},[637,647],[574,874,610],{"className":875},[651],[574,877],{"className":878,"style":725},[717],[574,880,677],{"className":881},[729],[574,883],{"className":884,"style":725},[717],[574,886,888,892,898,901,939],{"className":887},[624],[574,889],{"className":890,"style":891},[628],"height:1.0913em;vertical-align:-0.25em;",[574,893,895],{"className":894},[633],[574,896,596],{"className":897,"style":639},[637,638],[574,899,604],{"className":900},[643],[574,902,904,907],{"className":903},[637],[574,905,607],{"className":906},[637,647],[574,908,911],{"className":909},[910],"msupsub",[574,912,914],{"className":913},[752],[574,915,917],{"className":916},[756],[574,918,921],{"className":919,"style":920},[760],"height:0.8413em;",[574,922,924,928],{"style":923},"top:-3.063em;margin-right:0.05em;",[574,925],{"className":926,"style":927},[768],"height:2.7em;",[574,929,935],{"className":930},[931,932,933,934],"sizing","reset-size6","size3","mtight",[574,936,846],{"className":937,"style":938},[637,647,934],"margin-right:0.1389em;",[574,940,610],{"className":941},[651],". Deshalb gilt ",[574,944,946,989],{"className":945},[577],[574,947,949],{"className":948},[581],[583,950,951],{"xmlns":585},[587,952,953,986],{},[590,954,955,957,959,961,963,965,968,971,973,975,978,981,984],{},[593,956,596],{"mathvariant":595},[598,958,600],{},[598,960,604],{"stretchy":603},[593,962,607],{},[598,964,610],{"stretchy":603},[598,966,967],{},"≤",[593,969,970],{},"min",[598,972,600],{},[598,974,604],{"stretchy":603},[593,976,977],{},"m",[598,979,980],{"separator":620},",",[593,982,983],{},"n",[598,985,610],{"stretchy":603},[612,987,988],{"encoding":614},"\\operatorname{rg}(A) \\le \\min(m, n)",[574,990,992,1022],{"className":991,"ariaHidden":620},[619],[574,993,995,998,1004,1007,1010,1013,1016,1019],{"className":994},[624],[574,996],{"className":997,"style":629},[628],[574,999,1001],{"className":1000},[633],[574,1002,596],{"className":1003,"style":639},[637,638],[574,1005,604],{"className":1006},[643],[574,1008,607],{"className":1009},[637,647],[574,1011,610],{"className":1012},[651],[574,1014],{"className":1015,"style":725},[717],[574,1017,967],{"className":1018},[729],[574,1020],{"className":1021,"style":725},[717],[574,1023,1025,1028,1031,1034,1037,1041,1044,1047],{"className":1024},[624],[574,1026],{"className":1027,"style":629},[628],[574,1029,970],{"className":1030},[633],[574,1032,604],{"className":1033},[643],[574,1035,977],{"className":1036},[637,647],[574,1038,980],{"className":1039},[1040],"mpunct",[574,1042],{"className":1043,"style":718},[717],[574,1045,983],{"className":1046},[637,647],[574,1048,610],{"className":1049},[651],".",[565,1052,1053],{},"So bestimmt man den Rang:",[1055,1056,1057,1094],"ol",{},[1058,1059,1060,1089,1090,1093],"li",{},[574,1061,1063,1076],{"className":1062},[577],[574,1064,1066],{"className":1065},[581],[583,1067,1068],{"xmlns":585},[587,1069,1070,1074],{},[590,1071,1072],{},[593,1073,607],{},[612,1075,607],{"encoding":614},[574,1077,1079],{"className":1078,"ariaHidden":620},[619],[574,1080,1082,1086],{"className":1081},[624],[574,1083],{"className":1084,"style":1085},[628],"height:0.6833em;",[574,1087,607],{"className":1088},[637,647]," mit dem Gauß-Algorithmus in ",[569,1091,1092],{},"Zeilenstufenform"," bringen. Zeilenumformungen ändern den Rang nicht.",[1058,1095,1096],{},"Die Zahl der Zeilen zählen, die nicht nur Nullen enthalten. Das ist der Rang.",[565,1098,1099,1100,1103,1104,1135,1136,1240,1241,1313,1314,1418,1419,1447],{},"Alternativ ist der Rang die Größe des größten quadratischen Teilblocks (",[569,1101,1102],{},"Minor",") mit Determinante ungleich ",[574,1105,1107,1122],{"className":1106},[577],[574,1108,1110],{"className":1109},[581],[583,1111,1112],{"xmlns":585},[587,1113,1114,1120],{},[590,1115,1116],{},[1117,1118,1119],"mn",{},"0",[612,1121,1119],{"encoding":614},[574,1123,1125],{"className":1124,"ariaHidden":620},[619],[574,1126,1128,1132],{"className":1127},[624],[574,1129],{"className":1130,"style":1131},[628],"height:0.6444em;",[574,1133,1119],{"className":1134},[637],". Für eine quadratische Matrix ",[574,1137,1139,1171],{"className":1138},[577],[574,1140,1142],{"className":1141},[581],[583,1143,1144],{"xmlns":585},[587,1145,1146,1168],{},[590,1147,1148,1150,1153],{},[593,1149,607],{},[598,1151,1152],{},"∈",[840,1154,1155,1159],{},[593,1156,1158],{"mathvariant":1157},"double-struck","R",[590,1160,1161,1163,1166],{},[593,1162,983],{},[598,1164,1165],{},"×",[593,1167,983],{},[612,1169,1170],{"encoding":614},"A \\in \\mathbb{R}^{n \\times n}",[574,1172,1174,1193],{"className":1173,"ariaHidden":620},[619],[574,1175,1177,1181,1184,1187,1190],{"className":1176},[624],[574,1178],{"className":1179,"style":1180},[628],"height:0.7224em;vertical-align:-0.0391em;",[574,1182,607],{"className":1183},[637,647],[574,1185],{"className":1186,"style":725},[717],[574,1188,1152],{"className":1189},[729],[574,1191],{"className":1192,"style":725},[717],[574,1194,1196,1200],{"className":1195},[624],[574,1197],{"className":1198,"style":1199},[628],"height:0.7713em;",[574,1201,1203,1207],{"className":1202},[637],[574,1204,1158],{"className":1205},[637,1206],"mathbb",[574,1208,1210],{"className":1209},[910],[574,1211,1213],{"className":1212},[752],[574,1214,1216],{"className":1215},[756],[574,1217,1219],{"className":1218,"style":1199},[760],[574,1220,1221,1224],{"style":923},[574,1222],{"className":1223,"style":927},[768],[574,1225,1227],{"className":1226},[931,932,933,934],[574,1228,1230,1233,1237],{"className":1229},[637,934],[574,1231,983],{"className":1232},[637,647,934],[574,1234,1165],{"className":1235},[1236,934],"mbin",[574,1238,983],{"className":1239},[637,647,934]," gilt: ",[574,1242,1244,1270],{"className":1243},[577],[574,1245,1247],{"className":1246},[581],[583,1248,1249],{"xmlns":585},[587,1250,1251,1267],{},[590,1252,1253,1255,1257,1259,1261,1263,1265],{},[593,1254,596],{"mathvariant":595},[598,1256,600],{},[598,1258,604],{"stretchy":603},[593,1260,607],{},[598,1262,610],{"stretchy":603},[598,1264,677],{},[593,1266,983],{},[612,1268,1269],{"encoding":614},"\\operatorname{rg}(A) = n",[574,1271,1273,1303],{"className":1272,"ariaHidden":620},[619],[574,1274,1276,1279,1285,1288,1291,1294,1297,1300],{"className":1275},[624],[574,1277],{"className":1278,"style":629},[628],[574,1280,1282],{"className":1281},[633],[574,1283,596],{"className":1284,"style":639},[637,638],[574,1286,604],{"className":1287},[643],[574,1289,607],{"className":1290},[637,647],[574,1292,610],{"className":1293},[651],[574,1295],{"className":1296,"style":725},[717],[574,1298,677],{"className":1299},[729],[574,1301],{"className":1302,"style":725},[717],[574,1304,1306,1310],{"className":1305},[624],[574,1307],{"className":1308,"style":1309},[628],"height:0.4306em;",[574,1311,983],{"className":1312},[637,647]," genau dann, wenn ",[574,1315,1317,1341],{"className":1316},[577],[574,1318,1320],{"className":1319},[581],[583,1321,1322],{"xmlns":585},[587,1323,1324,1338],{},[590,1325,1326,1329,1331,1333,1336],{},[593,1327,1328],{},"det",[598,1330,600],{},[593,1332,607],{},[598,1334,1335],{"mathvariant":595},"≠",[1117,1337,1119],{},[612,1339,1340],{"encoding":614},"\\det A \\neq 0",[574,1342,1344,1409],{"className":1343,"ariaHidden":620},[619],[574,1345,1347,1351,1354,1357,1360,1363,1406],{"className":1346},[624],[574,1348],{"className":1349,"style":1350},[628],"height:0.8889em;vertical-align:-0.1944em;",[574,1352,1328],{"className":1353},[633],[574,1355],{"className":1356,"style":718},[717],[574,1358,607],{"className":1359},[637,647],[574,1361],{"className":1362,"style":725},[717],[574,1364,1366,1399,1403],{"className":1365},[729],[574,1367,1369],{"className":1368},[729],[574,1370,1373],{"className":1371},[637,1372],"vbox",[574,1374,1377],{"className":1375},[1376],"thinbox",[574,1378,1381,1384,1395],{"className":1379},[1380],"rlap",[574,1382],{"className":1383,"style":1350},[628],[574,1385,1388],{"className":1386},[1387],"inner",[574,1389,1391],{"className":1390},[637],[574,1392,1394],{"className":1393},[729],"",[574,1396],{"className":1397},[1398],"fix",[574,1400],{"className":1401},[717,1402],"nobreak",[574,1404,677],{"className":1405},[729],[574,1407],{"className":1408,"style":725},[717],[574,1410,1412,1415],{"className":1411},[624],[574,1413],{"className":1414,"style":1131},[628],[574,1416,1119],{"className":1417},[637],", also wenn ",[574,1420,1422,1435],{"className":1421},[577],[574,1423,1425],{"className":1424},[581],[583,1426,1427],{"xmlns":585},[587,1428,1429,1433],{},[590,1430,1431],{},[593,1432,607],{},[612,1434,607],{"encoding":614},[574,1436,1438],{"className":1437,"ariaHidden":620},[619],[574,1439,1441,1444],{"className":1440},[624],[574,1442],{"className":1443,"style":1085},[628],[574,1445,607],{"className":1446},[637,647]," invertierbar ist.",[560,1449,1451],{"id":1450},"beispiele","Beispiele",[574,1453,1456],{"className":1454},[1455],"katex-display",[574,1457,1459,1770],{"className":1458},[577],[574,1460,1462],{"className":1461},[581],[583,1463,1465],{"xmlns":585,"display":1464},"block",[587,1466,1467,1767],{},[590,1468,1469,1471,1473,1553,1557,1598,1600,1676,1678,1693,1695],{},[593,1470,607],{},[598,1472,677],{},[590,1474,1475,1477,1551],{},[598,1476,604],{"fence":620},[1478,1479,1483,1508,1529],"mtable",{"rowspacing":1480,"columnalign":1481,"columnspacing":1482},"0.16em","center center center","1em",[1484,1485,1486,1495,1502],"mtr",{},[1487,1488,1489],"mtd",{},[1490,1491,1492],"mstyle",{"scriptlevel":1119,"displaystyle":603},[1117,1493,1494],{},"1",[1487,1496,1497],{},[1490,1498,1499],{"scriptlevel":1119,"displaystyle":603},[1117,1500,1501],{},"2",[1487,1503,1504],{},[1490,1505,1506],{"scriptlevel":1119,"displaystyle":603},[1117,1507,1494],{},[1484,1509,1510,1516,1523],{},[1487,1511,1512],{},[1490,1513,1514],{"scriptlevel":1119,"displaystyle":603},[1117,1515,1501],{},[1487,1517,1518],{},[1490,1519,1520],{"scriptlevel":1119,"displaystyle":603},[1117,1521,1522],{},"4",[1487,1524,1525],{},[1490,1526,1527],{"scriptlevel":1119,"displaystyle":603},[1117,1528,1119],{},[1484,1530,1531,1538,1545],{},[1487,1532,1533],{},[1490,1534,1535],{"scriptlevel":1119,"displaystyle":603},[1117,1536,1537],{},"3",[1487,1539,1540],{},[1490,1541,1542],{"scriptlevel":1119,"displaystyle":603},[1117,1543,1544],{},"6",[1487,1546,1547],{},[1490,1548,1549],{"scriptlevel":1119,"displaystyle":603},[1117,1550,1494],{},[598,1552,610],{"fence":620},[1554,1555,1556],"mtext",{},"  ",[684,1558,1559,1563],{},[598,1560,1562],{"stretchy":620,"minsize":1561},"3.0em","→",[1564,1565,1568],"mpadded",{"width":1566,"lspace":1567},"+0.6em","0.3em",[590,1569,1570,1573,1576,1578,1581,1584,1586,1589,1591,1593,1595],{},[1554,1571,1572],{}," II",[598,1574,1575],{},"−",[1117,1577,1501],{},[1554,1579,1580],{}," ",[1554,1582,1583],{},"I",[598,1585,980],{"separator":620},[1554,1587,1588],{}," III",[598,1590,1575],{},[1117,1592,1537],{},[1554,1594,1580],{},[1554,1596,1597],{},"I ",[1554,1599,1556],{},[590,1601,1602,1604,1674],{},[598,1603,604],{"fence":620},[1478,1605,1606,1626,1650],{"rowspacing":1480,"columnalign":1481,"columnspacing":1482},[1484,1607,1608,1614,1620],{},[1487,1609,1610],{},[1490,1611,1612],{"scriptlevel":1119,"displaystyle":603},[1117,1613,1494],{},[1487,1615,1616],{},[1490,1617,1618],{"scriptlevel":1119,"displaystyle":603},[1117,1619,1501],{},[1487,1621,1622],{},[1490,1623,1624],{"scriptlevel":1119,"displaystyle":603},[1117,1625,1494],{},[1484,1627,1628,1634,1640],{},[1487,1629,1630],{},[1490,1631,1632],{"scriptlevel":1119,"displaystyle":603},[1117,1633,1119],{},[1487,1635,1636],{},[1490,1637,1638],{"scriptlevel":1119,"displaystyle":603},[1117,1639,1119],{},[1487,1641,1642],{},[1490,1643,1644],{"scriptlevel":1119,"displaystyle":603},[590,1645,1646,1648],{},[598,1647,1575],{},[1117,1649,1501],{},[1484,1651,1652,1658,1664],{},[1487,1653,1654],{},[1490,1655,1656],{"scriptlevel":1119,"displaystyle":603},[1117,1657,1119],{},[1487,1659,1660],{},[1490,1661,1662],{"scriptlevel":1119,"displaystyle":603},[1117,1663,1119],{},[1487,1665,1666],{},[1490,1667,1668],{"scriptlevel":1119,"displaystyle":603},[590,1669,1670,1672],{},[598,1671,1575],{},[1117,1673,1501],{},[598,1675,610],{"fence":620},[1554,1677,1556],{},[684,1679,1680,1682],{},[598,1681,1562],{"stretchy":620,"minsize":1561},[1564,1683,1684],{"width":1566,"lspace":1567},[590,1685,1686,1688,1690],{},[1554,1687,1588],{},[598,1689,1575],{},[1554,1691,1692],{},"II ",[1554,1694,1556],{},[590,1696,1697,1699,1765],{},[598,1698,604],{"fence":620},[1478,1700,1701,1721,1745],{"rowspacing":1480,"columnalign":1481,"columnspacing":1482},[1484,1702,1703,1709,1715],{},[1487,1704,1705],{},[1490,1706,1707],{"scriptlevel":1119,"displaystyle":603},[1117,1708,1494],{},[1487,1710,1711],{},[1490,1712,1713],{"scriptlevel":1119,"displaystyle":603},[1117,1714,1501],{},[1487,1716,1717],{},[1490,1718,1719],{"scriptlevel":1119,"displaystyle":603},[1117,1720,1494],{},[1484,1722,1723,1729,1735],{},[1487,1724,1725],{},[1490,1726,1727],{"scriptlevel":1119,"displaystyle":603},[1117,1728,1119],{},[1487,1730,1731],{},[1490,1732,1733],{"scriptlevel":1119,"displaystyle":603},[1117,1734,1119],{},[1487,1736,1737],{},[1490,1738,1739],{"scriptlevel":1119,"displaystyle":603},[590,1740,1741,1743],{},[598,1742,1575],{},[1117,1744,1501],{},[1484,1746,1747,1753,1759],{},[1487,1748,1749],{},[1490,1750,1751],{"scriptlevel":1119,"displaystyle":603},[1117,1752,1119],{},[1487,1754,1755],{},[1490,1756,1757],{"scriptlevel":1119,"displaystyle":603},[1117,1758,1119],{},[1487,1760,1761],{},[1490,1762,1763],{"scriptlevel":1119,"displaystyle":603},[1117,1764,1119],{},[598,1766,610],{"fence":620},[612,1768,1769],{"encoding":614},"A = \\begin{pmatrix} 1 & 2 & 1 \\\\ 2 & 4 & 0 \\\\ 3 & 6 & 1 \\end{pmatrix}\n\\;\\xrightarrow{\\ \\text{II} - 2\\,\\text{I},\\ \\text{III} - 3\\,\\text{I}\\ }\\;\n\\begin{pmatrix} 1 & 2 & 1 \\\\ 0 & 0 & -2 \\\\ 0 & 0 & -2 \\end{pmatrix}\n\\;\\xrightarrow{\\ \\text{III} - \\text{II}\\ }\\;\n\\begin{pmatrix} 1 & 2 & 1 \\\\ 0 & 0 & -2 \\\\ 0 & 0 & 0 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Zeilen sind ungleich null, also ",[574,2861,2863,2889],{"className":2862},[577],[574,2864,2866],{"className":2865},[581],[583,2867,2868],{"xmlns":585},[587,2869,2870,2886],{},[590,2871,2872,2874,2876,2878,2880,2882,2884],{},[593,2873,596],{"mathvariant":595},[598,2875,600],{},[598,2877,604],{"stretchy":603},[593,2879,607],{},[598,2881,610],{"stretchy":603},[598,2883,677],{},[1117,2885,1501],{},[612,2887,2888],{"encoding":614},"\\operatorname{rg}(A) = 2",[574,2890,2892,2922],{"className":2891,"ariaHidden":620},[619],[574,2893,2895,2898,2904,2907,2910,2913,2916,2919],{"className":2894},[624],[574,2896],{"className":2897,"style":629},[628],[574,2899,2901],{"className":2900},[633],[574,2902,596],{"className":2903,"style":639},[637,638],[574,2905,604],{"className":2906},[643],[574,2908,607],{"className":2909},[637,647],[574,2911,610],{"className":2912},[651],[574,2914],{"className":2915,"style":725},[717],[574,2917,677],{"className":2918},[729],[574,2920],{"className":2921,"style":725},[717],[574,2923,2925,2928],{"className":2924},[624],[574,2926],{"className":2927,"style":1131},[628],[574,2929,1501],{"className":2930},[637],". Gegenprobe: ",[574,2933,2935,2957],{"className":2934},[577],[574,2936,2938],{"className":2937},[581],[583,2939,2940],{"xmlns":585},[587,2941,2942,2954],{},[590,2943,2944,2946,2948,2950,2952],{},[593,2945,1328],{},[598,2947,600],{},[593,2949,607],{},[598,2951,677],{},[1117,2953,1119],{},[612,2955,2956],{"encoding":614},"\\det A = 0",[574,2958,2960,2984],{"className":2959,"ariaHidden":620},[619],[574,2961,2963,2966,2969,2972,2975,2978,2981],{"className":2962},[624],[574,2964],{"className":2965,"style":707},[628],[574,2967,1328],{"className":2968},[633],[574,2970],{"className":2971,"style":718},[717],[574,2973,607],{"className":2974},[637,647],[574,2976],{"className":2977,"style":725},[717],[574,2979,677],{"className":2980},[729],[574,2982],{"className":2983,"style":725},[717],[574,2985,2987,2990],{"className":2986},[624],[574,2988],{"className":2989,"style":1131},[628],[574,2991,1119],{"className":2992},[637],", aber der Minor aus Zeilen 1, 2 und Spalten 2, 3 ist ",[574,2995,2997,3057],{"className":2996},[577],[574,2998,3000],{"className":2999},[581],[583,3001,3002],{"xmlns":585},[587,3003,3004,3054],{},[590,3005,3006,3044,3046,3048,3050,3052],{},[590,3007,3008,3011,3042],{},[598,3009,3010],{"fence":620},"∣",[1478,3012,3014,3028],{"rowspacing":1480,"columnalign":3013,"columnspacing":1482},"center center",[1484,3015,3016,3022],{},[1487,3017,3018],{},[1490,3019,3020],{"scriptlevel":1119,"displaystyle":603},[1117,3021,1501],{},[1487,3023,3024],{},[1490,3025,3026],{"scriptlevel":1119,"displaystyle":603},[1117,3027,1494],{},[1484,3029,3030,3036],{},[1487,3031,3032],{},[1490,3033,3034],{"scriptlevel":1119,"displaystyle":603},[1117,3035,1522],{},[1487,3037,3038],{},[1490,3039,3040],{"scriptlevel":1119,"displaystyle":603},[1117,3041,1119],{},[598,3043,3010],{"fence":620},[598,3045,677],{},[598,3047,1575],{},[1117,3049,1522],{},[598,3051,1335],{"mathvariant":595},[1117,3053,1119],{},[612,3055,3056],{"encoding":614},"\\begin{vmatrix} 