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",[576,1063,1065,1095],{"className":1064},[579],[576,1066,1068],{"className":1067},[583],[585,1069,1070],{"xmlns":587},[589,1071,1072,1092],{},[592,1073,1074,1084],{},[1075,1076,1077,1081],"mfrac",{},[1078,1079,1080],"mn",{},"2",[1078,1082,1083],{},"3",[1085,1086,1087,1090],"msup",{},[595,1088,1089],{},"n",[1078,1091,1083],{},[599,1093,1094],{"encoding":601},"\\frac{2}{3}n^3",[576,1096,1098],{"className":1097,"ariaHidden":606},[605],[576,1099,1101,1105,1184],{"className":1100},[610],[576,1102],{"className":1103,"style":1104},[614],"height:1.1901em;vertical-align:-0.345em;",[576,1106,1108,1113,1180],{"className":1107},[619],[576,1109],{"className":1110},[1111,1112],"mopen","nulldelimiter",[576,1114,1116],{"className":1115},[1075],[576,1117,1119,1171],{"className":1118},[706,707],[576,1120,1122,1168],{"className":1121},[711],[576,1123,1126,1142,1153],{"className":1124,"style":1125},[715],"height:0.8451em;",[576,1127,1129,1133],{"style":1128},"top:-2.655em;",[576,1130],{"className":1131,"style":1132},[723],"height:3em;",[576,1134,1136],{"className":1135},[728,729,730,731],[576,1137,1139],{"className":1138},[619,731],[576,1140,1083],{"className":1141},[619,731],[576,1143,1145,1148],{"style":1144},"top:-3.23em;",[576,1146],{"className":1147,"style":1132},[723],[576,1149],{"className":1150,"style":1152},[1151],"frac-line","border-bottom-width:0.04em;",[576,1154,1156,1159],{"style":1155},"top:-3.394em;",[576,1157],{"className":1158,"style":1132},[723],[576,1160,1162],{"className":1161},[728,729,730,731],[576,1163,1165],{"className":1164},[619,731],[576,1166,1080],{"className":1167},[619,731],[576,1169,744],{"className":1170},[743],[576,1172,1174],{"className":1173},[711],[576,1175,1178],{"className":1176,"style":1177},[715],"height:0.345em;",[576,1179],{},[576,1181],{"className":1182},[1183,1112],"mclose",[576,1185,1187,1190],{"className":1186},[619],[576,1188,1089],{"className":1189},[619,620],[576,1191,1193],{"className":1192},[702],[576,1194,1196],{"className":1195},[706],[576,1197,1199],{"className":1198},[711],[576,1200,1203],{"className":1201,"style":1202},[715],"height:0.8141em;",[576,1204,1206,1209],{"style":1205},"top:-3.063em;margin-right:0.05em;",[576,1207],{"className":1208,"style":724},[723],[576,1210,1212],{"className":1211},[728,729,730,731],[576,1213,1083],{"className":1214},[619,731]," Rechenoperationen, jedes weitere Einsetzen nur etwa ",[576,1217,1219,1239],{"className":1218},[579],[576,1220,1222],{"className":1221},[583],[585,1223,1224],{"xmlns":587},[589,1225,1226,1236],{},[592,1227,1228,1230],{},[1078,1229,1080],{},[1085,1231,1232,1234],{},[595,1233,1089],{},[1078,1235,1080],{},[599,1237,1238],{"encoding":601},"2n^2",[576,1240,1242],{"className":1241,"ariaHidden":606},[605],[576,1243,1245,1248,1251],{"className":1244},[610],[576,1246],{"className":1247,"style":1202},[614],[576,1249,1080],{"className":1250},[619],[576,1252,1254,1257],{"className":1253},[619],[576,1255,1089],{"className":1256},[619,620],[576,1258,1260],{"className":1259},[702],[576,1261,1263],{"className":1262},[706],[576,1264,1266],{"className":1265},[711],[576,1267,1269],{"className":1268,"style":1202},[715],[576,1270,1271,1274],{"style":1205},[576,1272],{"className":1273,"style":724},[723],[576,1275,1277],{"className":1276},[728,729,730,731],[576,1278,1080],{"className":1279},[619,731],". Bei vielen Systemen mit derselben Matrix und verschiedenen rechten Seiten zerlegt man daher einmal und setzt mehrfach ein. Nebenbei ergibt sich ",[576,1282,1284,1337],{"className":1283},[579],[576,1285,1287],{"className":1286},[583],[585,1288,1289],{"xmlns":587},[589,1290,1291,1334],{},[592,1292,1293,1296,1299,1301,1303,1311,1314,1321,1324],{},[595,1294,1295],{},"det",[866,1297,1298],{},"⁡",[595,1300,597],{},[866,1302,868],{},[666,1304,1305,1308],{},[595,1306,1307],{},"u",[1078,1309,1310],{},"11",[866,1312,1313],{},"⋅",[666,1315,1316,1318],{},[595,1317,1307],{},[1078,1319,1320],{},"22",[866,1322,1323],{},"⋯",[666,1325,1326,1328],{},[595,1327,1307],{},[592,1329,1330,1332],{},[595,1331,1089],{},[595,1333,1089],{},[599,1335,1336],{"encoding":601},"\\det A = u_{11} \\cdot u_{22} \\cdots u_{nn}",[576,1338,1340,1365,1429],{"className":1339,"ariaHidden":606},[605],[576,1341,1343,1346,1350,1353,1356,1359,1362],{"className":1342},[610],[576,1344],{"className":1345,"style":908},[614],[576,1347,1295],{"className":1348},[1349],"mop",[576,1351],{"className":1352,"style":967},[893],[576,1354,597],{"className":1355},[619,620],[576,1357],{"className":1358,"style":894},[893],[576,1360,868],{"className":1361},[898],[576,1363],{"className":1364,"style":894},[893],[576,1366,1368,1372,1418,1422,1426],{"className":1367},[610],[576,1369],{"className":1370,"style":1371},[614],"height:0.5945em;vertical-align:-0.15em;",[576,1373,1375,1378],{"className":1374},[619],[576,1376,1307],{"className":1377},[619,620],[576,1379,1381],{"className":1380},[702],[576,1382,1384,1409],{"className":1383},[706,707],[576,1385,1387,1406],{"className":1386},[711],[576,1388,1391],{"className":1389,"style":1390},[715],"height:0.3011em;",[576,1392,1394,1397],{"style":1393},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[576,1395],{"className":1396,"style":724},[723],[576,1398,1400],{"className":1399},[728,729,730,731],[576,1401,1403],{"className":1402},[619,731],[576,1404,1310],{"className":1405},[619,731],[576,1407,744],{"className":1408},[743],[576,1410,1412],{"className":1411},[711],[576,1413,1416],{"className":1414,"style":1415},[715],"height:0.15em;",[576,1417],{},[576,1419],{"className":1420,"style":1421},[893],"margin-right:0.2222em;",[576,1423,1313],{"className":1424},[1425],"mbin",[576,1427],{"className":1428,"style":1421},[893],[576,1430,1432,1436,1479,1482,1486,1489],{"className":1431},[610],[576,1433],{"className":1434,"style":1435},[614],"height:0.5806em;vertical-align:-0.15em;",[576,1437,1439,1442],{"className":1438},[619],[576,1440,1307],{"className":1441},[619,620],[576,1443,1445],{"className":1444},[702],[576,1446,1448,1471],{"className":1447},[706,707],[576,1449,1451,1468],{"className":1450},[711],[576,1452,1454],{"className":1453,"style":1390},[715],[576,1455,1456,1459],{"style":1393},[576,1457],{"className":1458,"style":724},[723],[576,1460,1462],{"className":1461},[728,729,730,731],[576,1463,1465],{"className":1464},[619,731],[576,1466,1320],{"className":1467},[619,731],[576,1469,744],{"className":1470