Web skærmformat. - Aarhus Universitet

More documents

Recommendations

Info

V. Matricer – 3. Lineære ligninger 416 • Multiplikation af række med tal r = 0 1 0 a11 a12 = 0 r a21 a22 a11 a12 ra21 ra22 • Addition af et multiplum af en række til en anden 1 r a11 a12 a11 + ra21 a12 + ra22 = 0 1 a21 a22 3.28. Ensidig invers er tosidig ☞ [LA] 7 Rækkeoperations-matricer Sætning 10 Lad A,B vœre kvadratiske matricer af samme størelse. Så gœlder ”En højre invers er også en venstre invers.” a21 AB = I ⇔ BA = I Bevis Hvis AB = I har alle ligningssystemer Ax = b en løsning x = Bb. Den reducerede form af A kan da ikke have en 0-række og er derfor enhedsmatricen I. Altså findes en matrix C så CA = I. Til slut er C = C(AB) = (CA)B = B. 3.29. Invers ved operationer ☞ [LA] 7 Rækkeoperations-matricer Sætning a22
V. Matricer – 3. Lineære ligninger 417 En kvadratisk matrix A er invertibel, hvis og kun hvis dens reducerede form er enhedsmatricen I. I så fald er den augmenterede matrix (A|I) ∼ (I|A −1 ) ”Den inverse matrix beregnes ved rækkeoperationer på den augmenterede matrix.” 3.30. Invers 2x2-matrix Eksempel 4 ☞ [LA] 6 Løsningsteknik Løs matrixligningen, i.e. beregn invers, 2 1 x11 5 3 x21 x12 1 = x22 0 0 1 2 5 1 1 0 2 1 1 0 ∼ 1 3 0 1 0 2 −5 2 ∼ 2 1 0 2 0 6 −2 1 0 3 ∼ ∼ 0 1 −5 2 0 1 −5 1 1 1 −5 −1 2 0 2 3.31. Invers 2x2-matrix ☞ [LA] 6 Løsningsteknik Eksempel 4 - fortsat
Page 1:
CALCULUS "SLIDES" TIL CALCULUS 1 +
Page 4 and 5:
3. Generelle områder 201 4. Koordi
Page 7:
Forord "Slides" til forelæsningern
Page 10 and 11:
I. Differentiation - 1. Kontinuitet
Page 12 and 13:
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
I. Differentiation - 2. Partielle a
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
I. Differentiation - 3. Tangentplan
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
I. Differentiation - 4. Kædereglen
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
I. Differentiation - 5. Gradient 88
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
I. Differentiation - 6. Maksimum/mi
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
I. Differentiation - 7. Lagrangemet
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
II. Integration - 1. Dobbelt integr
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
II. Integration - 2. Itereret integ
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
II. Integration - 3. Generelle omr
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
II. Integration - 4. Koordinatskift
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
III. Potensrækker - 1. l’Hospita
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
III. Potensrækker - 2. Talfølger
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
III. Potensrækker - 3. Potensrækk
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
III. Potensrækker - 4. Taylorpolyn
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 311 and 312:
1. Grafiske/numeriske metoder IV Di
Page 313 and 314:
IV. Differentialligninger - 1. Graf
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
IV. Differentialligninger - 2. 1. o
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
IV. Differentialligninger - 3. Gene
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366: 1. Vektorer og matricer V Matricer
Page 367 and 368: V. Matricer - 1. Vektorer og matric
Page 385 and 386: V. Matricer - 2. Lineære afbildnin
Page 403 and 404: V. Matricer - 3. Lineære ligninger
Page 415: V. Matricer - 3. Lineære ligninger
Page 419 and 420: V. Matricer - 4. Determinanter 419
Page 423 and 424: V. Matricer - 4. Determinanter = (
Page 435 and 436: 1. Egenvektorer VI Egenvektorer og
Page 437 and 438: VI. Egenvektorer og diagonalisering
Page 467 and 468:
VI. Egenvektorer og diagonalisering
Page 469 and 470:
Page 471 and 472:
Page 473 and 474:
Page 475:
Page 478 and 479:
VII. Skalarprodukt og projektion -
Page 480 and 481:
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
VIII. Appendiks - 1. Polære koordi
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
Page 524 and 525:
IX. Opgaver - 1. August 2002 524 1.
Page 526 and 527:
IX. Opgaver - 1. August 2002 526 Op
Page 528 and 529:
IX. Opgaver - 1. August 2002 528 In
Page 530 and 531:
Page 532 and 533:
IX. Opgaver - 1. August 2002 ⎡
Page 534 and 535:
IX. Opgaver - 1. August 2002 534 Do
Page 536 and 537:
IX. Opgaver - 1. August 2002 536 An
Page 538 and 539:
Page 540 and 541:
IX. Opgaver - 1. August 2002 540 y
Page 542 and 543:
Page 544 and 545:
IX. Opgaver - 1. August 2002 544 2
Page 546 and 547:
IX. Opgaver - 1. August 2002 546 fu
Page 548 and 549:
IX. Opgaver - 2. Januar 2003 548 2.
Page 550 and 551:
IX. Opgaver - 2. Januar 2003 550 3)
Page 552 and 553:
IX. Opgaver - 2. Januar 2003 552 z
Page 554 and 555:
IX. Opgaver - 2. Januar 2003 554 y
Page 556 and 557:
IX. Opgaver - 2. Januar 2003 556 hv
Page 558 and 559:
IX. Opgaver - 2. Januar 2003 558 y
Page 560 and 561:
Page 562 and 563:
IX. Opgaver - 2. Januar 2003 562 Re
Page 564 and 565:
IX. Opgaver - 2. Januar 2003 564
Page 566 and 567:
Page 568 and 569:
IX. Opgaver - 2. Januar 2003 568 De
Page 570 and 571:
Page 572 and 573:
IX. Opgaver - 3. Januar 2004 572 2)
Page 574 and 575:
IX. Opgaver - 3. Januar 2004 574 L
Page 576 and 577:
Page 578 and 579:
Page 580 and 581:
IX. Opgaver - 3. Januar 2004 580 Eg
Page 582 and 583:
IX. Opgaver - 4. August 2004 582 =
Page 584 and 585:
IX. Opgaver - 4. August 2004 584 gi
Page 586 and 587:
IX. Opgaver - 4. August 2004 586 vi
Page 588 and 589:
Page 590 and 591:
IX. Opgaver - 4. August 2004 590 L
Page 592 and 593:
Page 594 and 595:
IX. Opgaver - 5. Januar 2005 594 (t
Page 596 and 597:
Page 598 and 599:
Page 600 and 601:
IX. Opgaver - 5. Januar 2005 600 Gr
Page 603 and 604:
Litteratur • Arnold, Ordinary dif
Page 605 and 606:
0-række/søjle, 423 n-te afsnitssu
Page 607 and 608:
Identitetsmatricen, 383 Identitetsm
Page 609:
etningsafledede, 89, 102 retningsfe
show all

Web skærmformat. - Aarhus Universitet

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?