2 & 1 \\\\ 4 & 0 \\end{vmatrix} = -4 \\neq 0",[574,3058,3060,3266,3320],{"className":3059,"ariaHidden":620},[619],[574,3061,3063,3067,3257,3260,3263],{"className":3062},[624],[574,3064],{"className":3065,"style":3066},[628],"height:2.4em;vertical-align:-0.95em;",[574,3068,3070,3116,3220],{"className":3069},[1801],[574,3071,3073],{"className":3072},[643],[574,3074,3076],{"className":3075},[1808,1809],[574,3077,3079,3107],{"className":3078},[752,1813],[574,3080,3082,3104],{"className":3081},[756],[574,3083,3086],{"className":3084,"style":3085},[760],"height:1.45em;",[574,3087,3089,3093],{"style":3088},"top:-3.45em;",[574,3090],{"className":3091,"style":3092},[768],"height:4.4em;",[574,3094,3096],{"style":3095},"width:0.333em;height:2.4em;",[789,3097,3101],{"xmlns":791,"width":3098,"height":3099,"viewBox":3100},"0.333em","2.4em","0 0 333 2400",[798,3102],{"d":3103},"M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z",[574,3105,1843],{"className":3106},[1842],[574,3108,3110],{"className":3109},[756],[574,3111,3114],{"className":3112,"style":3113},[760],"height:0.95em;",[574,3115],{},[574,3117,3119],{"className":3118},[637],[574,3120,3122,3169,3172,3175],{"className":3121},[1478],[574,3123,3125],{"className":3124},[1862],[574,3126,3128,3161],{"className":3127},[752,1813],[574,3129,3131,3158],{"className":3130},[756],[574,3132,3134,3146],{"className":3133,"style":3085},[760],[574,3135,3137,3140],{"style":3136},"top:-3.61em;",[574,3138],{"className":3139,"style":769},[768],[574,3141,3143],{"className":3142},[637],[574,3144,1501],{"className":3145},[637],[574,3147,3149,3152],{"style":3148},"top:-2.41em;",[574,3150],{"className":3151,"style":769},[768],[574,3153,3155],{"className":3154},[637],[574,3156,1522],{"className":3157},[637],[574,3159,1843],{"className":3160},[1842],[574,3162,3164],{"className":3163},[756],[574,3165,3167],{"className":3166,"style":3113},[760],[574,3168],{},[574,3170],{"className":3171,"style":1923},[1922],[574,3173],{"className":3174,"style":1923},[1922],[574,3176,3178],{"className":3177},[1862],[574,3179,3181,3212],{"className":3180},[752,1813],[574,3182,3184,3209],{"className":3183},[756],[574,3185,3187,3198],{"className":3186,"style":3085},[760],[574,3188,3189,3192],{"style":3136},[574,3190],{"className":3191,"style":769},[768],[574,3193,3195],{"className":3194},[637],[574,3196,1494],{"className":3197},[637],[574,3199,3200,3203],{"style":3148},[574,3201],{"className":3202,"style":769},[768],[574,3204,3206],{"className":3205},[637],[574,3207,1119],{"className":3208},[637],[574,3210,1843],{"className":3211},[1842],[574,3213,3215],{"className":3214},[756],[574,3216,3218],{"className":3217,"style":3113},[760],[574,3219],{},[574,3221,3223],{"className":3222},[651],[574,3224,3226],{"className":3225},[1808,1809],[574,3227,3229,3249],{"className":3228},[752,1813],[574,3230,3232,3246],{"className":3231},[756],[574,3233,3235],{"className":3234,"style":3085},[760],[574,3236,3237,3240],{"style":3088},[574,3238],{"className":3239,"style":3092},[768],[574,3241,3242],{"style":3095},[789,3243,3244],{"xmlns":791,"width":3098,"height":3099,"viewBox":3100},[798,3245],{"d":3103},[574,3247,1843],{"className":3248},[1842],[574,3250,3252],{"className":3251},[756],[574,3253,3255],{"className":3254,"style":3113},[760],[574,3256],{},[574,3258],{"className":3259,"style":725},[717],[574,3261,677],{"className":3262},[729],[574,3264],{"className":3265,"style":725},[717],[574,3267,3269,3272,3275,3278,3281,3317],{"className":3268},[624],[574,3270],{"className":3271,"style":1350},[628],[574,3273,1575],{"className":3274},[637],[574,3276,1522],{"className":3277},[637],[574,3279],{"className":3280,"style":725},[717],[574,3282,3284,3311,3314],{"className":3283},[729],[574,3285,3287],{"className":3286},[729],[574,3288,3290],{"className":3289},[637,1372],[574,3291,3293],{"className":3292},[1376],[574,3294,3296,3299,3308],{"className":3295},[1380],[574,3297],{"className":3298,"style":1350},[628],[574,3300,3302],{"className":3301},[1387],[574,3303,3305],{"className":3304},[637],[574,3306,1394],{"className":3307},[729],[574,3309],{"className":3310},[1398],[574,3312],{"className":3313},[717,1402],[574,3315,677],{"className":3316},[729],[574,3318],{"className":3319,"style":725},[717],[574,3321,3323,3326],{"className":3322},[624],[574,3324],{"className":3325,"style":1131},[628],[574,3327,1119],{"className":3328},[637],". Man sieht es auch direkt: Spalte 2 ist das Doppelte von Spalte 1.",[565,3331,3332,3333,3487,3488,3697,3698,3726,3727,3792,3793,4113,4114,1050],{},"Jede Matrix der Form ",[574,3334,3336,3372],{"className":3335},[577],[574,3337,3339],{"className":3338},[581],[583,3340,3341],{"xmlns":585},[587,3342,3343,3369],{},[590,3344,3345,3352,3354],{},[684,3346,3347,3350],{"accent":620},[593,3348,3349],{},"u",[598,3351,691],{},[1554,3353,1580],{},[840,3355,3356,3363],{},[684,3357,3358,3361],{"accent":620},[593,3359,3360],{},"v",[598,3362,691],{},[590,3364,3365,3367],{},[1554,3366,1580],{},[593,3368,846],{},[612,3370,3371],{"encoding":614},"\\vec u\\, \\vec