Linear Algebra

More documents

Recommendations

Info

$MATLAB C++ Math Library Reference$

Index accuracy of Gauss’ method, 67–71 rounding error, 68 addition vector, 80 additive inverse, 80 adjoint matrix, 328 angle, 42 antipodal, 340 antisymmetric matrix, 139 arrow diagram, 216, 232, 238, 242, 351 augmented matrix, 14 automorphism, 163 dilation, 164 reflection, 164 rotation, 164 back-substitution, 5 basis, 113–124 change of, 238 definition, 113 orthogonal, 256 orthogonalization, 257 orthonormal, 258 standard, 114, 350 standard over the complex numbers, 350 string, 371 best fit line, 269 block matrix, 311 box, 321 orientation, 321 sense, 321 volume, 321 C language, 67 canonical form for matrix equivalence, 245 for nilpotent matrices, 374 for row equivalence, 57 for similarity, 392 canonical representative, A-12 Cauchy-Schwartz Inequality, 41 Cayley-Hamilton theorem, 382 central projection, 337 change of basis, 238–249 characteristic vectors, values, 357 characteristic equation, 360 characteristic polynomial, 360 characterized, 172 characterizes, 246 Chemistry problem, 1, 9 chemistry problem, 22 circuits parallel, 73 series, 73 series-parallel, 74 closure, 95 of nullspace, 367 of rangespace, 367 codomain, A-8 cofactor, 327 column, 13 rank, 126 vector, 15 column rank full, 131 column space, 126 complementary subspaces, 136 orthogonal, 263 complex numbers vector space over, 91 component, 15 composition self, 365 computer algebra systems, 61–62 concatenation, 134 conditioning number, 70 congruent figures, 286
congruent plane figures, 286 contradiction, A-6 convex set, 183 coordinates homogeneous, 340 with respect to a basis, 116 correspondence, 161, A-9 coset, 193 Cramer’s rule, 331–333 cross product, 298 crystals, 143–146 diamond, 144 graphite, 144 salt, 143 unit cell, 144 da Vinci, Leonardo, 337 determinant, 294, 299–318 cofactor, 327 Cramer’s rule, 332 definition, 299 exists, 309, 315 Laplace expansion, 327 minor, 327 permutation expansion, 308, 312, 334 diagonal matrix, 209, 225 diagonalizable, 354–357 difference equation, 406 homogeneous, 406 dilation, 164, 275 representing, 203 dimension, 121 direct sum, 131 ), 140 definition, 135 external, 168 of two subspaces, 136 direction vector, 35 distance-preserving, 286 division theorem, 348 dot product, 39 dual space, 193 echelon form, 5 free variable, 12 leading variable, 5 reduced, 46 eigenspace, 361 eigenvalue, eigenvector of a matrix, 358 of a transformation, 357 elementary matrix, 226, 275 elementary reduction operations, 4 pivoting, 4 rescaling, 4 swapping, 4 elementary row operations, 4 entry, 13 equivalence class, A-11 canonical representative, A-12 representative, A-11 equivalence relation, A-10, A-11 isomorphism, 169 matrix equivalence, 244 matrix similarity, 351 row equivalence, 50 equivalent statements, A-3 Erlanger Program, 286 Euclid, 286 even functions, 99, 138 even polynomials, 398 external direct sum, 168 Fibonacci sequence, 405 field, 141–142 definition, 141 finite-dimensional vector space, 119 flat, 36 form, 55 free variable, 12 full column rank, 131 full row rank, 131 function, A-8 inverse image, 185 codomain, A-8 composition, 215, A-8 correspondence, A-9 domain, A-8 even, 99 identity, A-8 inverse, 231, A-9 left inverse, 231 multilinear, 305 odd, 99 one-to-one, A-9 onto, A-9 range, A-8
Page 1 and 2:
� � � �1 2� � �3 1�
Page 3 and 4:
Preface In most mathematics program
Page 5 and 6:
dent projects for individuals or sm
Page 7 and 8:
Contents 1 Linear Systems 1 1.I Sol
Page 9:
5.II.1 Definition and Examples ....
Page 12 and 13:
2 Chapter 1. Linear Systems To fini
Page 14 and 15:
4 Chapter 1. Linear Systems Conside
Page 16 and 17:
6 Chapter 1. Linear Systems the sec
Page 18 and 19:
8 Chapter 1. Linear Systems Don’t
Page 20 and 21:
10 Chapter 1. Linear Systems 1.25 G
Page 22 and 23:
12 Chapter 1. Linear Systems can al
Page 24 and 25:
14 Chapter 1. Linear Systems Matric
Page 26 and 27:
16 Chapter 1. Linear Systems Scalar