},[743],[576,1472,1474],{"className":1473},[711],[576,1475,1477],{"className":1476,"style":1415},[715],[576,1478],{},[576,1480],{"className":1481,"style":967},[893],[576,1483,1323],{"className":1484},[1485],"minner",[576,1487],{"className":1488,"style":967},[893],[576,1490,1492,1495],{"className":1491},[619],[576,1493,1307],{"className":1494},[619,620],[576,1496,1498],{"className":1497},[702],[576,1499,1501,1526],{"className":1500},[706,707],[576,1502,1504,1523],{"className":1503},[711],[576,1505,1508],{"className":1506,"style":1507},[715],"height:0.1514em;",[576,1509,1510,1513],{"style":1393},[576,1511],{"className":1512,"style":724},[723],[576,1514,1516],{"className":1515},[728,729,730,731],[576,1517,1519],{"className":1518},[619,731],[576,1520,1522],{"className":1521},[619,620,731],"nn",[576,1524,744],{"className":1525},[743],[576,1527,1529],{"className":1528},[711],[576,1530,1532],{"className":1531,"style":1415},[715],[576,1533],{},[565,1535,1536,1537,1540,1541,1601,1602,1630],{},"Ohne Zeilentausch existiert die Zerlegung nur, wenn bei der Elimination kein Pivotelement null wird. Im Allgemeinen und aus Stabilitätsgründen tauscht man Zeilen (",[569,1538,1539],{},"Spaltenpivotisierung",") und erhält ",[576,1542,1544,1566],{"className":1543},[579],[576,1545,1547],{"className":1546},[583],[585,1548,1549],{"xmlns":587},[589,1550,1551,1564],{},[592,1552,1553,1556,1558,1560,1562],{},[595,1554,1555],{},"P",[595,1557,597],{},[866,1559,868],{},[595,1561,829],{},[595,1563,636],{},[599,1565,553],{"encoding":601},[576,1567,1569,1591],{"className":1568,"ariaHidden":606},[605],[576,1570,1572,1575,1579,1582,1585,1588],{"className":1571},[610],[576,1573],{"className":1574,"style":615},[614],[576,1576,1555],{"className":1577,"style":1578},[619,620],"margin-right:0.1389em;",[576,1580,597],{"className":1581},[619,620],[576,1583],{"className":1584,"style":894},[893],[576,1586,868],{"className":1587},[898],[576,1589],{"className":1590,"style":894},[893],[576,1592,1594,1597],{"className":1593},[610],[576,1595],{"className":1596,"style":615},[614],[576,1598,1600],{"className":1599,"style":651},[619,620],"LU"," mit einer Permutationsmatrix ",[576,1603,1605,1618],{"className":1604},[579],[576,1606,1608],{"className":1607},[583],[585,1609,1610],{"xmlns":587},[589,1611,1612,1616],{},[592,1613,1614],{},[595,1615,1555],{},[599,1617,1555],{"encoding":601},[576,1619,1621],{"className":1620,"ariaHidden":606},[605],[576,1622,1624,1627],{"className":1623},[610],[576,1625],{"className":1626,"style":615},[614],[576,1628,1555],{"className":1629,"style":1578},[619,620],". Diese Zerlegung existiert für jede invertierbare Matrix.",[560,1632,1634],{"id":1633},"beispiele","Beispiele",[576,1636,1639],{"className":1637},[1638],"katex-display",[576,1640,1642,1743],{"className":1641},[579],[576,1643,1645],{"className":1644},[583],[585,1646,1648],{"xmlns":587,"display":1647},"block",[589,1649,1650,1740],{},[592,1651,1652,1654,1656],{},[595,1653,597],{},[866,1655,868],{},[592,1657,1658,1661,1737],{},[866,1659,1660],{"fence":606},"(",[1662,1663,1667,1693,1714],"mtable",{"rowspacing":1664,"columnalign":1665,"columnspacing":1666},"0.16em","center center center","1em",[1668,1669,1670,1680,1687],"mtr",{},[1671,1672,1673],"mtd",{},[1674,1675,1678],"mstyle",{"scriptlevel":1676,"displaystyle":1677},"0","false",[1078,1679,1080],{},[1671,1681,1682],{},[1674,1683,1684],{"scriptlevel":1676,"displaystyle":1677},[1078,1685,1686],{},"1",[1671,1688,1689],{},[1674,1690,1691],{"scriptlevel":1676,"displaystyle":1677},[1078,1692,1686],{},[1668,1694,1695,1702,1708],{},[1671,1696,1697],{},[1674,1698,1699],{"scriptlevel":1676,"displaystyle":1677},[1078,1700,1701],{},"4",[1671,1703,1704],{},[1674,1705,1706],{"scriptlevel":1676,"displaystyle":1677},[1078,1707,1083],{},[1671,1709,1710],{},[1674,1711,1712],{"scriptlevel":1676,"displaystyle":1677},[1078,1713,1083],{},[1668,1715,1716,1723,1730],{},[1671,1717,1718],{},[1674,1719,1720],{"scriptlevel":1676,"displaystyle":1677},[1078,1721,1722],{},"8",[1671,1724,1725],{},[1674,1726,1727],{"scriptlevel":1676,"displaystyle":1677},[1078,1728,1729],{},"7",[1671,1731,1732],{},[1674,1733,1734],{"scriptlevel":1676,"displaystyle":1677},[1078,1735,1736],{},"9",[866,1738,1739],{"fence":606},")",[599,1741,1742],{"encoding":601},"A = \\begin{pmatrix} 2 & 1 & 1 \\\\ 4 & 3 & 3 \\\\ 8 & 7 & 9 \\end{pmatrix}",[576,1744,1746,1764],{"className":1745,"ariaHidden":606},[605],[576,1747,1749,1752,1755,1758,1761],{"className":1748},[610],[576,1750],{"className":1751,"style":615},[614],[576,1753,597],{"className":1754},[619,620],[576,1756],{"className":1757,"style":894},[893],[576,1759,868],{"className":1760},[898],[576,1762],{"className":1763,"style":894},[893],[576,1765,1767,1771],{"className":1766},[610],[576,1768],{"className":1769,"style":1770},[614],"height:3.6em;vertical-align:-1.55em;",[576,1772,1774,1825,2017],{"className":1773},[1485],[576,1775,1777],{"className":1776},[1111],[576,1778,1782],{"className":1779},[1780,1781],"delimsizing","mult",[576,1783,1785,1816],{"className":1784},[706,707],[576,1786,1788,1813],{"className":1787},[711],[576,1789,1792],{"className":1790,"style":1791},[715],"height:2.05em;",[576,1793,1795,1799],{"style":1794},"top:-4.05em;",[576,1796],{"className":1797,"style":1798},[723],"height:5.6em;",[576,1800,1802],{"style":1801},"width:0.875em;height:3.6em;",[1803,1804,1809],"svg",{"xmlns":1805,"width":1806,"height":1807,"viewBox":1808},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.875em","3.6em","0 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= \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 4 & 3 & 1 \\end{pmatrix}, \\qquad U = \\begin{pmatrix} 2 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 2 \\end{pmatrix}",[576,2534,2536,2554,2848],{"className":2535,"ariaHidden":606},[605],[576,2537,2539,2542,2545,2548,2551],{"className":2538},[610],[576,2540],{"className":2541,"style":615},[614],[576,2543,829],{"className":2544},[619,620],[576,2546],{"className":2547,"style":894},[893],[576,2549,868],{"className":2550},[898],[576,2552],{"className":2553,"style":894},[893],[576,2555,2557,2560,2823,2826,2829,2833,2836,2839,2842,2845],{"className":2556},[610],[576,2558],{"className":2559,"style":1770},[614],[576,2561,2563,2600,2786],{"className":2562},[1485],[576,2564,2566],{"className":2565},[1111],[576,2567,2569],{"className":2568},[1780,1781],[576,2570,2572,2592],{"className":2571},[706,707],[576,2573,2575,2589],{"className":2574},[711],[576,2576,2578],{"className":2577,"style":1791},[715],[576,2579,2580,2583],{"style":1794},[576,2581],{"className":2582,"style":1798},[723],[576,2584,2585],{"style":1801},[1803,2586,2587],{"xmlns":1805,"width":1806,"height":1807,"viewBox":1808},[1810,2588],{"d":1812},[576,2590,744],{"className":2591},[743],[576,2593,2595],{"className":2594},[711],[576,2596,2598],{"className":2597,"style":1822},[715],[576,2599],{},[576,2601,2603],{"className":2602},[619],[576,2604,2606,2662,2665,2668,2724,2727,2730],{"className":2605},[1662],[576,2607,2609],{"className":2608},[1834],[576,2610,2612,2654],{"className":2611},[706,707],[576,2613,2615,2651],{"className":2614},[711],[576,2616,2618,2629,2640],{"className":2617,"style":1791},[715],[576,2619,2620,2623],{"style":1846},[576,2621],{"className":2622,"style":1132},[723],[576,2624,2626],{"className":2625},[619],[576,2627,1686],{"className":2628},[619],[576,2630,2631,2634],{"style":1858},[576,2632],{"className":2633,"style":1132},[723],[576,2635,2637],{"className":2636},[619],[576,2638,1080],{"className":2639},[619],[576,2641,2642,2645],{"style":1870},[576,2643],{"className":2644,"style":1132},[723],[576,2646,2648],{"className":2647},[619],[576,2649,1701],{"className":2650},[619],[576,2652,744],{"className":2653},[743],[576,2655,2657],{"className":2656},[711],[576,2658,2660],{"className":2659,"style":1822},[715],[576,2661],{},[576,2663],{"className":2664,"style":1895},[1894],[576,2666],{"className":2667,"style":1895},[1894],[576,2669,2671],{"className":2670},[1834],[576,2672,2674,2716],{"className":2673},[706,707],[576,2675,2677,2713],{"className":2676},[711],[576,2678,2680,2691,2702],{"className":2679,"style":1791},[715],[576,2681,2682,2685],{"style":1846},[576,2683],{"className":2684,"style":1132},[723],[576,2686,2688],{"className":2687},[619],[576,2689,1676],{"className":2690},[619],[576,2692,2693,2696],{"style":1858},[576,2694],{"className":2695,"style":1132},[723],[576,2697,2699],{"className":2698},[619],[576,2700,1686],{"className":2701},[619],[576,2703,2704,2707],{"style":1870},[576,2705],{"className":2706,"style":1132},[723],[576,2708,2710],{"className":2709},[619],[576,2711,1083],{"className":2712},[619],[576,2714,744],{"className":2715},[743],[576,2717,2719],{"className":2718},[711],[576,2720,2722],{"className":2721,"style":1822},[715],[576,2723],{},[576,2725],{"className":2726,"style":1895},[1894],[576,2728],{"className":2729,"style":1895},[1894],[576,2731,2733],{"className":2732},[1834],[576,2734,2736,2778],{"className":2735},[706,707],[576,2737,2739,2775],{"className":2738},[711],[576,2740,2742,2753,2764],{"className":2741,"style":1791},[715],[576,2743,2744,2747],{"style":1846},[576,2745],{"className":2746,"style":1132},[723],[576,2748,2750],{"className":2749},[619],[576,2751,1676],{"className":2752},[619],[576,2754,2755,2758],{"style":1858},[576,2756],{"className":2757,"style":1132},[723],[576,2759,2761],{"className":2760},[619],[576,2762,1676],{"className":2763},[619],[576,2765,2766,2769],{"style":1870},[576,2767],{"className":2768,"style":1132},[723],[576,2770,2772],{"className":2771},[619],[576,2773,1686],{"className":2774},[619],[576,2776,744],{"className":2777},[743],[576,2779,2781],{"className":2780},[711],[576,2782,2784],{"className":2783,"style":1822},[715],[576,2785],{},[576,2787,2789],{"className":2788},[1183],[576,2790,2792],{"className":2791},[1780,1781],[576,2793,2795,2815],{"className":2794},[706,707],[576,2796,2798,2812],{"className":2797},[711],[576,2799,2801],{"className":2800,"style":1791},[715],[576,2802,2803,2806],{"style":1794},[576,2804],{"className":2805,"style":1798},[723],[576,2807,2808],{"style":1801},[1803,2809,2810],{"xmlns":1805,"width":1806,"height":1807,"viewBox":1808},[1810,2811],{"d":2043},[576,2813,744],{"className":2814},[743],[576,2816,2818],{"className":2817},[711],[576,2819,2821],{"className":2820,"style":1822},[715],[576,2822],{},[576,2824],{"className":2825,"style":967},[893],[576,2827,2147],{"className":2828},[2178],[576,2830],{"className":2831,"style":2832},[893],"margin-right:2em;",[576,2834],{"className":2835,"style":967},[893],[576,2837,636],{"className":2838,"style":651},[619,620],[576,2840],{"className":2841,"style":894},[893],[576,2843,868],{"className":2844},[898],[576,2846],{"className":2847,"style":894},[893],[576,2849,2851,2854],{"className":2850},[610],[576,2852],{"className":2853,"style":1770},[614],[576,2855,2857,2894,3080],{"className":2856},[1485],[576,2858,2860],{"className":2859},[1111],[576,2861,2863],{"className":2862},[1780,1781],[576,2864,2866,2886],{"className":2865},[706,707],[576,2867,2869,2883],{"className":2868},[711],[576,2870,2872],{"className":2871,"style":1791},[715],[576,2873,2874,2877],{"style":1794},[576,2875],{"className":2876,"style":1798},[723],[576,2878,2879],{"style":1801},[1803,2880,2881],{"xmlns":1805,"width":1806,"height":1807,"viewBox":1808},[1810,2882],{"d":1812},[576,2884,744],{"className":2885},[743],[576,2887,2889],{"className":2888},[711],[576,2890,2892],{"className":2891,"style":1822},[715],[576,2893],{},[576,2895,2897],{"className":2896},[619],[576,2898,2900,2956,2959,2962,3018,3021,3024],{"className":2899},[1662],[576,2901,2903],{"className":2902},[1834],[576,2904,2906,2948],{"className":2905},[706,707],[576,2907,2909,2945],{"className":2908},[711],[576,2910,2912,2923,2934],{"className":2911,"style":1791},[715],[576,2913,2914,2917],{"style":1846},[576,2915],{"className":2916,"style":1132},[723],[576,2918,2920],{"className":2919},[619],[576,2921,1080],{"className":2922},[619],[576,2924,2925,2928],{"style":1858},[576,2926],{"className":2927,"style":1132},[723],[576,2929,2931],{"className":2930},[619],[576,2932,1676],{"className":2933},[619],[576,2935,2936,2939],{"style":1870},[576,2937],{"className":2938,"style":1132},[723],[576,2940,2942],{"className":2941},[619],[576,2943,1676],{"className":2944},[619],[576,2946,744],{"className":2947},[743],[576,2949,2951],{"className":2950},[711],[576,2952,2954],{"className":2953,"style":1822},[715],[576,2955],{},[576,2957],{"className":2958,"style":1895},[1894],[576,2960],{"className":2961,"style":1895},[1894],[576,2963,2965],{"className":2964},[1834],[576,2966,2968,3010],{"className":2967},[706,707],[576,2969,2971,3007],{"className":2970},[711],[576,2972,2974,2985,2996],{"className":2973,"style":1791},[715],[576,2975,2976,2979],{"style":1846},[576,2977],{"className":2978,"style":1132},