A First Course in Linear Algebra, 2017a

More documents

Recommendations

Info

516 Vector Spaces dimension of im(T ) is m. If rank(T )=m, then by Theorem 9.83, since im(T ) is a subspace of W, it follows that im(T )=W. ♠ Example 9.85: One to One Transformation Let S : P 2 → M 22 be a linear transformation defined by [ ] S(ax 2 a + b a+ c + bx + c)= for all ax 2 + bx + c ∈ P b − c b+ c 2 . Prove that S is one to one but not onto. Solution. You may recall this example from earlier in Example 9.65. Here we will determine that S is one to one, but not onto, using the method provided in Corollary 9.84. By definition, ker(S)={ax 2 + bx + c ∈ P 2 | a + b = 0,a + c = 0,b − c = 0,b + c = 0}. Suppose p(x)=ax 2 + bx + c ∈ ker(S). This leads to a homogeneous system of four equations in three variables. Putting the augmented matrix in reduced row-echelon form: ⎡ ⎢ ⎣ 1 1 0 0 1 0 1 0 0 1 −1 0 0 1 1 0 ⎤ ⎡ ⎥ ⎦ →···→ ⎢ ⎣ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 Since the unique solution is a = b = c = 0, ker(S)={⃗0}, and thus S is one-to-one by Corollary 9.84. Similarly, by Corollary 9.84,ifS is onto it will have rank(S)=dim(M 22 )=4. The image of S is given by {[ ]} {[ ] [ ] [ ]} a + b a+ c 1 1 1 0 0 1 im(S)= = span , , b − c b+ c 0 0 1 1 −1 1 These matrices are linearly independent whichmeansthissetformsabasisforim(S). Therefore the dimension of im(S), also called rank(S), is equal to 3. It follows that S is not onto. ♠ Exercises ⎤ ⎥ ⎦ . Exercise 9.8.1 Let V = R 3 and let ⎧⎡ ⎨ W = span(S), where S = ⎣ ⎩ Find a basis of W consisting of vectors in S. 1 −1 1 ⎤ ⎡ ⎦, ⎣ −2 2 −2 ⎤ ⎡ ⎦, ⎣ −1 1 1 ⎤ ⎡ ⎦, ⎣ 1 −1 3 ⎤⎫ ⎬ ⎦ ⎭ Exercise 9.8.2 Let T be a linear transformation given by [ ] [ ][ ] x 1 1 x T = y 1 1 y
9.9. The Matrix of a Linear Transformation 517 Find a basis for ker(T ) and im(T ). Exercise 9.8.3 Let T be a linear transformation given by [ ] [ ][ x 1 0 x T = y 1 1 y ] Find a basis for ker(T ) and im(T ). Exercise 9.8.4 Let V = R 3 and let ⎧⎡ ⎨ W = span ⎣ ⎩ 1 1 1 ⎤ ⎡ ⎦, ⎣ −1 2 −1 ⎤⎫ ⎬ ⎦ ⎭ Extend this basis of W to a basis of V . Exercise 9.8.5 Let T be a linear transformation given by ⎡ ⎤ x [ T ⎣ y ⎦ 1 1 1 = 1 1 1 z What is dim(ker(T ))? ] ⎡ ⎣ x y z ⎤ ⎦ 9.9 The Matrix of a Linear Transformation Outcomes A. Find the matrix of a linear transformation with respect to general bases in vector spaces. You may recall from R n that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another. Let T : V ↦→ W be an isomorphism where V and W are vector spaces. Recall from Lemma 9.74 that T maps a basis in V to a basis in W. When discussing this Lemma, we were not specific on what this basis looked like. In this section we will make such a distinction. Consider now an important definition.
