linear-guest

Recommendations

Info

6 9.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 202 10 Linear Independence 203 10.1 Showing Linear Dependence . . . . . . . . . . . . . . . . . . . 204 10.2 Showing Linear Independence . . . . . . . . . . . . . . . . . . 207 10.3 From Dependent Independent . . . . . . . . . . . . . . . . . . 209 10.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 210 11 Basis and Dimension 213 11.1 Bases in R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 11.2 Matrix of a Linear Transformation (Redux) . . . . . . . . . . 218 11.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 221 12 Eigenvalues and Eigenvectors 225 12.1 Invariant Directions . . . . . . . . . . . . . . . . . . . . . . . . 227 12.2 The Eigenvalue–Eigenvector Equation . . . . . . . . . . . . . . 233 12.3 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 12.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 238 13 Diagonalization 241 13.1 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . 241 13.2 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 242 13.3 Changing to a Basis of Eigenvectors . . . . . . . . . . . . . . . 246 13.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 248 14 Orthonormal Bases and Complements 253 14.1 Properties of the Standard Basis . . . . . . . . . . . . . . . . . 253 14.2 Orthogonal and Orthonormal Bases . . . . . . . . . . . . . . . 255 14.2.1 Orthonormal Bases and Dot Products . . . . . . . . . . 256 14.3 Relating Orthonormal Bases . . . . . . . . . . . . . . . . . . . 258 14.4 Gram-Schmidt & Orthogonal Complements . . . . . . . . . . 261 14.4.1 The Gram-Schmidt Procedure . . . . . . . . . . . . . . 264 14.5 QR Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 265 14.6 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . 267 14.7 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 272 15 Diagonalizing Symmetric Matrices 277 15.1 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 281 6
7 16 Kernel, Range, Nullity, Rank 285 16.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 16.2 Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 16.2.1 One-to-one and Onto . . . . . . . . . . . . . . . . . . 289 16.2.2 Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 16.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 16.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 299 17 Least squares and Singular Values 303 17.1 Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . 306 17.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 308 17.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 312 A List of Symbols 315 B Fields 317 C Online Resources 319 D Sample First Midterm 321 E Sample Second Midterm 331 F Sample Final Exam 341 G Movie Scripts 367 G.1 What is Linear Algebra? . . . . . . . . . . . . . . . . . . . . . 367 G.2 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . 367 G.3 Vectors in Space n-Vectors . . . . . . . . . . . . . . . . . . . . 377 G.4 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 G.5 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 383 G.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 G.7 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 G.8 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . 403 G.9 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 404 G.10 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . 407 G.11 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 409 G.12 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 415 G.13 Orthonormal Bases and Complements . . . . . . . . . . . . . . 421 7
Page 1 and 2: Linear Algebra David Cherney, Tom D