[723],[576,2980,2982],{"className":2981},[619],[576,2983,1686],{"className":2984},[619],[576,2986,2987,2990],{"style":1858},[576,2988],{"className":2989,"style":1132},[723],[576,2991,2993],{"className":2992},[619],[576,2994,1686],{"className":2995},[619],[576,2997,2998,3001],{"style":1870},[576,2999],{"className":3000,"style":1132},[723],[576,3002,3004],{"className":3003},[619],[576,3005,1676],{"className":3006},[619],[576,3008,744],{"className":3009},[743],[576,3011,3013],{"className":3012},[711],[576,3014,3016],{"className":3015,"style":1822},[715],[576,3017],{},[576,3019],{"className":3020,"style":1895},[1894],[576,3022],{"className":3023,"style":1895},[1894],[576,3025,3027],{"className":3026},[1834],[576,3028,3030,3072],{"className":3029},[706,707],[576,3031,3033,3069],{"className":3032},[711],[576,3034,3036,3047,3058],{"className":3035,"style":1791},[715],[576,3037,3038,3041],{"style":1846},[576,3039],{"className":3040,"style":1132},[723],[576,3042,3044],{"className":3043},[619],[576,3045,1686],{"className":3046},[619],[576,3048,3049,3052],{"style":1858},[576,3050],{"className":3051,"style":1132},[723],[576,3053,3055],{"className":3054},[619],[576,3056,1686],{"className":3057},[619],[576,3059,3060,3063],{"style":1870},[576,3061],{"className":3062,"style":1132},[723],[576,3064,3066],{"className":3065},[619],[576,3067,1080],{"className":3068},[619],[576,3070,744],{"className":3071},[743],[576,3073,3075],{"className":3074},[711],[576,3076,3078],{"className":3077,"style":1822},[715],[576,3079],{},[576,3081,3083],{"className":3082},[1183],[576,3084,3086],{"className":3085},[1780,1781],[576,3087,3089,3109],{"className":3088},[706,707],[576,3090,3092,3106],{"className":3091},[711],[576,3093,3095],{"className":3094,"style":1791},[715],[576,3096,3097,3100],{"style":1794},[576,3098],{"className":3099,"style":1798},[723],[576,3101,3102],{"style":1801},[1803,3103,3104],{"xmlns":1805,"width":1806,"height":1807,"viewBox":1808},[1810,3105],{"d":2043},[576,3107,744],{"className":3108},[743],[576,3110,3112],{"className":3111},[711],[576,3113,3115],{"className":3114,"style":1822},[715],[576,3116],{},[565,3118,3119,3120,844],{},"und ",[576,3121,3123,3157],{"className":3122},[579],[576,3124,3126],{"className":3125},[583],[585,3127,3128],{"xmlns":587},[589,3129,3130,3154],{},[592,3131,3132,3134,3136,3138,3140,3142,3144,3146,3148,3150,3152],{},[595,3133,1295],{},[866,3135,1298],{},[595,3137,597],{},[866,3139,868],{},[1078,3141,1080],{},[866,3143,1313],{},[1078,3145,1686],{},[866,3147,1313],{},[1078,3149,1080],{},[866,3151,868],{},[1078,3153,1701],{},[599,3155,3156],{"encoding":601},"\\det A = 2 \\cdot 1 \\cdot 2 = 4",[576,3158,3160,3184,3202,3220,3238],{"className":3159,"ariaHidden":606},[605],[576,3161,3163,3166,3169,3172,3175,3178,3181],{"className":3162},[610],[576,3164],{"className":3165,"style":908},[614],[576,3167,1295],{"className":3168},[1349],[576,3170],{"className":3171,"style":967},[893],[576,3173,597],{"className":3174},[619,620],[576,3176],{"className":3177,"style":894},[893],[576,3179,868],{"className":3180},[898],[576,3182],{"className":3183,"style":894},[893],[576,3185,3187,3190,3193,3196,3199],{"className":3186},[610],[576,3188],{"className":3189,"style":2086},[614],[576,3191,1080],{"className":3192},[619],[576,3194],{"className":3195,"style":1421},[893],[576,3197,1313],{"className":3198},[1425],[576,3200],{"className":3201,"style":1421},[893],[576,3203,3205,3208,3211,3214,3217],{"className":3204},[610],[576,3206],{"className":3207,"style":2086},[614],[576,3209,1686],{"className":3210},[619],[576,3212],{"className":3213,"style":1421},[893],[576,3215,1313],{"className":3216},[1425],[576,3218],{"className":3219,"style":1421},[893],[576,3221,3223,3226,3229,3232,3235],{"className":3222},[610],[576,3224],{"className":3225,"style":2086},[614],[576,3227,1080],{"className":3228},[619],[576,3230],{"className":3231,"style":894},[893],[576,3233,868],{"className":3234},[898],[576,3236],{"className":3237,"style":894},[893],[576,3239,3241,3244],{"className":3240},[610],[576,3242],{"className":3243,"style":2086},[614],[576,3245,1701],{"className":3246},[619],[565,3248,3249,3250,3339,3340,3433,3434,3433,3594,3796,3797,3433,3889,3433,4025,4230,4231,844],{},"Für ",[576,3251,3253,3285],{"className":3252},[579],[576,3254,3256],{"className":3255},[583],[585,3257,3258],{"xmlns":587},[589,3259,3260,3282],{},[592,3261,3262,3264,3266,3268,3270,3272,3275,3277,3280],{},[595,3263,871],{},[866,3265,868],{},[866,3267,1660],{"stretchy":1677},[1078,3269,1701],{},[866,3271,2147],{"separator":606},[1078,3273,3274],{},"10",[866,3276,2147],{"separator":606},[1078,3278,3279],{},"24",[866,3281,1739],{"stretchy":1677},[599,3283,3284],{"encoding":601},"b = (4, 10, 24)",[576,3286,3288,3306],{"className":3287,"ariaHidden":606},[605],[576,3289,3291,3294,3297,3300,3303],{"className":3290},[610],[576,3292],{"className":3293,"style":908},[614],[576,3295,871],{"className":3296},[619,620],[576,3298],{"className":3299,"style":894},[893],[576,3301,868],{"className":3302},[898],[576,3304],{"className":3305,"style":894},[893],[576,3307,3309,3312,3315,3318,3321,3324,3327,3330,3333,3336],{"className":3308},[610],[576,3310],{"className":3311,"style":2168},[614],[576,3313,1660],{"className":3314},[1111],[576,3316,1701],{"className":3317},[619],[576,3319,2147],{"className":3320},[2178],[576,3322],{"className":3323,"style":967},[893],[576,3325,3274],{"className":3326},[619],[576,3328,2147],{"className":3329},[2178],[576,3331],{"className":3332,"style":967},[893],[576,3334,3279],{"className":3335},[619],[576,3337,1739],{"className":3338},[1183]," liefert Vorwärtseinsetzen ",[576,3341,3343,3365],{"className":3342},[579],[576,3344,3346],{"className":3345},[583],[585,3347,3348],{"xmlns":587},[589,3349,3350,3362],{},[592,3351,3352,3358,3360],{},[666,3353,3354,3356],{},[595,3355,943],{},[1078,3357,1686],{},[866,3359,868],{},[1078,3361,1701],{},[599,3363,3364],{"encoding":601},"y_1 = 