v^{\\,T}",[574,3373,3375],{"className":3374,"ariaHidden":620},[619],[574,3376,3378,3381,3416,3419],{"className":3377},[624],[574,3379],{"className":3380,"style":920},[628],[574,3382,3384],{"className":3383},[637,748],[574,3385,3387],{"className":3386},[752],[574,3388,3390],{"className":3389},[756],[574,3391,3393,3401],{"className":3392,"style":761},[760],[574,3394,3395,3398],{"style":764},[574,3396],{"className":3397,"style":769},[768],[574,3399,3349],{"className":3400},[637,647],[574,3402,3403,3406],{"style":764},[574,3404],{"className":3405,"style":769},[768],[574,3407,3409],{"className":3408,"style":782},[781],[574,3410,3412],{"className":3411,"style":787},[786],[789,3413,3414],{"xmlns":791,"width":792,"height":793,"style":794,"viewBox":795,"preserveAspectRatio":796},[798,3415],{"d":800},[574,3417],{"className":3418,"style":718},[717],[574,3420,3422,3458],{"className":3421},[637],[574,3423,3425],{"className":3424},[637,748],[574,3426,3428],{"className":3427},[752],[574,3429,3431],{"className":3430},[756],[574,3432,3434,3443],{"className":3433,"style":761},[760],[574,3435,3436,3439],{"style":764},[574,3437],{"className":3438,"style":769},[768],[574,3440,3360],{"className":3441,"style":3442},[637,647],"margin-right:0.0359em;",[574,3444,3445,3448],{"style":764},[574,3446],{"className":3447,"style":769},[768],[574,3449,3451],{"className":3450,"style":782},[781],[574,3452,3454],{"className":3453,"style":787},[786],[789,3455,3456],{"xmlns":791,"width":792,"height":793,"style":794,"viewBox":795,"preserveAspectRatio":796},[798,3457],{"d":800},[574,3459,3461],{"className":3460},[910],[574,3462,3464],{"className":3463},[752],[574,3465,3467],{"className":3466},[756],[574,3468,3470],{"className":3469,"style":920},[760],[574,3471,3472,3475],{"style":923},[574,3473],{"className":3474,"style":927},[768],[574,3476,3478],{"className":3477},[931,932,933,934],[574,3479,3481,3484],{"className":3480},[637,934],[574,3482],{"className":3483,"style":2140},[717,934],[574,3485,846],{"className":3486,"style":938},[637,647,934]," mit ",[574,3489,3491,3525],{"className":3490},[577],[574,3492,3494],{"className":3493},[581],[583,3495,3496],{"xmlns":585},[587,3497,3498,3522],{},[590,3499,3500,3506,3508,3514,3516],{},[684,3501,3502,3504],{"accent":620},[593,3503,3349],{},[598,3505,691],{},[598,3507,980],{"separator":620},[684,3509,3510,3512],{"accent":620},[593,3511,3360],{},[598,3513,691],{},[598,3515,1335],{"mathvariant":595},[684,3517,3518,3520],{"accent":620},[1117,3519,1119],{},[598,3521,691],{},[612,3523,3524],{"encoding":614},"\\vec u, \\vec v \\neq \\vec 0",[574,3526,3528,3653],{"className":3527,"ariaHidden":620},[619],[574,3529,3531,3535,3570,3573,3576,3611,3614,3650],{"className":3530},[624],[574,3532],{"className":3533,"style":3534},[628],"height:0.9084em;vertical-align:-0.1944em;",[574,3536,3538],{"className":3537},[637,748],[574,3539,3541],{"className":3540},[752],[574,3542,3544],{"className":3543},[756],[574,3545,3547,3555],{"className":3546,"style":761},[760],[574,3548,3549,3552],{"style":764},[574,3550],{"className":3551,"style":769},[768],[574,3553,3349],{"className":3554},[637,647],[574,3556,3557,3560],{"style":764},[574,3558],{"className":3559,"style":769},[768],[574,3561,3563],{"className":3562,"style":782},[781],[574,3564,3566],{"className":3565,"style":787},[786],[789,3567,3568],{"xmlns":791,"width":792,"height":793,"style":794,"viewBox":795,"preserveAspectRatio":796},[798,3569],{"d":800},[574,3571,980],{"className":3572},[1040],[574,3574],{"className":3575,"style":718},[717],[574,3577,3579],{"className":3578},[637,748],[574,3580,3582],{"className":3581},[752],[574,3583,3585],{"className":3584},[756],[574,3586,3588,3596],{"className":3587,"style":761},[760],[574,3589,3590,3593],{"style":764},[574,3591],{"className":3592,"style":769},[768],[574,3594,3360],{"className":3595,"style":3442},[637,647],[574,3597,3598,3601],{"style":764},[574,3599],{"className":3600,"style":769},[768],[574,3602,3604],{"className":3603,"style":782},[781],[574,3605,3607],{"className":3606,"style":787},[786],[789,3608,3609],{"xmlns":791,"width":792,"height":793,"style":794,"viewBox":795,"preserveAspectRatio":796},[798,3610],{"d":800},[574,3612],{"className":3613,"style":725},[717],[574,3615,3617,3644,3647],{"className":3616},[729],[574,3618,3620],{"className":3619},[729],[574,3621,3623],{"className":3622},[637,1372],[574,3624,3626],{"className":3625},[1376],[574,3627,3629,3632,3641],{"className":3628},[1380],[574,3630],{"className":3631,"style":1350},[628],[574,3633,3635],{"className":3634},[1387],[574,3636,3638],{"className":3637},[637],[574,3639,1394],{"className":3640},[729],[574,3642],{"className":3643},[1398],[574,3645],{"className":3646},[717,1402],[574,3648,677],{"className":3649},[729],[574,3651],{"className":3652,"style":725},[717],[574,3654,3656,3660],{"className":3655},[624],[574,3657],{"className":3658,"style":3659},[628],"height:0.9274em;",[574,3661,3663],{"className":3662},[637,748],[574,3664,3666],{"className":3665},[752],[574,3667,3669],{"className":3668},[756],[574,3670,3672,3680],{"className":3671,"style":3659},[760],[574,3673,3674,3677],{"style":764},[574,3675],{"className":3676,"style":769},[768],[574,3678,1119],{"className":3679},[637],[574,3681,3683,3686],{"style":3682},"top:-3.2134em;",[574,3684],{"className":3685,"style":769},[768],[574,3687,3690],{"className":3688,"style":3689},[781],"left:-0.2355em;",[574,3691,3693],{"className":3692,"style":787},[786],[789,3694,3695],{"xmlns":791,"width":792,"height":793,"style":794,"viewBox":795,"preserveAspectRatio":796},[798,3696],{"d":800}," hat Rang ",[574,3699,3701,3714],{"className":3700},[577],[574,3702,3704],{"className":3703},[581],[583,3705,3706],{"xmlns":585},[587,3707,3708,3712],{},[590,3709,3710],{},[1117,3711,1494],{},[612,3713,1494],{"encoding":614},[574,3715,3717],{"className":3716,"ariaHidden":620},[619],[574,3718,3720,3723],{"className":3719},[624],[574,3721],{"className":3722,"style":1131},[628],[574,3724,1494],{"className":3725},[637],", denn