Page 3 and 4: Contents 1 What is Linear Algebra?
Page 5: 5 7.3 Properties of Matrices . . .
Page 9 and 10: What is Linear Algebra? 1 Many diff
Page 11 and 12: 1.1 Organizing Information 11 ⎛
Page 13 and 14: 1.2 What are Vectors? 13 (C) Polyno
Page 15 and 16: 1.3 What are Linear Functions? 15 1
Page 17 and 18: 1.3 What are Linear Functions? 17 W
Page 19 and 20: 1.3 What are Linear Functions? 19 F
Page 21 and 22: 1.4 So, What is a Matrix? 21 Each b
Page 23 and 24: 1.4 So, What is a Matrix? 23 This i
Page 25 and 26: 1.4 So, What is a Matrix? 25 We hav
Page 27 and 28: 1.4 So, What is a Matrix? 27 = (2ax
Page 29 and 30: 1.4 So, What is a Matrix? 29 Exampl
Page 31 and 32: 1.5 Review Problems 31 Remember tha
Page 33 and 34: 1.5 Review Problems 33 6. Matrix Mu
Page 35 and 36: 1.5 Review Problems 35 (iv) Your an
Page 37 and 38: Systems of Linear Equations 2 2.1 G
Page 39 and 40: 2.1 Gaussian Elimination 39 Entries
Page 41 and 42: 2.1 Gaussian Elimination 41 called
Page 43 and 44: 2.1 Gaussian Elimination 43 Algorit
Page 45 and 46: 2.1 Gaussian Elimination 45 Advance
Page 47 and 48: 2.1 Gaussian Elimination 47 ⎧ ⎪
Page 49 and 50: 2.2 Review Problems 49 2. Solve the
Page 51 and 52: 2.2 Review Problems 51 8. Show that
Page 53 and 54: 2.3 Elementary Row Operations 53 Ex
Page 55 and 56: 2.3 Elementary Row Operations 55 Ex
Page 57 and 58:
2.3 Elementary Row Operations 57
Page 59 and 60:
2.3 Elementary Row Operations 59 wh
Page 61 and 62:
2.4 Review Problems 61 The LDU fact
Page 63 and 64:
2.5 Solution Sets for Systems of Li
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
2.6 Review Problems 69 so that a 2
Page 71 and 72:
The Simplex Method 3 In Chapter 2,
Page 73 and 74:
3.2 Graphical Solutions 73 3.2 Grap
Page 75 and 76:
3.3 Dantzig’s Algorithm 75 Here w
Page 77 and 78:
3.3 Dantzig’s Algorithm 77 ⎛
Page 79 and 80:
3.4 Pablo Meets Dantzig 79 These ar
Page 81 and 82:
3.5 Review Problems 81 their profit
Page 83 and 84:
Vectors in Space, n-Vectors 4 To co
Page 85 and 86:
4.2 Hyperplanes 85 4.2 Hyperplanes
Page 87 and 88:
4.2 Hyperplanes 87 unless any of th
Page 89 and 90:
4.3 Directions and Magnitudes 89 Th
Page 91 and 92:
4.3 Directions and Magnitudes 91 Ex
Page 93 and 94:
4.3 Directions and Magnitudes 93 Th
Page 95 and 96:
4.4 Vectors, Lists and Functions: R
Page 97 and 98:
4.5 Review Problems 97 4.5 Review P
Page 99 and 100:
4.5 Review Problems 99 where the ve
Page 101 and 102:
Vector Spaces 5 As suggested at the
Page 103 and 104:
5.1 Examples of Vector Spaces 103 E
Page 105 and 106:
5.1 Examples of Vector Spaces 105 E
Page 107 and 108:
5.2 Other Fields 107 {( ∣∣∣ }
Page 109 and 110:
5.3 Review Problems 109 5.3 Review
Page 111 and 112:
Linear Transformations 6 The main o
Page 113 and 114:
6.1 The Consequence of Linearity 11
Page 115 and 116:
6.3 Linear Differential Operators 1
Page 117 and 118:
6.4 Bases (Take 1) 117 sum of multi
Page 119 and 120:
6.5 Review Problems 119 2. If f is
Page 121 and 122:
Matrices 7 Matrices are a powerful
Page 123 and 124:
7.1 Linear Transformations and Matr
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
7.2 Review Problems 129 Linear oper
Page 131 and 132:
7.2 Review Problems 131 (c) Find th
Page 133 and 134:
7.3 Properties of Matrices 133 (m)
Page 135 and 136:
7.3 Properties of Matrices 135 Exam
Page 137 and 138:
7.3 Properties of Matrices 137 This
Page 139 and 140:
7.3 Properties of Matrices 139 The
Page 141 and 142:
7.3 Properties of Matrices 141 Some
Page 143 and 144:
7.3 Properties of Matrices 143 •
Page 145 and 146:
7.3 Properties of Matrices 145 7.3.
Page 147 and 148:
7.4 Review Problems 147 ⎛ ⎞ ⎛
Page 149 and 150:
7.4 Review Problems 149 Give a few