Page 1:
with Open Texts Linear A Algebra wi
Page 5:
A First Course in Linear Algebra an
Page 9:
A First Course in Linear Algebra an
Page 12 and 13:
iv Table of Contents 3 Determinants
Page 14 and 15:
vi Table of Contents 7.3.5 The Matr
Page 17 and 18:
Chapter 1 Systems of Equations 1.1
Page 19 and 20:
1.1. Systems of Equations, Geometry
Page 21 and 22:
1.2. Systems of Equations, Algebrai
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65:
Page 68 and 69:
54 Matrices Consider the following
Page 70 and 71:
56 Matrices Solution. Notice that b
Page 72 and 73:
58 Matrices Proposition 2.10: Prope
Page 74 and 75:
60 Matrices on the left by a vector
Page 76 and 77:
62 Matrices will satisfy the equati
Page 78 and 79:
64 Matrices Solution. First check i
Page 80 and 81:
66 Matrices Solution. First check i
Page 82 and 83:
68 Matrices Proposition 2.25: Prope
Page 84 and 85:
70 Matrices Definition 2.29: Symmet
Page 86 and 87:
72 Matrices Theorem 2.34: Uniquenes
Page 88 and 89:
74 Matrices Let’s solve the first
Page 90 and 91:
76 Matrices Notice that the left ha
Page 92 and 93:
78 Matrices What if the right side,
Page 94 and 95:
80 Matrices First, consider the fol
Page 96 and 97:
82 Matrices You can see that B is t
Page 98 and 99:
84 Matrices = = ⎡ ⎣ ⎡ ⎣ 1 0
Page 100 and 101:
86 Matrices First: ⎡ ⎣ 0 1 0 1
Page 102 and 103:
88 Matrices Proof. Suppose that A a
Page 104 and 105:
90 Matrices It follows that the red
Page 106 and 107:
92 Matrices (h) DE ⎡ Exercise 2.1
Page 108 and 109:
94 Matrices (a) (A − B) 2 = A 2
Page 110 and 111:
96 Matrices Exercise 2.1.42 Let ⎡
Page 112 and 113:
98 Matrices (a) Find the elementary
Page 114 and 115:
100 Matrices One way to find the LU
Page 116 and 117:
102 Matrices which yields very quic
Page 118 and 119:
104 Matrices By Lemma 2.70, this im
Page 120 and 121:
106 Matrices Exercise 2.2.12 Find t
Page 122 and 123:
108 Determinants The 2×2 determina
Page 124 and 125:
110 Determinants Definition 3.7: Th
Page 126 and 127:
112 Determinants The following prov
Page 128 and 129:
114 Determinants 3.1.3 Properties o
Page 130 and 131:
116 Determinants Example 3.23: Mult
Page 132 and 133:
118 Determinants Solution. Consider
Page 134 and 135:
120 Determinants = ∑a 1,l (cof(B)
Page 136 and 137:
122 Determinants Theorem 3.35: Let
Page 138 and 139:
124 Determinants Here, det(C)=−3d
Page 140 and 141:
126 Determinants (a) minor(A) 11 (b
Page 142 and 143:
128 Determinants Exercise 3.1.13 An
Page 144 and 145:
130 Determinants 3.2.1 A Formula fo
Page 146 and 147:
132 Determinants You can verify tha
Page 148 and 149:
134 Determinants The cofactor matri
Page 150 and 151:
136 Determinants Therefore, ∣ ∣
Page 152 and 153:
138 Determinants After row operatio
Page 154 and 155:
140 Determinants Using the given po
Page 156 and 157:
142 Determinants Does there exist a
Page 158 and 159:
144 R n Hence, every element in R 2
Page 160 and 161:
146 R n components are equal. Preci
Page 162 and 163:
148 R n Theorem 4.6: Properties of
Page 164 and 165:
150 R n 4.3 Geometric Meaning of Ve
Page 166 and 167:
152 R n 4.4 Length of a Vector Outc
Page 168 and 169:
154 R n Solution. We will use the f
Page 170 and 171:
156 R n = √ 1 + 9 + 16 = √ 26 I
Page 172 and 173:
158 R n ⃗u 2⃗v ⃗u + 2⃗v ⃗
Page 174 and 175:
160 R n ⎡ Let ⃗q = ⎣ Then, x
Page 176 and 177:
162 R n Let t = x−2 y−1 3 ,t =
Page 178 and 179:
164 R n Example 4.26: Compute a Dot
Page 180 and 181:
166 R n Notice that this proof was
Page 182 and 183:
168 R n Therefore, the cosine of th
Page 184 and 185:
170 R n 4.7.3 Projections In some a
Page 186 and 187:
172 R n We will conclude this secti
Page 188 and 189:
174 R n Exercise 4.7.11 Find proj
Page 190 and 191:
176 R n Solution. By Definition 4.4
Page 192 and 193:
178 R n From the above scalar equat
Page 194 and 195:
180 R n Definition 4.48: Geometric
Page 196 and 197:
182 R n We will now look at an exam
Page 198 and 199:
184 R n 4.9.1 The Box Product Recal
Page 200 and 201:
186 R n = a ∣ e h ⎡ = det⎣ f
Page 202 and 203:
188 R n 4.10 Spanning, Linear Indep
Page 204 and 205:
190 R n 4.10.2 Linearly Independent
Page 206 and 207:
192 R n Theorem 4.66: Linear Indepe
Page 208 and 209:
194 R n Therefore we can write ⎡
Page 210 and 211:
196 R n Now suppose that⃗u ∉ sp
Page 212 and 213:
198 R n A subspace is simply a set
Page 214 and 215:
200 R n Definition 4.80: Basis of a
Page 216 and 217:
202 R n ⃗u 1 = Furthermore, ⎡
Page 218 and 219:
204 R n ⎡ ⎢ ⎣ 1 2 1 1 3 0 1 3
Page 220 and 221:
206 R n Solution. An easy way to do
Page 222 and 223:
208 R n Since {⃗r 1 ,..., p⃗r j
Page 224 and 225:
210 R n Example 4.101: Rank, Column
Page 226 and 227:
212 R n Definition 4.106: Image of
Page 228 and 229:
214 R n Example 4.110: Null Space o
Page 230 and 231:
216 R n Theorem 4.114: Let A be an
Page 232 and 233:
218 R n Exercise 4.10.7 Here are so
Page 234 and 235:
220 R n Exercise 4.10.17 Are the fo
Page 236 and 237:
222 R n Thse vectors can’t possib
Page 238 and 239:
224 R n ⎧⎡ ⎪⎨ Exercise 4.10
Page 240 and 241:
226 R n Exercise 4.10.52 Consider t
Page 242 and 243:
228 R n Exercise 4.10.68 Find the r
Page 244 and 245:
230 R n Solution. We already verifi
Page 246 and 247:
232 R n Thus to find a correspondin
Page 248 and 249:
234 R n That is: We readily compute
Page 250 and 251:
236 R n We will say that the column
Page 252 and 253:
238 R n To show that {⃗v 1 ,··
Page 254 and 255:
240 R n is an orthogonal basis of U
Page 256 and 257:
242 R n = = = ⎡ ⎣ ⎡ ⎣ ⎡
Page 258 and 259:
244 R n In order to find W ⊥ , we
Page 260 and 261:
246 R n Now, we need to write ⃗y
Page 262 and 263:
248 R n Theorem 4.150: Existence of
Page 264 and 265:
250 R n Solution. First, consider w
Page 266 and 267:
252 R n −1 1 2 3 4 5 6 y 4 2 x Re
Page 268 and 269:
254 R n Exercise 4.11.7 For U an or
Page 270 and 271:
256 R n and here is a point: (1,1,2
Page 272 and 273:
258 R n What does addition of vecto
Page 274 and 275:
260 R n 4 3 First we want to know t
Page 276 and 277:
262 R n = −58 Newton meters Note
Page 278 and 279:
264 R n Exercise 4.12.19 A girl dra
Page 280 and 281:
266 Linear Transformations numerica
Page 282 and 283:
268 Linear Transformations Exercise
Page 284 and 285:
270 Linear Transformations In this
Page 286 and 287:
272 Linear Transformations ⎡ ⎣
Page 288 and 289:
274 Linear Transformations ⎡ T
Page 290 and 291:
276 Linear Transformations ⎡ (b)
Page 292 and 293:
278 Linear Transformations The nece
Page 294 and 295:
280 Linear Transformations Then, co
Page 296 and 297:
282 Linear Transformations More gen
Page 298 and 299:
284 Linear Transformations Solution
Page 300 and 301:
Page 302 and 303:
288 Linear Transformations Definiti
Page 304 and 305:
290 Linear Transformations This tel
Page 306 and 307:
292 Linear Transformations Example
Page 308 and 309:
294 Linear Transformations 2. T is
Page 310 and 311:
296 Linear Transformations Proposit
Page 312 and 313:
298 Linear Transformations Proof. S
Page 314 and 315:
300 Linear Transformations You can
Page 316 and 317:
Page 318 and 319:
304 Linear Transformations for exam
Page 320 and 321:
306 Linear Transformations Example
Page 322 and 323:
308 Linear Transformations Since {T
Page 324 and 325:
310 Linear Transformations Find a b
Page 326 and 327:
312 Linear Transformations Theorem
Page 328 and 329:
314 Linear Transformations and one
Page 330 and 331:
316 Linear Transformations Hence
Page 332 and 333:
318 Linear Transformations system.
Page 334 and 335:
320 Linear Transformations Solution
Page 336 and 337:
Page 338 and 339:
324 Complex Numbers Theorem 6.1: Pr
Page 340 and 341:
326 Complex Numbers Example 6.5: Co
Page 342 and 343:
328 Complex Numbers Definition 6.10
Page 344 and 345:
330 Complex Numbers Exercise 6.1.5
Page 346 and 347:
332 Complex Numbers Example 6.14: P
Page 348 and 349:
334 Complex Numbers This requires r
Page 350 and 351:
336 Complex Numbers Therefore, x 3
Page 352 and 353:
338 Complex Numbers Consider the fo
Page 355 and 356:
Chapter 7 Spectral Theory 7.1 Eigen
Page 357 and 358:
7.1. Eigenvalues and Eigenvectors o
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
7.2. Diagonalization 355 One eigenv
Page 371 and 372:
7.2. Diagonalization 357 In words,
Page 373 and 374:
7.2. Diagonalization 359 Conversely
Page 375 and 376:
7.2. Diagonalization 361 Theorem 7.
Page 377 and 378:
7.2. Diagonalization 363 7.2.3 Comp
Page 379 and 380:
7.2. Diagonalization 365 Exercise 7
Page 381 and 382:
7.3. Applications of Spectral Theor
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
Page 391 and 392:
Page 393 and 394:
Page 395 and 396:
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Page 405 and 406:
Page 407 and 408:
Page 409 and 410:
7.4. Orthogonality 395 [ x(0) y(0)
Page 411 and 412:
7.4. Orthogonality 397 Solution. Fi
Page 413 and 414:
7.4. Orthogonality 399 Therefore an
Page 415 and 416:
7.4. Orthogonality 401 In the above
Page 417 and 418:
7.4. Orthogonality 403 Theorem 7.62
Page 419 and 420:
7.4. Orthogonality 405 Theorem 7.66
Page 421 and 422:
7.4. Orthogonality 407 λ 1 = 16: s
Page 423 and 424:
7.4. Orthogonality 409 [ ] [ ] AV1
Page 425 and 426:
7.4. Orthogonality 411 = [ 4 0 0 0
Page 427 and 428:
7.4. Orthogonality 413 by the assum
Page 429 and 430:
7.4. Orthogonality 415 operations)
Page 431 and 432:
7.4. Orthogonality 417 Notice that
Page 433 and 434:
7.4. Orthogonality 419 insignifican
Page 435 and 436:
7.4. Orthogonality 421 Usually it w
Page 437 and 438:
7.4. Orthogonality 423 We now turn
Page 439 and 440:
7.4. Orthogonality 425 Solution. No
Page 441 and 442:
7.4. Orthogonality 427 = 2y 2 1 + 8
Page 443 and 444:
7.4. Orthogonality 429 ⎡ A = ⎢
Page 445 and 446:
7.4. Orthogonality 431 Exercise 7.4
Page 447 and 448:
Chapter 8 Some Curvilinear Coordina
Page 449 and 450:
8.1. Polar Coordinates and Polar Gr
Page 451 and 452:
Page 453 and 454:
Page 455 and 456:
Page 457 and 458:
8.2. Spherical and Cylindrical Coor
Page 459 and 460:
Page 461 and 462:
Page 463 and 464:
Chapter 9 Vector Spaces 9.1 Algebra
Page 465 and 466:
9.1. Algebraic Considerations 451 E
Page 467 and 468:
9.1. Algebraic Considerations 453 H
Page 469 and 470:
9.1. Algebraic Considerations 455
Page 471 and 472:
9.1. Algebraic Considerations 457 =
Page 473 and 474:
Page 475 and 476:
Page 477 and 478:
Page 479 and 480: 9.2. Spanning Sets 465 Show that, w
Page 481 and 482: 9.2. Spanning Sets 467 Clearlynoval
Page 483 and 484: 9.3. Linear Independence 469 9.3 Li
Page 485 and 486: 9.3. Linear Independence 471 Soluti
Page 487 and 488: 9.3. Linear Independence 473 Exerci
Page 489 and 490: 9.3. Linear Independence 475 Exerci
Page 491 and 492: 9.4. Subspaces and Basis 477 (b) {
Page 493 and 494: 9.4. Subspaces and Basis 479 1. U
Page 495 and 496: 9.4. Subspaces and Basis 481 It fol
Page 497 and 498: 9.4. Subspaces and Basis 483 But th
Page 499 and 500: 9.4. Subspaces and Basis 485 Then {
Page 501 and 502: 9.4. Subspaces and Basis 487 Then f
Page 503 and 504: 9.4. Subspaces and Basis 489 Theref
Page 505 and 506: 9.4. Subspaces and Basis 491 The re
Page 507 and 508: 9.6. Linear Transformations 493 2.
