A First Course in Linear Algebra, 2017a

More documents

Recommendations

Info

4 Systems of Equations Example 1.1: A Graphical Solution Use a graph to find the solution to the following system of equations x + y = 3 y − x = 5 Solution. Through graphing the above equations and identifying the point of intersection, we can find the solution(s). Remember that we must have either one solution, infinitely many, or no solutions at all. The following graph shows the two equations, as well as the intersection. Remember, the point of intersection represents the solution of the two equations, or the (x,y) which satisfy both equations. In this case, there is one point of intersection at (−1,4) which means we have one unique solution, x = −1,y = 4. 6 y (x,y)=(−1,4) 4 2 −4 −3 −2 −1 1 x In the above example, we investigated the intersection point of two equations in two variables, x and y. Now we will consider the graphical solutions of three equations in two variables. Consider a system of three equations in two variables. Again, these equations can be graphed as straight lines in the plane, so that the resulting graph contains three straight lines. Recall the three possible types of solutions; no solution, one solution, and infinitely many solutions. There are now more complex ways of achieving these situations, due to the presence of the third line. For example, you can imagine the case of three intersecting lines having no common point of intersection. Perhaps you can also imagine three intersecting lines which do intersect at a single point. These two situations are illustrated below. ♠ y y x No Solution x One Solution
1.1. Systems of Equations, Geometry 5 Consider the first picture above. While all three lines intersect with one another, there is no common point of intersection where all three lines meet at one point. Hence, there is no solution to the three equations. Remember, a solution is a point (x,y) which satisfies all three equations. In the case of the second picture, the lines intersect at a common point. This means that there is one solution to the three equations whose graphs are the given lines. You should take a moment now to draw the graph of a system which results in three parallel lines. Next, try the graph of three identical lines. Which type of solution is represented in each of these graphs? We have now considered the graphical solutions of systems of two equations in two variables, as well as three equations in two variables. However, there is no reason to limit our investigation to equations in two variables. We will now consider equations in three variables. You may recall that equations in three variables, such as 2x + 4y − 5z = 8, form a plane. Above, we were looking for intersections of lines in order to identify any possible solutions. When graphically solving systems of equations in three variables, we look for intersections of planes. These points of intersection give the (x,y,z) that satisfy all the equations in the system. What types of solutions are possible when working with three variables? Consider the following picture involving two planes, which are given by two equations in three variables. Notice how these two planes intersect in a line. This means that the points (x,y,z) on this line satisfy both equations in the system. Since the line contains infinitely many points, this system has infinitely many solutions. It could also happen that the two planes fail to intersect. However, is it possible to have two planes intersect at a single point? Take a moment to attempt drawing this situation, and convince yourself that it is not possible! This means that when we have only two equations in three variables, there is no way to have a unique solution! Hence, the types of solutions possible for two equations in three variables are no solution or infinitely many solutions. Now imagine adding a third plane. In other words, consider three equations in three variables. What types of solutions are now possible? Consider the following diagram. New Plane ✠ In this diagram, there is no point which lies in all three planes. There is no intersection between all
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: Chapter 1 Systems of Equations 1.1
Page 21 and 22: 1.2. Systems of Equations, Algebrai
Page 65: 1.2. Systems of Equations, Algebrai
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 and 530:
9.8. The Kernel And Image Of A Line
Page 531 and 532:
9.9. The Matrix of a Linear Transfo
Page 533 and 534:
Page 535 and 536:
Page 537 and 538:
Page 539 and 540:
Page 541 and 542:
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?