Page 28 and 29:
18 Chapter 1. Linear Systems Must t
Page 30 and 31:
20 Chapter 1. Linear Systems 2.30 [
Page 32 and 33:
22 Chapter 1. Linear Systems Studyi
Page 34 and 35:
24 Chapter 1. Linear Systems the bo
Page 36 and 37:
26 Chapter 1. Linear Systems shows
Page 38 and 39:
28 Chapter 1. Linear Systems 3.13 E
Page 40 and 41:
30 Chapter 1. Linear Systems (a) 2x
Page 42 and 43:
32 Chapter 1. Linear Systems 1.II L
Page 44 and 45:
34 Chapter 1. Linear Systems positi
Page 46 and 47:
36 Chapter 1. Linear Systems In R3
Page 48 and 49:
38 Chapter 1. Linear Systems � 1.
Page 50 and 51:
40 Chapter 1. Linear Systems Notice
Page 52 and 53:
42 Chapter 1. Linear Systems 2.7 De
Page 54 and 55:
44 Chapter 1. Linear Systems 2.25 P
Page 56 and 57:
46 Chapter 1. Linear Systems We can
Page 58 and 59:
48 Chapter 1. Linear Systems each e
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
54 Chapter 1. Linear Systems x1 has
Page 66 and 67:
56 Chapter 1. Linear Systems First,
Page 68 and 69:
58 Chapter 1. Linear Systems 2.8 Ex
Page 70 and 71:
60 Chapter 1. Linear Systems (2) Ca
Page 72 and 73:
62 Chapter 1. Linear Systems (b) Th
Page 74 and 75:
64 Chapter 1. Linear Systems For th
Page 76 and 77:
66 Chapter 1. Linear Systems (a) Fi
Page 78 and 79:
68 Chapter 1. Linear Systems 4 3 2
Page 80 and 81:
70 Chapter 1. Linear Systems expert
Page 82 and 83:
72 Chapter 1. Linear Systems Topic:
Page 84 and 85:
74 Chapter 1. Linear Systems We beg
Page 86 and 87:
76 Chapter 1. Linear Systems 1 Calc
Page 88 and 89:
78 Chapter 1. Linear Systems parall
Page 90 and 91:
80 Chapter 2. Vector Spaces 2.I Def
Page 92 and 93:
82 Chapter 2. Vector Spaces For the
Page 94 and 95:
84 Chapter 2. Vector Spaces A vecto
Page 96 and 97:
86 Chapter 2. Vector Spaces 1.12 Ex
Page 98 and 99:
88 Chapter 2. Vector Spaces Chapter
Page 100 and 101:
90 Chapter 2. Vector Spaces (d) The
Page 102 and 103:
92 Chapter 2. Vector Spaces 2.2 Exa
Page 104 and 105:
94 Chapter 2. Vector Spaces second.
Page 106 and 107:
96 Chapter 2. Vector Spaces Proof.
Page 108 and 109:
98 Chapter 2. Vector Spaces The sub
Page 110 and 111:
100 Chapter 2. Vector Spaces S. Mem
Page 112 and 113:
102 Chapter 2. Vector Spaces 2.II L
Page 114 and 115:
104 Chapter 2. Vector Spaces 1.5 Ex
Page 116 and 117:
106 Chapter 2. Vector Spaces Lookin
Page 118 and 119:
108 Chapter 2. Vector Spaces Proof.
Page 120 and 121:
110 Chapter 2. Vector Spaces (a) {2
Page 122 and 123:
112 Chapter 2. Vector Spaces � 1.
Page 124 and 125:
114 Chapter 2. Vector Spaces 1.5 De
Page 126 and 127:
116 Chapter 2. Vector Spaces 1.13 D
Page 128 and 129:
118 Chapter 2. Vector Spaces � 1.
Page 130 and 131:
120 Chapter 2. Vector Spaces we stu
Page 132 and 133:
122 Chapter 2. Vector Spaces 2.11 C
Page 134 and 135:
124 Chapter 2. Vector Spaces subspa
Page 136 and 137:
126 Chapter 2. Vector Spaces 3.6 De
Page 138 and 139:
128 Chapter 2. Vector Spaces Anothe
Page 140 and 141:
130 Chapter 2. Vector Spaces (a)
Page 142 and 143:
132 Chapter 2. Vector Spaces by fin
Page 144 and 145:
134 Chapter 2. Vector Spaces has a
Page 146 and 147:
136 Chapter 2. Vector Spaces 4.12 E
Page 148 and 149:
138 Chapter 2. Vector Spaces 4.22 I
Page 150 and 151:
140 Chapter 2. Vector Spaces (c) Le
Page 152 and 153:
142 Chapter 2. Vector Spaces We cou
Page 154 and 155:
144 Chapter 2. Vector Spaces Then w
Page 156 and 157:
146 Chapter 2. Vector Spaces (d) Pl
Page 158 and 159:
148 Chapter 2. Vector Spaces howeve
Page 160 and 161:
150 Chapter 2. Vector Spaces positi
Page 162 and 163:
152 Chapter 2. Vector Spaces Topic:
Page 164 and 165:
154 Chapter 2. Vector Spaces The cl
Page 166 and 167:
156 Chapter 2. Vector Spaces Remark
Page 168 and 169:
158 Chapter 2. Vector Spaces (b) Sh
Page 170 and 171:
160 Chapter 3. Maps Between Spaces
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
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:
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:
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:
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:
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:
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:
Page 300 and 301:
Page 303 and 304:
Chapter 4 Determinants In the first
Page 305 and 306:
Section I. Definition 295 determine
Page 307 and 308:
Section I. Definition 297 Exercises
Page 309 and 310:
Section I. Definition 299 1.15 Prov
Page 311 and 312:
Section I. Definition 301 That resu
Page 313 and 314:
Section I. Definition 303 (b) Prove
Page 315 and 316:
Section I. Definition 305 3.2 Defin
Page 317 and 318:
Section I. Definition 307 Therefore
Page 319 and 320:
Section I. Definition 309 read alou
Page 321 and 322:
Section I. Definition 311 � 3.23
Page 323 and 324:
Section I. Definition 313 could be
Page 325 and 326:
Section I. Definition 315 We still
Page 327 and 328:
Section I. Definition 317 (terms wi
Page 329 and 330:
Section II. Geometry of Determinant
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Section III. Other Formulas 327 The
Page 339 and 340:
Section III. Other Formulas 329 1.9
Page 341 and 342:
Topic: Cramer’s Rule 331 Topic: C
Page 343 and 344:
Topic: Cramer’s Rule 333 4 Suppos
Page 345 and 346:
Topic: Speed of Calculating Determi
Page 347 and 348:
Topic: Projective Geometry 337 Topi
Page 349 and 350:
Topic: Projective Geometry 339 P P
Page 351 and 352:
Topic: Projective Geometry 341 of t
Page 353 and 354:
Topic: Projective Geometry 343 The
Page 355 and 356:
Topic: Projective Geometry 345 the
Page 357 and 358:
Chapter 5 Similarity While studying
Page 359 and 360:
Section I. Complex Vector Spaces 34
Page 361 and 362:
Section II. Similarity 351 5.II Sim
Page 363 and 364:
Section II. Similarity 353 1.7 Cons
Page 365 and 366:
Section II. Similarity 355 2.4 Coro
Page 367 and 368:
Section II. Similarity 357 2.12 The
Page 369 and 370:
Section II. Similarity 359 and the
Page 371 and 372:
Section II. Similarity 361 Proof. A
Page 373 and 374:
Section II. Similarity 363 3.19 Cor
Page 375 and 376:
Section III. Nilpotence 365 5.III N
Page 377 and 378:
Section III. Nilpotence 367 and thi
Page 379 and 380:
Section III. Nilpotence 369 Proof.
Page 381 and 382:
Section III. Nilpotence 371 The new
Page 383 and 384:
Section III. Nilpotence 373 Now, wi
Page 385 and 386:
Section III. Nilpotence 375 We can
Page 387 and 388:
Section III. Nilpotence 377 � −
Page 389 and 390: Section IV. Jordan Form 379 5.IV Jo
Page 391 and 392: Section IV. Jordan Form 381 Using t
Page 393 and 394: Section IV. Jordan Form 383 Equate
Page 395 and 396: Section IV. Jordan Form 385 � 1.1
Page 397 and 398: Section IV. Jordan Form 387 is (x
Page 399 and 400: Section IV. Jordan Form 389 steps t
Page 401 and 402: Section IV. Jordan Form 391 each te
Page 403 and 404: Section IV. Jordan Form 393 2.13 Ex
Page 405 and 406: Section IV. Jordan Form 395 2.15 Ex
Page 407 and 408: Section IV. Jordan Form 397 � 5 4
Page 409 and 410: Topic: Computing Eigenvalues—the
Page 411 and 412: Topic: Computing Eigenvalues—the
Page 413 and 414: Topic: Stable Populations 403 Topic
Page 415 and 416: Topic: Linear Recurrences 405 Topic
Page 417 and 418: Topic: Linear Recurrences 407 And,
Page 419 and 420: Topic: Linear Recurrences 409 diamo
Page 421 and 422: Topic: Linear Recurrences 411 Compu
Page 423 and 424: Appendix Introduction Mathematics i
Page 425 and 426: A-3 ‘if a number is divisible by
Page 427 and 428: Techniques of Proof A-5 Induction.
Page 429 and 430: A-7 The Principle of Extensionality
Page 431 and 432: A-9 So an identity map plays the sa
Page 433 and 434: A-11 Similarly, the equivalence rel
Page 435 and 436: Bibliography [Abbot] Edwin Abbott,
Page 437 and 438: [Feller] William Feller, An Introdu
Page 439: [Programmer’s Ref.] Microsoft Pro
Page 443 and 444: Gauss’ method, 3 Gauss-Jordan, 46
Page 445 and 446: educed echelon form, 46 reflection,
show all

Linear Algebra

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

Delete template?

Save as template?