Page 509 and 510: 9.6. Linear Transformations 495 2.
Page 511 and 512: 9.6. Linear Transformations 497 The
Page 513 and 514: 9.7. Isomorphisms 499 9.7 Isomorphi
Page 515 and 516: 9.7. Isomorphisms 501 Example 9.66:
Page 517 and 518: 9.7. Isomorphisms 503 Example 9.71:
Page 519 and 520: 9.7. Isomorphisms 505 Since T is on
Page 521 and 522: 9.7. Isomorphisms 507 for ∑ n i=1
Page 523 and 524: 9.7. Isomorphisms 509 Exercises Exe
Page 525 and 526: 9.7. Isomorphisms 511 for example.
Page 527 and 528: 9.8. The Kernel And Image Of A Line
Page 529: 9.8. The Kernel And Image Of A Line
Page 533 and 534: 9.9. The Matrix of a Linear Transfo
Page 543 and 544: Appendix A Some Prerequisite Topics
Page 545 and 546: A.2. Well Ordering and Induction 53
Page 547 and 548: A.2. Well Ordering and Induction 53
Page 549 and 550: Appendix B Selected Exercise Answer
Page 551 and 552: 537 1.2.33 Solution is: [x = 2 −
Page 553 and 554: 539 2.1.7 0A =(0 + 0)A = 0A + 0A.No
Page 555 and 556: 541 [ 2.1.18 Let A = 2.1.20 Let A =
Page 557 and 558: 543 2.1.40 2.1.41 ⎡ ⎣ ⎡ ⎣ 1
Page 559 and 560: 545 2.2.10 An LU factorization of t
Page 561 and 562: 547 3.1.14 If the determinant is no
Page 563 and 564: 549 ⎡ 3.2.2 det⎣ 1 2 0 0 2 1 3
Page 565 and 566: 551 3.2.16 This follows because det
Page 567 and 568: 553 4.7.16 ( ) ( ) ⃗u •⃗v ⃗
Page 569 and 570: 555 It follows that the expression
Page 571 and 572: 557 4.11.6 (a) Orthogonal. (b) Symm
Page 573 and 574: 559 4.11.14 Then a solution is ⎡
Page 575 and 576: 561 4.12.23 ⎡ ⎢ ⃗F 1 • ⎣
Page 577 and 578: 563 5.2.13 ⎡ 1 ⎣ 10 1 0 3 0 0 0
Page 579 and 580: 565 Now to write in terms of (a,b),
Page 581 and 582:
567 ⎡ 5.9.7 Solution is: ⎣ −
Page 583 and 584:
569 6.1.6 If p(z)=0, then you have
Page 585 and 586:
571 7.1.2 Say AX = λX.ThencAX = c
Page 587 and 588:
573 7.3.1 First we write A = PDP
Page 589 and 590:
575 7.3.19 The solution is [ e At 8
Page 591 and 592:
577 7.4.12 The eigenvectors and eig
Page 593 and 594:
579 9.3.30 No. They can’t be. 9.3
Page 595 and 596:
581 9.8.5 We can easily see that di
Page 597 and 598:
Index ∩, 529 ∪, 529 \, 529 row-
Page 599 and 600:
INDEX 585 null space, 211 orthogona
Page 601:
www.lyryx.com
show all

A First Course in Linear Algebra, 2017a

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

Delete template?

Save as template?