Page 151 and 152:
7.5 Inverse Matrix 151 Figure 7.1:
Page 153 and 154:
7.5 Inverse Matrix 153 At this poin
Page 155 and 156:
7.6 Review Problems 155 Notice that
Page 157 and 158:
7.6 Review Problems 157 (a) Compute
Page 159 and 160:
7.7 LU Redux 159 7.7 LU Redux Certa
Page 161 and 162:
7.7 LU Redux 161 ⎛ ⎞ u • Step
Page 163 and 164:
7.7 LU Redux 163 so we would like t
Page 165 and 166:
7.7 LU Redux 165 Reading homework:
Page 167 and 168:
7.8 Review Problems 167 (k) Try to
Page 169 and 170:
Determinants 8 Given a square matri
Page 171 and 172:
8.1 The Determinant Formula 171 We
Page 173 and 174:
8.1 The Determinant Formula 173 Thi
Page 175 and 176:
8.2 Elementary Matrices and Determi
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
8.3 Review Problems 183 Use row ope
Page 185 and 186:
8.3 Review Problems 185 express you
Page 187 and 188:
8.4 Properties of the Determinant 1
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
8.5 Review Problems 193 Figure 8.8:
Page 195 and 196:
Subspaces and Spanning Sets 9 It is
Page 197 and 198:
9.2 Building Subspaces 197 Note tha
Page 199 and 200:
9.2 Building Subspaces 199 of the f
Page 201 and 202:
9.2 Building Subspaces 201 Hence, t
Page 203 and 204:
10 Linear Independence Consider a p
Page 205 and 206:
10.1 Showing Linear Dependence 205
Page 207 and 208:
10.2 Showing Linear Independence 20
Page 209 and 210:
10.3 From Dependent Independent 209
Page 211 and 212:
10.4 Review Problems 211 (b) If pos
Page 213 and 214:
11 Basis and Dimension In chapter 1
Page 215 and 216:
215 Worked Example Next, we would l
Page 217 and 218:
11.1 Bases in R n . 217 a basis for
Page 219 and 220:
11.2 Matrix of a Linear Transformat
Page 221 and 222:
11.3 Review Problems 221 Alternativ
Page 223 and 224:
11.3 Review Problems 223 6. Let S n
Page 225 and 226:
12 Eigenvalues and Eigenvectors In
Page 227 and 228:
12.1 Invariant Directions 227 as we
Page 229 and 230:
12.1 Invariant Directions 229 Figur
Page 231 and 232:
12.1 Invariant Directions 231 Readi
Page 233 and 234:
12.2 The Eigenvalue-Eigenvector Equ
Page 235 and 236:
12.2 The Eigenvalue-Eigenvector Equ
Page 237 and 238:
12.3 Eigenspaces 237 is also an eig
Page 239 and 240:
12.4 Review Problems 239 4. Let L b
Page 241 and 242:
13 Diagonalization Given a linear t
Page 243 and 244:
13.2 Change of Basis 243 Here, the
Page 245 and 246:
13.2 Change of Basis 245 Meanwhile,
Page 247 and 248:
13.3 Changing to a Basis of Eigenve
Page 249 and 250:
13.4 Review Problems 249 (t n , . .
Page 251 and 252:
13.4 Review Problems 251 (a) Show t
Page 253 and 254:
14 Orthonormal Bases and Complement
Page 255 and 256:
14.2 Orthogonal and Orthonormal Bas
Page 257 and 258:
14.2 Orthogonal and Orthonormal Bas
Page 259 and 260:
14.3 Relating Orthonormal Bases 259
Page 261 and 262:
14.4 Gram-Schmidt & Orthogonal Comp
Page 263 and 264:
14.4 Gram-Schmidt & Orthogonal Comp
Page 265 and 266:
14.5 QR Decomposition 265 First, we
Page 267 and 268:
14.6 Orthogonal Complements 267 vec
Page 269 and 270:
14.6 Orthogonal Complements 269 The
Page 271 and 272:
14.6 Orthogonal Complements 271 Exa
Page 273 and 274:
14.7 Review Problems 273 (c) Suppos
Page 275 and 276:
14.7 Review Problems 275 7. (a) Sho
Page 277 and 278:
15 Diagonalizing Symmetric Matrices
Page 279 and 280:
279 Example 141 The matrix M = ( )
Page 281 and 282:
15.1 Review Problems 281 To diagona
Page 283 and 284:
15.1 Review Problems 283 (b) Explai
Page 285 and 286:
16 Kernel, Range, Nullity, Rank Giv
Page 287 and 288:
16.2 Image 287 It might occur to yo
Page 289 and 290:
16.2 Image 289 Figure 16.1: For the
Page 291 and 292:
16.2 Image 291 Theorem 16.2.1. A fu
Page 293 and 294:
16.2 Image 293 Example 148 of calcu
Page 295 and 296:
16.2 Image 295 Thus ⎧⎛ ⎞ ⎛
Page 297 and 298:
16.3 Summary 297 Reading homework:
Page 299 and 300:
16.4 Review Problems 299 16.4 Revie
Page 301 and 302:
16.4 Review Problems 301 Find the d
Page 303 and 304:
17 Least squares and Singular Value
Page 305 and 306:
305 However, the converse is often
Page 307 and 308:
17.1 Projection Matrices 307 MX = V
Page 309 and 310:
17.2 Singular Value Decomposition 3
Page 311 and 312:
17.2 Singular Value Decomposition 3
Page 313 and 314:
17.3 Review Problems 313 • v T M
Page 315 and 316:
A List of Symbols ∈ ∼ R I n Pn
Page 317 and 318:
B Fields Definition A field F is a
Page 319 and 320:
Online Resources C Here are some in
Page 321 and 322:
D Sample First Midterm Here are som
Page 323 and 324:
323 8. What are the four main thing
Page 325 and 326:
325 2. (a) The augmented matrix ⎛
Page 327 and 328:
327 Since M T M −1 ≠ I, it foll
Page 329 and 330:
329 (b) This is a vector space. Alt
Page 331 and 332:
E Sample Second Midterm Here are so
Page 333 and 334:
333 6. Suppose λ is an eigenvalue
Page 335 and 336:
335 Thus ⎛ ⎞ ⎛ ⎞ ⎛ 3 1
Page 337 and 338:
337 8. The associated eigenvalues s
Page 339 and 340:
339 10. (i) We say that the vectors
Page 341 and 342:
F Sample Final Exam Here are some w
Page 343 and 344:
343 (h) Is M T M diagonalizable? (i
Page 345 and 346:
345 after one orbit the satellite w
Page 347 and 348:
347 (e) Is there a direction in whi
Page 349 and 350:
349 well into the future. Then F 3
Page 351 and 352:
351 Solutions (a) Write down a line
Page 353 and 354:
353 Now solve UX = W by back substi
Page 355 and 356:
355 Now zero out the top row by sub
Page 357 and 358:
357 Finally, for λ = 3 2 ⎛ − 3
Page 359 and 360:
359 11. (a) d2 X dt 2 (b) Since thi
Page 361 and 362:
361 (b) Yes, the matrix M is symmet
Page 363 and 364:
363 and since ker L = {0 V } we hav
Page 365 and 366:
365 (b) The rows of M span (kerL)
Page 367 and 368:
G Movie Scripts G.1 What is Linear
Page 369 and 370:
G.2 Systems of Linear Equations 369
Page 371 and 372:
Page 373 and 374:
Page 375 and 376:
Page 377 and 378:
G.3 Vectors in Space n-Vectors 377
Page 379 and 380:
G.4 Vector Spaces 379 idea is to pl
Page 381 and 382:
G.4 Vector Spaces 381 We are half-w
Page 383 and 384:
G.5 Linear Transformations 383 You
Page 385 and 386:
G.6 Matrices 385 G.6 Matrices Adjac
Page 387 and 388:
G.6 Matrices 387 so this pair of ma
Page 389 and 390:
G.6 Matrices 389 Hint for Review Qu
Page 391 and 392:
G.6 Matrices 391 So this means that
Page 393 and 394:
G.6 Matrices 393 Using an LU Decomp
Page 395 and 396:
G.7 Determinants 395 we could fit t
Page 397 and 398:
G.7 Determinants 397 2. As a matrix
Page 399 and 400:
G.7 Determinants 399 ⎛ 1 Sj(λ) i
Page 401 and 402:
G.7 Determinants 401 Now note that
Page 403 and 404:
G.8 Subspaces and Spanning Sets 403
Page 405 and 406:
G.9 Linear Independence 405 notion
Page 407 and 408:
G.10 Basis and Dimension 407 G.10 B
Page 409 and 410:
G.11 Eigenvalues and Eigenvectors 4
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
G.12 Diagonalization 415 G.12 Diago
Page 417 and 418:
G.12 Diagonalization 417 fruit (s,
Page 419 and 420:
G.12 Diagonalization 419 Note that
Page 421 and 422:
G.13 Orthonormal Bases and Compleme
Page 423 and 424:
Page 425 and 426:
Page 427 and 428:
Page 429 and 430:
G.14 Diagonalizing Symmetric Matric
Page 431 and 432:
G.15 Kernel, Range, Nullity, Rank 4
Page 433 and 434:
Index Action, 389 Angle between vec
Page 435 and 436:
INDEX 435 Linearly independent, 204
show all

linear-guest

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

Delete template?

Save as template?