alle Spalten sind Vielfache von ",[574,3728,3730,3748],{"className":3729},[577],[574,3731,3733],{"className":3732},[581],[583,3734,3735],{"xmlns":585},[587,3736,3737,3745],{},[590,3738,3739],{},[684,3740,3741,3743],{"accent":620},[593,3742,3349],{},[598,3744,691],{},[612,3746,3747],{"encoding":614},"\\vec u",[574,3749,3751],{"className":3750,"ariaHidden":620},[619],[574,3752,3754,3757],{"className":3753},[624],[574,3755],{"className":3756,"style":761},[628],[574,3758,3760],{"className":3759},[637,748],[574,3761,3763],{"className":3762},[752],[574,3764,3766],{"className":3765},[756],[574,3767,3769,3777],{"className":3768,"style":761},[760],[574,3770,3771,3774],{"style":764},[574,3772],{"className":3773,"style":769},[768],[574,3775,3349],{"className":3776},[637,647],[574,3778,3779,3782],{"style":764},[574,3780],{"className":3781,"style":769},[768],[574,3783,3785],{"className":3784,"style":782},[781],[574,3786,3788],{"className":3787,"style":787},[786],[789,3789,3790],{"xmlns":791,"width":792,"height":793,"style":794,"viewBox":795,"preserveAspectRatio":796},[798,3791],{"d":800},". Beispiel: ",[574,3794,3796,3883],{"className":3795},[577],[574,3797,3799],{"className":3798},[581],[583,3800,3801],{"xmlns":585},[587,3802,3803,3880],{},[590,3804,3805,3830,3832,3834,3837,3839,3841,3843],{},[590,3806,3807,3809,3828],{},[598,3808,604],{"fence":620},[1478,3810,3812,3820],{"rowspacing":1480,"columnalign":3811,"columnspacing":1482},"center",[1484,3813,3814],{},[1487,3815,3816],{},[1490,3817,3818],{"scriptlevel":1119,"displaystyle":603},[1117,3819,1494],{},[1484,3821,3822],{},[1487,3823,3824],{},[1490,3825,3826],{"scriptlevel":1119,"displaystyle":603},[1117,3827,1501],{},[598,3829,610],{"fence":620},[598,3831,604],{"stretchy":603},[1117,3833,1537],{},[1554,3835,3836],{},"  ",[1117,3838,1522],{},[598,3840,610],{"stretchy":603},[598,3842,677],{},[590,3844,3845,3847,3878],{},[598,3846,604],{"fence":620},[1478,3848,3849,3863],{"rowspacing":1480,"columnalign":3013,"columnspacing":1482},[1484,3850,3851,3857],{},[1487,3852,3853],{},[1490,3854,3855],{"scriptlevel":1119,"displaystyle":603},[1117,3856,1537],{},[1487,3858,3859],{},[1490,3860,3861],{"scriptlevel":1119,"displaystyle":603},[1117,3862,1522],{},[1484,3864,3865,3871],{},[1487,3866,3867],{},[1490,3868,3869],{"scriptlevel":1119,"displaystyle":603},[1117,3870,1544],{},[1487,3872,3873],{},[1490,3874,3875],{"scriptlevel":1119,"displaystyle":603},[1117,3876,3877],{},"8",[598,3879,610],{"fence":620},[612,3881,3882],{"encoding":614},"\\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} (3 \\ \\ 4) = \\begin{pmatrix} 3 & 4 \\\\ 6 & 8 \\end{pmatrix}",[574,3884,3886,3990],{"className":3885,"ariaHidden":620},[619],[574,3887,3889,3892,3960,3963,3966,3969,3972,3975,3978,3981,3984,3987],{"className":3888},[624],[574,3890],{"className":3891,"style":3066},[628],[574,3893,3895,3903,3954],{"className":3894},[1801],[574,3896,3900],{"className":3897,"style":3899},[643,3898],"delimcenter","top:0em;",[574,3901,604],{"className":3902},[1808,933],[574,3904,3906],{"className":3905},[637],[574,3907,3909],{"className":3908},[1478],[574,3910,3912],{"className":3911},[1862],[574,3913,3915,3946],{"className":3914},[752,1813],[574,3916,3918,3943],{"className":3917},[756],[574,3919,3921,3932],{"className":3920,"style":3085},[760],[574,3922,3923,3926],{"style":3136},[574,3924],{"className":3925,"style":769},[768],[574,3927,3929],{"className":3928},[637],[574,3930,1494],{"className":3931},[637],[574,3933,3934,3937],{"style":3148},[574,3935],{"className":3936,"style":769},[768],[574,3938,3940],{"className":3939},[637],[574,3941,1501],{"className":3942},[637],[574,3944,1843],{"className":3945},[1842],[574,3947,3949],{"className":3948},[756],[574,3950,3952],{"className":3951,"style":3113},[760],[574,3953],{},[574,3955,3957],{"className":3956,"style":3899},[651,3898],[574,3958,610],{"className":3959},[1808,933],[574,3961],{"className":3962,"style":718},[717],[574,3964,604],{"className":3965},[643],[574,3967,1537],{"className":3968},[637],[574,3970,2122],{"className":3971},[717],[574,3973,2122],{"className":3974},[717],[574,3976,1522],{"className":3977},[637],[574,3979,610],{"className":3980},[651],[574,3982],{"className":3983,"style":725},[717],[574,3985,677],{"className":3986},[729],[574,3988],{"className":3989,"style":725},[717],[574,3991,3993,3996],{"className":3992},[624],[574,3994],{"className":3995,"style":3066},[628],[574,3997,3999,4005,4107],{"className":3998},[1801],[574,4000,4002],{"className":4001,"style":3899},[643,3898],[574,4003,604],{"className":4004},[1808,933],[574,4006,4008],{"className":4007},[637],[574,4009,4011,4056,4059,4062],{"className":4010},[1478],[574,4012,4014],{"className":4013},[1862],[574,4015,4017,4048],{"className":4016},[752,1813],[574,4018,4020,4045],{"className":4019},[756],[574,4021,4023,4034],{"className":4022,"style":3085},[760],[574,4024,4025,4028],{"style":3136},[574,4026],{"className":4027,"style":769},[768],[574,4029,4031],{"className":4030},[637],[574,4032,1537],{"className":4033},[637],[574,4035,4036,4039],{"style":3148},[574,4037],{"className":4038,"style":769},[768],[574,4040,4042],{"className":4041},[637],[574,4043,1544],{"className":4044},[637],[574,4046,1843],{"className":4047},[1842],[574,4049,4051],{"className":4050},[756],[574,4052,4054],{"className":4053,"style":3113},[760],[574,4055],{},[574,4057],{"className":4058,"style":1923},[1922],[574,4060],{"className":4061,"style":1923},[1922],[574,4063,4065],{"className":4064},[1862],[574,4066,4068,4099],{"className":4067},[752,1813],[574,4069,4071,4096],{"className":4070},[756],[574,4072,4074,4085],{"className":4073,"style":3085},[760],[574,4075,4076,4079],{"style":3136},[574,4077],{"className":4078,"style":769},[768],[574,4080,4082],{"className":4081},[637],[574,4083,1522],{"className":4084},[637],[574,4086,4087,4090],{"style":3148},[574,4088],{"className":4089,"style":769},[768],[574,4091,4093],{"className":4092},[637],[574,4094,3877],{"className":4095},[637],[574,4097,1843],{"className":4098},[1842],[574,4100,4102],{"className":4101},[756],[574,4103,4105],{"className":4104,"style":3113},[760],[574,4106],{},[574,4108,4110],{"className":4109,"style":3899},[651,3898],[574,4111,610],{"className":4112},[1808,933]," mit