4",[576,3366,3368,3424],{"className":3367,"ariaHidden":606},[605],[576,3369,3371,3374,3415,3418,3421],{"className":3370},[610],[576,3372],{"className":3373,"style":1054},[614],[576,3375,3377,3380],{"className":3376},[619],[576,3378,943],{"className":3379,"style":971},[619,620],[576,3381,3383],{"className":3382},[702],[576,3384,3386,3407],{"className":3385},[706,707],[576,3387,3389,3404],{"className":3388},[711],[576,3390,3392],{"className":3391,"style":1390},[715],[576,3393,3395,3398],{"style":3394},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[576,3396],{"className":3397,"style":724},[723],[576,3399,3401],{"className":3400},[728,729,730,731],[576,3402,1686],{"className":3403},[619,731],[576,3405,744],{"className":3406},[743],[576,3408,3410],{"className":3409},[711],[576,3411,3413],{"className":3412,"style":1415},[715],[576,3414],{},[576,3416],{"className":3417,"style":894},[893],[576,3419,868],{"className":3420},[898],[576,3422],{"className":3423,"style":894},[893],[576,3425,3427,3430],{"className":3426},[610],[576,3428],{"className":3429,"style":2086},[614],[576,3431,1701],{"className":3432},[619],", ",[576,3435,3437,3472],{"className":3436},[579],[576,3438,3440],{"className":3439},[583],[585,3441,3442],{"xmlns":587},[589,3443,3444,3469],{},[592,3445,3446,3452,3454,3456,3459,3461,3463,3465,3467],{},[666,3447,3448,3450],{},[595,3449,943],{},[1078,3451,1080],{},[866,3453,868],{},[1078,3455,3274],{},[866,3457,3458],{},"−",[1078,3460,1080],{},[866,3462,1313],{},[1078,3464,1701],{},[866,3466,868],{},[1078,3468,1080],{},[599,3470,3471],{"encoding":601},"y_2 = 10 - 2 \\cdot 4 = 2",[576,3473,3475,3530,3549,3567,3585],{"className":3474,"ariaHidden":606},[605],[576,3476,3478,3481,3521,3524,3527],{"className":3477},[610],[576,3479],{"className":3480,"style":1054},[614],[576,3482,3484,3487],{"className":3483},[619],[576,3485,943],{"className":3486,"style":971},[619,620],[576,3488,3490],{"className":3489},[702],[576,3491,3493,3513],{"className":3492},[706,707],[576,3494,3496,3510],{"className":3495},[711],[576,3497,3499],{"className":3498,"style":1390},[715],[576,3500,3501,3504],{"style":3394},[576,3502],{"className":3503,"style":724},[723],[576,3505,3507],{"className":3506},[728,729,730,731],[576,3508,1080],{"className":3509},[619,731],[576,3511,744],{"className":3512},[743],[576,3514,3516],{"className":3515},[711],[576,3517,3519],{"className":3518,"style":1415},[715],[576,3520],{},[576,3522],{"className":3523,"style":894},[893],[576,3525,868],{"className":3526},[898],[576,3528],{"className":3529,"style":894},[893],[576,3531,3533,3537,3540,3543,3546],{"className":3532},[610],[576,3534],{"className":3535,"style":3536},[614],"height:0.7278em;vertical-align:-0.0833em;",[576,3538,3274],{"className":3539},[619],[576,3541],{"className":3542,"style":1421},[893],[576,3544,3458],{"className":3545},[1425],[576,3547],{"className":3548,"style":1421},[893],[576,3550,3552,3555,3558,3561,3564],{"className":3551},[610],[576,3553],{"className":3554,"style":2086},[614],[576,3556,1080],{"className":3557},[619],[576,3559],{"className":3560,"style":1421},[893],[576,3562,1313],{"className":3563},[1425],[576,3565],{"className":3566,"style":1421},[893],[576,3568,3570,3573,3576,3579,3582],{"className":3569},[610],[576,3571],{"className":3572,"style":2086},[614],[576,3574,1701],{"className":3575},[619],[576,3577],{"className":3578,"style":894},[893],[576,3580,868],{"className":3581},[898],[576,3583],{"className":3584,"style":894},[893],[576,3586,3588,3591],{"className":3587},[610],[576,3589],{"className":3590,"style":2086},[614],[576,3592,1080],{"className":3593},[619],[576,3595,3597,3639],{"className":3596},[579],[576,3598,3600],{"className":3599},[583],[585,3601,3602],{"xmlns":587},[589,3603,3604,3636],{},[592,3605,3606,3612,3614,3616,3618,3620,3622,3624,3626,3628,3630,3632,3634],{},[666,3607,3608,3610],{},[595,3609,943],{},[1078,3611,1083],{},[866,3613,868],{},[1078,3615,3279],{},[866,3617,3458],{},[1078,3619,1701],{},[866,3621,1313],{},[1078,3623,1701],{},[866,3625,3458],{},[1078,3627,1083],{},[866,3629,1313],{},[1078,3631,1080],{},[866,3633,868],{},[1078,3635,1080],{},[599,3637,3638],{"encoding":601},"y_3 = 24 - 4 \\cdot 4 - 3 \\cdot 2 = 2",[576,3640,3642,3697,3715,3733,3751,3769,3787],{"className":3641,"ariaHidden":606},[605],[576,3643,3645,3648,3688,3691,3694],{"className":3644},[610],[576,3646],{"className":3647,"style":1054},[614],[576,3649,3651,3654],{"className":3650},[619],[576,3652,943],{"className":3653,"style":971},[619,620],[576,3655,3657],{"className":3656},[702],[576,3658,3660,3680],{"className":3659},[706,707],[576,3661,3663,3677],{"className":3662},[711],[576,3664,3666],{"className":3665,"style":1390},[715],[576,3667,3668,3671],{"style":3394},[576,3669],{"className":3670,"style":724},[723],[576,3672,3674],{"className":3673},[728,729,730,731],[576,3675,1083],{"className":3676},[619,731],[576,3678,744],{"className":3679},[743],[576,3681,3683],{"className":3682},[711],[576,3684,3686],{"className":3685,"style":1415},[715],[576,3687],{},[576,3689],{"className":3690,"style":894},[893],[576,3692,868],{"className":3693},[898],[576,3695],{"className":3696,"style":894},[893],[576,3698,3700,3703,3706,3709,3712],{"className":3699},[610],[576,3701],{"className":3702,"style":3536},[614],[576,3704,3279],{"className":3705},[619],[576,3707],{"className":3708,"style":1421},[893],[576,3710,3458],{"className":3711},[1425],[576,3713],{"className":3714,"style":1421},[893],[576,3716,3718,3721,3724,3727,3730],{"className":3717},[610],[576,3719],{"className":3720,"style":2086},[614],[576,3722,1701],{"className":3723},[619],[576,3725],{"className":3726,"style":1421},[893],[576,3728,1313],{"className":3729},[1425],[576,3731],{"className":3732,"style":1421},[893],[576,3734,3736,3739,3742,3745,3748],{"className":3735},[610],[576,3737],{"className":3738,"style":3536},[614],[576,3740,1701],{"className":3741},[619],[576,3743],{"className":3744,"style":1421},[893],[576,3746,3458],{"className":3747},[1425],[576,3749],{"className":3750,"style":1421},[893],[576,3752,3754,3757,3760,3763,3766],{"className":3753},[610],[576,3755],{"className":3756,"style":2086},[614],[576,3758,1083],{"className":3759},[619],[576,3761],{"className":3762,"style":1421},[893],[576,3764,1313],{"className":3765},[1425],[576,3767],{"className":3768,"style":1421},[893],[576,3770,3772,3775,3778,3781,3784],{"className":3771},[610],[576,3773],{"className":3774,"style":2086},[614],[576,3776,1080],{"className":3777},[619],[576,3779],{"className":3780,"style":894},[893],[576,3782,868],{"className":3783},[898],[576,3785],{"className":3786,"style":894},[893],[576,3788,3790,3793],{"className":3789},[610],[576,3791],{"className":3792,"style":2086},[614],[576,3794,1080],{"className":3795},[619],". Rückwärtseinsetzen ergibt ",[576,3798,3800,3822],{"className":3799},[579],[576,3801,3803],{"className":3802},[583],[585,3804,3805],{"xmlns":587},[589,3806,3807,3819],{},[592,3808,3809,3815,3817],{},[666,3810,3811,3813],{},[595,3812,864],{},[1078,3814,1083],{},[866,3816,868],{},[1078,3818,1686],{},[599,3820,3821],{"encoding":601},"x_3 = 1",[576,3823,3825,3880],{"className":3824,"ariaHidden":606},[605],[576,3826,3828,3831,3871,3874,3877],{"className":3827},[610],[576,3829],{"className":3830,"style":1435},[614],[576,3832,3834,3837],{"className":3833},[619],[576,3835,864],{"className":3836},[619,620],[576,3838,3840],{"className":3839},[702],[576,3841,3843,3863],{"className":3842},[706,707],[576,3844,3846,3860],{"className":3845},[711],[576,3847,3849],{"className":3848,"style":1390},[715],[576,3850,3851,3854],{"style":1393},[576,3852],{"className":3853,"style":724},[723],[576,3855,3857],{"className":3856},[728,729,730,731],[576,3858,1083],{"className":3859},[619,731],[576,3861,744],{"className":3862},[743],[576,3864,3866],{"className":3865},[711],[576,3867,3869],{"className":3868,"style":1415},[715],[576,3870],{},[576,3872],{"className":3873,"style":894},[893],[576,3875,868],{"className":3876},[898],[576,3878],{"className":3879,"style":894},[893],[576,3881,3883,3886],{"className":3882},[610],[576,3884],{"className":3885,"style":2086},[614],[576,3887,1686],{"className":3888},[619],[576,3890,3892,3922],{"className":3891},[579],[576,3893,3895],{"className":3894},[583],[585,3896,3897],{"xmlns":587},[589,3898,3899,3919],{},[592,3900,3901,3907,3909,3911,3913,3915,3917],{},[666,3902,3903,3905],{},[595,3904,864],{},[1078,3906,1080],{},[866,3908,868],{},[1078,3910,1080],{},[866,3912,3458],{},[1078,3914,1686],{},[866,3916,868],{},[1078,3918,1686],{},[599,3920,3921],{"encoding":601},"x_2 = 2 - 1 = 1",[576,3923,3925,3980,3998,4016],{"className":3924,"ariaHidden":606},[605],[576,3926,3928,3931,3971,3974,3977],{"className":3927},[610],[576,3929],{"className":3930,"style":1435},[614],[576,3932,3934,3937],{"className":3933},[619],[576,3935,864],{"className":3936},[619,620],[576,3938,3940],{"className":3939},[702],[576,3941,3943,3963],{"className":3942},[706,707],[576,3944,3946,3960],{"className":3945},[711],[576,3947,3949],{"className":3948,"style":1390},[715],[576,3950,3951,3954],{"style":1393},[576,3952],{"className":3953,"style":724},[723],[576,3955,3957],{"className":3956},[728,729,730,731],[576,3958,1080],{"className":3959},[619,731],[576,3961,744],{"className":3962},[743],[576,3964,3966],{"className":3965},[711],[576,3967,3969],{"className":3968,"style":1415},[715],[576,3970],{},[576,3972],{"className":3973,"style":894},[893],[576,3975,868],{"className":3976},[898],[576,3978],{"className":3979,"style":894},[893],[576,3981,3983,3986,3989,3992,3995],{"className":3982},[610],[576,3984],{"className":3985,"style":3536},[614],[576,3987,1080],{"className":3988},[619],[576,3990],{"className":3991,"style":1421},[893],[576,3993,3458],{"className":3994},[1425],[576,3996],{"className":3997,"style":1421},[893],[576,3999,4001,4004,4007,4010,4013],{"className":4000},[610],[576,4002],{"className":4003,"style":2086},[614],[576,4005,1686],{"className":4006},[619],[576,4008],{"className":4009,"style":894},[893],[576,4011,868],{"className":4012},[898],[576,4014],{"className":4015,"style":894},[893],[576,4017,4019,4022],{"className":4018},[610],[576,4020],{"className":4021,"style":2086},[614],[576,4023,1686],{"className":4024},[619],[576,4026,4028,4068],{"className":4027},[579],[576,4029,4031],{"className":4030},[583],[585,4032,4033],{"xmlns":587},[589,4034,4035,4065],{},[592,4036,4037,4043,4045,4061,4063],{},[666,4038,4039,4041],{},[595,4040,864],{},[1078,4042,1686],{},[866,4044,868],{},[1075,4046,4047,4059],{},[592,4048,4049,4051,4053,4055,4057],{},[1078,4050,1701],{},[866,4052,3458],{},[1078,4054,1686],{},[866,4056,3458],{},[1078,4058,1686],{},[1078,4060,1080],{},[866,4062,868],{},[1078,4064,1686],{},[599,4066,4067],{"encoding":601},"x_1 = \\frac{4 - 1 - 1}{2} = 1",[576,4069,4071,4126,4221],{"className":4070,"ariaHidden":606},[605],[576,4072,4074,4077,4117,4120,4123],{"className":4073},[610],[576,4075],{"className":4076,"style":1435},[614],[576,4078,4080,4083],{"className":4079},[619],[576,4081,864],{"className":4082},[619,620],[576,4084,4086],{"className":4085},[702],[576,4087,4089,4109],{"className":4088},[706,707],[576,4090,4092,4106],{"className":4091},[711],[576,4093,4095],{"className":4094,"style":1390},[715],[576,4096,4097,4100],{"style":1393},[576,4098],{"className":4099,"style":724},[723],[576,4101,4103],{"className":4102},[728,729,730,731],[576,4104,1686],{"className":4105},[619,731],[576,4107,744],{"className":4108},[743],[576,4110,4112],{"className":4111},[711],[576,4113,4115],{"className":4114,"style":1415},[715],[576,4116],{},[576,4118],{"className":4119,"style":894},[893],[576,4121,868],{"className":4122},[898],[576,4124],{"className":4125,"style":894},[893],[576,4127,4129,4132,4212,4215,4218],{"className":4128},[610],[576,4130],{"className":4131,"style":1104},[614],[576,4133,4135,4138,4209],{"className":4134},[619],[576,4136],{"className":4137},[1111,1112],[576,4139,4141],{"className":4140},[1075],[576,4142,4144,4201],{"className":4143},[706,707],[576,4145,4147,4198],{"className":4146},[711],[576,4148,4150,4164,4172],{"className":4149,"style":1125},[715],[576,4151,4152,4155],{"style":1128},[576,4153],{"className":4154,"style":1132},[723],[576,4156,4158],{"className":4157},[728,729,730,731],[576,4159,4161],{"className":4160},[619,731],[576,4162,1080],{"className":4163},[619,731],[576,4165,4166,4169],{"style":1144},[576,4167],{"className":4168,"style":1132},[723],[576,4170],{"className":4171,"style":1152},[1151],[576,4173,4174,4177],{"style":1155},[576,4175],{"className":4176,"style":1132},[723],[576,4178,4180],{"className":4179},[728,729,730,731],[576,4181,4183,4186,4189,4192,4195],{"className":4182},[619,731],[576,4184,1701],{"className":4185},[619,731],[576,4187,3458],{"className":4188},[1425,731],[576,4190,1686],{"className":4191},[619,731],[576,4193,3458],{"className":4194},[1425,731],[576,4196,1686],{"className":4197},[619,731],[576,4199,744],{"className":4200},[743],[576,4202,4204],{"className":4203},[711],[576,4205,4207],{"className":4206,"style":1177},[715],[576,4208],{},[576,4210],{"className":4211},[1183,1112],[576,4213],{"className":4214,"style":894},[893],[576,4216,868],{"className":4217},[898],[576,4219],{"className":4220,"style":894},[893],[576,4222,4224,4227],{"className":4223},[610],[576,4225],{"className":4226,"style":2086},[614],[576,4228,1686],{"className":4229},[619],", also ",[576,4232,4234,4264],{"className":4233},[579],[576,4235,4237],{"className":4236},[583],[585,4238,4239],{"xmlns":587},[589,4240,4241,4261],{},[592,4242,4243,4245,4247,4249,4251,4253,4255,4257,4259],{},[595,4244,864],{},[866,4246,868],{},[866,4248,1660],{"stretchy":1677},[1078,4250,1686],{},[866,4252,2147],{"separator":606},[1078,4254,1686],{},[866,4256,2147],{"separator":606},[1078,4258,1686],{},[866,4260,1739],{"stretchy":1677},[599,4262,4263],{"encoding":601},"x = (1, 1, 1)",[576,4265,4267,4286],{"className":4266,"ariaHidden":606},[605],[576,4268,4270,4274,4277,4280,4283],{"className":4269},[610],[576,4271],{"className":4272,"style":4273},[614],"height:0.4306em;",[576,4275,864],{"className":4276},[619,620],[576,4278],{"className":4279,"style":894},[893],[576,4281,868],{"className":4282},[898],[576,4284],{"className":4285,"style":894},[893],[576,4287,4289,4292,4295,4298,4301,4304,4307,4310,4313,4316],{"className":4288},[610],[576,4290],{"className":4291,"style":2168},[614],[576,4293,1660],{"className":4294},[1111],[576,4296,1686],{"className":4297},[619],[576,4299,2147],{"className":4300},[2178],[576,4302],{"className":4303,"style":967},[893],[576,4305,1686],{"className":4306},[619],[576,4308,2147],{"className":4309},[2178],[576,4311],{"className":4312,"style":967},[893],[576,4314,1686],{"className":4315},[619],[576,4317,1739],{"className":4318},[1183],[560,4320,4322],{"id":4321},"typische-fehler","Typische Fehler",[4324,4325,4326,4455,4458,4645],"ul",{},[917,4327,4328,4329,4357,4358,4386,4387,4390,4391,4419,4420,844],{},"Die Faktoren mit falschem Vorzeichen in ",[576,4330,4332,4345],{"className":4331},[579],[576,4333,4335],{"className":4334},[583],[585,4336,4337],{"xmlns":587},[589,4338,4339,4343],{},[592,4340,4341],{},[595,4342,829],{},[599,4344,829],{"encoding":601},[576,4346,4348],{"className":4347,"ariaHidden":606},[605],[576,4349,4351,4354],{"className":4350},[610],[576,4352],{"className":4353,"style":615},[614],[576,4355,829],{"className":4356},[619,620]," eintragen. Wird das ",[576,4359,4361,4374],{"className":4360},[579],[576,4362,4364],{"className":4363},[583],[585,4365,4366],{"xmlns":587},[589,4367,4368,4372],{},[592,4369,4370],{},[1078,4371,1080],{},[599,4373,1080],{"encoding":601},[576,4375,4377],{"className":4376,"ariaHidden":606},[605],[576,4378,4380,4383],{"className":4379},[610],[576,4381],{"className":4382,"style":2086},[614],[576,4384,1080],{"className":4385},[619],"-Fache ",[569,4388,4389],{},"abgezogen",", steht in ",[576,4392,4394,4407],{"className":4393},[579],[576,4395,4397],{"className":4396},[583],[585,4398,4399],{"xmlns":587},[589,4400,4401,4405],{},[592,4402,4403],{},[595,4404,829],{},[599,4406,829],{"encoding":601},[576,4408,4410],{"className":4409,"ariaHidden":606},[605],[576,4411,4413,4416],{"className":4412},[610],[576,4414],{"className":4415,"style":615},[614],[576,4417,829],{"className":4418},[619,620]," eine ",[576,4421,4423,4440],{"className":4422},[579],[576,4424,4426],{"className":4425},[583],[585,4427,4428],{"xmlns":587},[589,4429,4430,4437],{},[592,4431,4432,4435],{},[866,4433,4434],{},"+",[1078,4436,1080],{},[599,4438,4439],{"encoding":601},"+2",[576,4441,4443],{"className":4442,"ariaHidden":606},[605],[576,4444,4446,4449,4452],{"className":4445},[610],[576,4447],{"className":4448,"style":3536},[614],[576,4450,4434],{"className":4451},[619],[576,4453,1080],{"className":4454},[619],[917,4456,4457],{},"Für die Faktoren der zweiten Spalte die ursprünglichen statt der bereits umgeformten Zeilen verwenden.",[917,4459,4460,4461,4644],{},"Die Zerlegung erzwingen, obwohl ein Pivot null ist. ",[576,4462,4464,4511],{"className":4463},[579],[576,4465,4467],{"className":4466},[583],[585,4468,4469],{"xmlns":587},[589,4470,4471,4508],{},[592,4472,4473,4475,4506],{},[866,4474,1660],{"fence":606},[1662,4476,4478,4492],{"rowspacing":1664,"columnalign":4477,"columnspacing":1666},"center center",[1668,4479,4480,4486],{},[1671,4481,4482],{},[1674,4483,4484],{"scriptlevel":1676,"displaystyle":1677},[1078,4485,1676],{},[1671,4487,4488],{},[1674,4489,4490],{"scriptlevel":1676,"displaystyle":1677},[1078,4491,1686],{},[1668,4493,4494,4500],{},[1671,4495,4496],{},[1674,4497,4498],{"scriptlevel":1676,"displaystyle":1677},[1078,4499,1686],{},[1671,4501,4502],{},[1674,4503,4504],{"scriptlevel":1676,"displaystyle":1677},[1078,4505,1686],{},[866,4507,1739],{"fence":606},[599,4509,4510],{"encoding":601},"\\begin{pmatrix} 0 & 1 \\\\ 1 & 1 \\end{pmatrix}",[576,4512,4514],{"className":4513,"ariaHidden":606},[605],[576,4515,4517,4521],{"className":4516},[610],[576,4518],{"className":4519,"style":4520},[614],"height:2.4em;vertical-align:-0.95em;",[576,4522,4524,4532,4638],{"className":4523},[1485],[576,4525,4529],{"className":4526,"style":4528},[1111,4527],"delimcenter","top:0em;",[576,4530,1660],{"className":4531},[1780,730],[576,4533,4535],{"className":4534},[619],[576,4536,4538,4587,4590,4593],{"className":4537},[1662],[576,4539,4541],{"className":4540},[1834],[576,4542,4544,4578],{"className":4543},[706,707],[576,4545,4547,4575],{"className":4546},[711],[576,4548,4551,4563],{"className":4549,"style":4550},[715],"height:1.45em;",[576,4552,4554,4557],{"style":4553},"top:-3.61em;",[576,4555],{"className":4556,"style":1132},[723],[576,4558,4560],{"className":4559},[619],[576,4561,1676],{"className":4562},[619],[576,4564,4566,4569],{"style":4565},"top:-2.41em;",[576,4567],{"className":4568,"style":1132},[723],[576,4570,4572],{"className":4571},[619],[576,4573,1686],{"className":4574},[619],[576,4576,744],{"className":4577},[743],[576,4579,4581],{"className":4580},[711],[576,4582,4585],{"className":4583,"style":4584},[715],"height:0.95em;",[576,4586],{},[576,4588],{"className":4589,"style":1895},[1894],[576,4591],{"className":4592,"style":1895},[1894],[576,4594,4596],{"className":4595},[1834],[576,4597,4599,4630],{"className":4598},[706,707],[576,4600,4602,4627],{"className":4601},[711],[576,4603,4605,4616],{"className":4604,"style":4550},[715],[576,4606,4607,4610],{"style":4553},[576,4608],{"className":4609,"style":1132},[723],[576,4611,4613],{"className":4612},[619],[576,4614,1686],{"className":4615},[619],[576,4617,4618,4621],{"style":4565},[576,4619],{"className":4620,"style":1132},[723],[576,4622,4624],{"className":4623},[619],[576,4625,1686],{"className":4626},[619],[576,4628,744],{"className":4629},[743],[576,4631,4633],{"className":4632},[711],[576,4634,4636],{"className":4635,"style":4584},[715],[576,4637],{},[576,4639,4641],{"className":4640,"style":4528},[1183,4527],[576,4642,1739],{"className":4643},[1780,730]," hat keine LU-Zerlegung ohne Zeilentausch.",[917,4646,4647,4648,4705,4706,4734,4735,844],{},"Bei ",[576,4649,4651,4672],{"className":4650},[579],[576,4652,4654],{"className":4653},[583],[585,4655,4656],{"xmlns":587},[589,4657,4658,4670],{},[592,4659,4660,4662,4664,4666,4668],{},[595,4661,1555],{},[595,4663,597],{},[866,4665,868],{},[595,4667,829],{},[595,4669,636],{},[599,4671,553],{"encoding":601},[576,4673,4675,4696],{"className":4674,"ariaHidden":606},[605],[576,4676,4678,4681,4684,4687,4690,4693],{"className":4677},[610],[576,4679],{"className":4680,"style":615},[614],[576,4682,1555],{"className":4683,"style":1578},[619,620],[576,4685,597],{"className":4686},[619,620],[576,4688],{"className":4689,"style":894},[893],[576,4691,868],{"className":4692},[898],[576,4694],{"className":4695,"style":894},[893],[576,4697,4699,4702],{"className":4698},[610],[576,4700],{"className":4701,"style":615},[614],[576,4703,1600],{"className":4704,"style":651},[619,620]," vergessen, auch ",[576,4707,4709,4722],{"className":4708},[579],[576,4710,4712],{"className":4711},[583],[585,4713,4714],{"xmlns":587},[589,4715,4716,4720],{},[592,4717,4718],{},[595,4719,871],{},[599,4721,871],{"encoding":601},[576,4723,4725],{"className":4724,"ariaHidden":606},[605],[576,4726,4728,4731],{"className":4727},[610],[576,4729],{"className":4730,"style":908},[614],[576,4732,871],{"className":4733},[619,620]," zu permutieren: Man löst ",[576,4736,4738,4760],{"className":4737},[579],[576,4739,4741],{"className":4740},[583],[585,4742,4743],{"xmlns":587},[589,4744,4745,4757],{},[592,4746,4747,4749,4751,4753,4755],{},[595,4748,829],{},[595,4750,943],{},[866,4752,868],{},[595,4754,1555],{},[595,4756,871],{},[599,4758,4759],{"encoding":601},"Ly = Pb",[576,4761,4763,4784],{"className":4762,"ariaHidden":606},[605],[576,4764,4766,4769,4772,4775,4778,4781],{"className":4765},[610],[576,4767],{"className":4768,"style":960},[614],[576,4770,829],{"className":4771},[619,620],[576,4773,943],{"className":4774,"style":971},[619,620],[576,4776],{"className":4777,"style":894},[893],[576,4779,868],{"className":4780},[898],[576,4782],{"className":4783,"style":894},[893],[576,4785,4787,4790,4793],{"className":4786},[610],[576,4788],{"className":4789,"style":908},[614],[576,4791,1555],{"className":4792,"style":1578},[619,620],[576,4794,871],{"className":4795},[619,620],{"title":4797,"searchDepth":4798,"depth":4798,"links":4799},"",2,[4800,4801,4802],{"id":562,"depth":4798,"text":563},{"id":1633,"depth":4798,"text":1634},{"id":4321,"depth":4798,"text":4322},[4804,4806,4808,4810,4812,4814,4817,4819,4821,4823,4825,4827],{"lhs":4805,"rhs":1676},"sum(abs(multiply([[1, 0, 0], [2, 1, 0], [4, 3, 1]], [[2, 1, 1], [0, 1, 1], [0, 0, 2]]) - [[2, 1, 1], [4, 3, 3], [8, 7, 9]]))",{"lhs":4807,"rhs":1080},"4\u002F2",{"lhs":4809,"rhs":1701},"8\u002F2",{"lhs":4811,"rhs":1083},"(7 - 4*1)\u002F(3 - 2*1)",{"lhs":4813,"rhs":1080},"(9 - 4*1) - 3*(3 - 2*1)",{"lhs":4815,"rhs":4816},"det([[2, 1, 1], [4, 3, 3], [8, 7, 9]])","2*1*2",{"lhs":4818,"rhs":1080},"10 - 2*4",{"lhs":4820,"rhs":1080},"24 - 4*4 - 3*2",{"lhs":4822,"rhs":1686},"2\u002F2",{"lhs":4824,"rhs":1686},"2 - 1*1",{"lhs":4826,"rhs":1686},"(4 - 1*1 - 1*1)\u002F2",{"lhs":4828,"rhs":3279},"multiply([[2, 1, 1], [4, 3, 3], [8, 7, 9]], [1, 1, 1])[3]","A \\in \\mathbb{R}^{n \\times n},\\ \\text{Gauß-Elimination ohne Zeilentausch durchführbar}","\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>n\u003C\u002Fmi>\u003Cmo>×\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A \\in \\mathbb{R}^{n \\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:0.9658em;vertical-align:-0.1944em;\">\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\">n\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>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">Gauß-Elimination ohne Zeilentausch durchführbar\u003C\u002Fspan>",null,"md","lu-decomposition","A = L \\cdot U","\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>A\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>U\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A = L \\cdot U\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>\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:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">U\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fnumerics\u002Flu-zerlegung",[],[4842,4843,4844,4845],"gaussian-elimination","determinant-3x3","matrix-multiplication","gauss-seidel-method",{"title":548,"description":4797},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Flu-zerlegung","Die LU-Zerlegung schreibt eine Matrix als Produkt einer unteren und einer oberen Dreiecksmatrix, wie sie die Gauß-Elimination liefert.",[4850,4851,4852,4853],"lineares gleichungssystem","matrixzerlegung","dreiecksmatrix","gauß-verfahren",[4855,4858,4861],{"symbol":597,"meaning":4856,"symbolHtml":4857},"quadratische Koeffizientenmatrix","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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":829,"meaning":4859,"symbolHtml":4860},"untere Dreiecksmatrix mit Einsen auf der Diagonalen; enthält die Eliminationsfaktoren","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L\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\">L\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":636,"meaning":4862,"symbolHtml":4863},"obere Dreiecksmatrix, das Ergebnis der Gauß-Elimination","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>U\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">U\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\" style=\"margin-right:0.109em;\">U\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","bCs_qINtYnbaCbSukcutP0CQeH3A7Nly93DC3F0628g",[4866,4869,4872,4875],{"stem":4867,"title":4868,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fdeterminante-3x3","Determinante einer 3×3-Matrix (Regel von Sarrus)",{"stem":4870,"title":4871,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fgauss-algorithmus","Gauß-Algorithmus",{"stem":4873,"title":4874,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fmatrizenmultiplikation","Matrizenmultiplikation",{"stem":4876,"title":4877,"category":391},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Fgauss-seidel-verfahren","Gauß-Seidel-Verfahren",1793824265884]