Determinante ",[574,4115,4117,4130],{"className":4116},[577],[574,4118,4120],{"className":4119},[581],[583,4121,4122],{"xmlns":585},[587,4123,4124,4128],{},[590,4125,4126],{},[1117,4127,1119],{},[612,4129,1119],{"encoding":614},[574,4131,4133],{"className":4132,"ariaHidden":620},[619],[574,4134,4136,4139],{"className":4135},[624],[574,4137],{"className":4138,"style":1131},[628],[574,4140,1119],{"className":4141},[637],[560,4143,4145],{"id":4144},"typische-fehler","Typische Fehler",[4147,4148,4149,4152,4155,4410],"ul",{},[1058,4150,4151],{},"Die Zahl der Nullzeilen statt der Nicht-Nullzeilen zählen.",[1058,4153,4154],{},"Spaltenumformungen und Zeilenumformungen mischen und dann die Lösung eines Gleichungssystems ablesen. Für den Rang sind beide erlaubt, für das Lösen nur Zeilenumformungen.",[1058,4156,4157,4158,4218,4219,4290,4291,4351,4352,4380,4381,4409],{},"Aus ",[574,4159,4161,4182],{"className":4160},[577],[574,4162,4164],{"className":4163},[581],[583,4165,4166],{"xmlns":585},[587,4167,4168,4180],{},[590,4169,4170,4172,4174,4176,4178],{},[593,4171,1328],{},[598,4173,600],{},[593,4175,607],{},[598,4177,677],{},[1117,4179,1119],{},[612,4181,2956],{"encoding":614},[574,4183,4185,4209],{"className":4184,"ariaHidden":620},[619],[574,4186,4188,4191,4194,4197,4200,4203,4206],{"className":4187},[624],[574,4189],{"className":4190,"style":707},[628],[574,4192,1328],{"className":4193},[633],[574,4195],{"className":4196,"style":718},[717],[574,4198,607],{"className":4199},[637,647],[574,4201],{"className":4202,"style":725},[717],[574,4204,677],{"className":4205},[729],[574,4207],{"className":4208,"style":725},[717],[574,4210,4212,4215],{"className":4211},[624],[574,4213],{"className":4214,"style":1131},[628],[574,4216,1119],{"className":4217},[637]," schließen, dass ",[574,4220,4222,4248],{"className":4221},[577],[574,4223,4225],{"className":4224},[581],[583,4226,4227],{"xmlns":585},[587,4228,4229,4245],{},[590,4230,4231,4233,4235,4237,4239,4241,4243],{},[593,4232,596],{"mathvariant":595},[598,4234,600],{},[598,4236,604],{"stretchy":603},[593,4238,607],{},[598,4240,610],{"stretchy":603},[598,4242,677],{},[1117,4244,1119],{},[612,4246,4247],{"encoding":614},"\\operatorname{rg}(A) = 0",[574,4249,4251,4281],{"className":4250,"ariaHidden":620},[619],[574,4252,4254,4257,4263,4266,4269,4272,4275,4278],{"className":4253},[624],[574,4255],{"className":4256,"style":629},[628],[574,4258,4260],{"className":4259},[633],[574,4261,596],{"className":4262,"style":639},[637,638],[574,4264,604],{"className":4265},[643],[574,4267,607],{"className":4268},[637,647],[574,4270,610],{"className":4271},[651],[574,4273],{"className":4274,"style":725},[717],[574,4276,677],{"className":4277},[729],[574,4279],{"className":4280,"style":725},[717],[574,4282,4284,4287],{"className":4283},[624],[574,4285],{"className":4286,"style":1131},[628],[574,4288,1119],{"className":4289},[637]," ist. ",[574,4292,4294,4315],{"className":4293},[577],[574,4295,4297],{"className":4296},[581],[583,4298,4299],{"xmlns":585},[587,4300,4301,4313],{},[590,4302,4303,4305,4307,4309,4311],{},[593,4304,1328],{},[598,4306,600],{},[593,4308,607],{},[598,4310,677],{},[1117,4312,1119],{},[612,4314,2956],{"encoding":614},[574,4316,4318,4342],{"className":4317,"ariaHidden":620},[619],[574,4319,4321,4324,4327,4330,4333,4336,4339],{"className":4320},[624],[574,4322],{"className":4323,"style":707},[628],[574,4325,1328],{"className":4326},[633],[574,4328],{"className":4329,"style":718},[717],[574,4331,607],{"className":4332},[637,647],[574,4334],{"className":4335,"style":725},[717],[574,4337,677],{"className":4338},[729],[574,4340],{"className":4341,"style":725},[717],[574,4343,4345,4348],{"className":4344},[624],[574,4346],{"className":4347,"style":1131},[628],[574,4349,1119],{"className":4350},[637]," heißt nur, dass der Rang kleiner als ",[574,4353,4355,4368],{"className":4354},[577],[574,4356,4358],{"className":4357},[581],[583,4359,4360],{"xmlns":585},[587,4361,4362,4366],{},[590,4363,4364],{},[593,4365,983],{},[612,4367,983],{"encoding":614},[574,4369,4371],{"className":4370,"ariaHidden":620},[619],[574,4372,4374,4377],{"className":4373},[624],[574,4375],{"className":4376,"style":1309},[628],[574,4378,983],{"className":4379},[637,647]," ist. Rang ",[574,4382,4384,4397],{"className":4383},[577],[574,4385,4387],{"className":4386},[581],[583,4388,4389],{"xmlns":585},[587,4390,4391,4395],{},[590,4392,4393],{},[1117,4394,1119],{},[612,4396,1119],{"encoding":614},[574,4398,4400],{"className":4399,"ariaHidden":620},[619],[574,4401,4403,4406],{"className":4402},[624],[574,4404],{"className":4405,"style":1131},[628],[574,4407,1119],{"className":4408},[637]," hat nur die Nullmatrix.",[1058,4411,4412,4413,4466,4467,4495,4496,1050],{},"Annehmen, der Rang einer ",[574,4414,4416,4434],{"className":4415},[577],[574,4417,4419],{"className":4418},[581],[583,4420,4421],{"xmlns":585},[587,4422,4423,4431],{},[590,4424,4425,4427,4429],{},[1117,4426,1501],{},[598,4428,1165],{},[1117,4430,1537],{},[612,4432,4433],{"encoding":614},"2 \\times 3",[574,4435,4437,4457],{"className":4436,"ariaHidden":620},[619],[574,4438,4440,4444,4447,4451,4454],{"className":4439},[624],[574,4441],{"className":4442,"style":4443},[628],"height:0.7278em;vertical-align:-0.0833em;",[574,4445,1501],{"className":4446},[637],[574,4448],{"className":4449,"style":4450},[717],"margin-right:0.2222em;",[574,4452,1165],{"className":4453},[1236],[574,4455],{"className":4456,"style":4450},[717],[574,4458,4460,4463],{"className":4459},[624],[574,4461],{"className":4462,"style":1131},[628],[574,4464,1537],{"className":4465},[637],"-Matrix könne ",[574,4468,4470,4483],{"className":4469},[577],[574,4471,4473],{"className":4472},[581],[583,4474,4475],{"xmlns":585},[587,4476,4477,4481],{},[590,4478,4479],{},[1117,4480,1537],{},[612,4482,1537],{"encoding":614},[574,4484,4486],{"className":4485,"ariaHidden":620},[619],[574,4487,4489,4492],{"className":4488},[624],[574,4490],{"className":4491,"style":1131},[628],[574,4493,1537],{"className":4494},[637]," sein. Er ist höchstens ",[574,4497,4499,4529],{"className":4498},[577],[574,4500,4502],{"className":4501},[581],[583,4503,4504],{"xmlns":585},[587,4505,4506,4526],{},[590,4507,4508,4510,4512,4514,4516,4518,4520,4522,4524],{},[593,4509,970],{},[598,4511,600],{},[598,4513,604],{"stretchy":603},[1117,4515,1501],{},[598,4517,980],{"separator":620},[1117,4519,1537],{},[598,4521,610],{"stretchy":603},[598,4523,677],{},[1117,4525,1501],{},[612,4527,4528],{"encoding":614},"\\min(2, 3) = 2",[574,4530,4532,4568],{"className":4531,"ariaHidden":620},[619],[574,4533,4535,4538,4541,4544,4547,4550,4553,4556,4559,4562,4565],{"className":4534},[624],[574,4536],{"className":4537,"style":629},[628],[574,4539,970],{"className":4540},[633],[574,4542,604],{"className":4543},[643],[574,4545,1501],{"className":4546},[637],[574,4548,980],{"className":4549},[1040],[574,4551],{"className":4552,"style":718},[717],[574,4554,1537],{"className":4555},[637],[574,4557,610],{"className":4558},[651],[574,4560],{"className":4561,"style":725},[717],[574,4563,677],{"className":4564},[729],[574,4566],{"className":4567,"style":725},[717],[574,4569,4571,4574],{"className":4570},[624],[574,4572],{"className":4573,"style":1131},[628],[574,4575,1501],{"className":4576},[637],{"title":4578,"searchDepth":4579,"depth":4579,"links":4580},"",2,[4581,4582,4583],{"id":562,"depth":4579,"text":563},{"id":1450,"depth":4579,"text":1451},{"id":4144,"depth":4579,"text":4145},[4585,4587,4589,4592,4594,4596,4598,4600,4603,4606,4608],{"lhs":4586,"rhs":1119},"2 - 2*1",{"lhs":4588,"rhs":1119},"4 - 2*2",{"lhs":4590,"rhs":4591},"0 - 2*1","-2",{"lhs":4593,"rhs":1119},"3 - 3*1",{"lhs":4595,"rhs":1119},"6 - 3*2",{"lhs":4597,"rhs":4591},"1 - 3*1",{"lhs":4599,"rhs":1119},"det([[1, 2, 1], [2, 4, 0], [3, 6, 1]])",{"lhs":4601,"rhs":4602},"det([[2, 1], [4, 0]])","-4",{"lhs":4604,"rhs":4605},"2*1 + 2*2 + 2*3","2 + 4 + 6",{"lhs":4607,"rhs":1119},"dot(2*[1, 2, 3] - [2, 4, 6], 2*[1, 2, 3] - [2, 4, 6])",{"lhs":4609,"rhs":1119},"det([[1*3, 1*4], [2*3, 2*4]])","A \\in \\mathbb{R}^{m \\times n}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmsup>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmrow>\u003Cmi>m\u003C\u002Fmi>\u003Cmo>×\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A \\in \\mathbb{R}^{m \\times n}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7713em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">m\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">×\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","matrix-rank","\\operatorname{rg}(A) = \\dim \\operatorname{Bild} A = \\operatorname{rg}(A^{T}) \\le \\min(m, n)","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">rg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>dim\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">Bild\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">rg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>A\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\operatorname{rg}(A) = \\dim \\operatorname{Bild} A = \\operatorname{rg}(A^{T}) \\le \\min(m, n)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\" style=\"margin-right:0.0139em;\">rg\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">dim\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">Bild\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.1413em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\" style=\"margin-right:0.0139em;\">rg\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8913em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≤\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":615,"meaning":4640,"symbolHtml":4641},"Rang von A, die maximale Anzahl linear unabhängiger Spalten (oder Zeilen)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">rg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">Bild\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\operatorname{Bild} A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">Bild\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4647,"meaning":4648,"symbolHtml":4649},"A^{T}","transponierte Matrix von A","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4651,"meaning":4652,"symbolHtml":4653},"m, n","Zeilen- und Spaltenzahl von A","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>m\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">m, n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan 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