Elementary Linear Algebra, Sixth Edition provides a solid introduction to both the computational and theoretical aspects of linear algebra, covering many important real-world applications, including graph theory, circuit theory, Markov chains, elementary coding theory, least-squares polynomials and least-squares solutions for inconsistent systems, differential equations, computer graphics and quadratic forms. In addition, many computational techniques in linear algebra are presented, including iterative methods for solving linear systems, LDU Decomposition, the Power Method for finding eigenvalues, QR Decomposition, and Singular Value Decomposition and its usefulness in digital imaging.
1. 1.3 An Introduction to Proof Techniques - p. 22 2. Proof Technique: Direct Proof - p. 22 3. “If A Then B ” Proofs - p. 23 4. Working “Backward” to Discover a Proof - p. 23 5. “ A If and Only If B ” Proofs - p. 24 6. “If A Then ( B or C )” Proofs - p. 25 7. Proof Technique: Proof by Contrapositive - p. 26 8. Converse and Inverse - p. 26 9. Proof Technique: Proof by Contradiction - p. 27 10. Proof Technique: Proof by Induction - p. 28 11. Proof Technique: Reducing to a Previous Result - p. 29 12. Negating Statements With Quantifiers and Connectives - p. 30 13. Disproving Statements - p. 30 14. New Vocabulary - p. 32 15. Highlights - p. 32 16. Exercises for Section 1.3 - p. 32 17. 1.4 Fundamental Operations With Matrices - p. 35 18. Definition of a Matrix - p. 35 19. Special Types of Matrices - p. 36 20. Addition and Scalar Multiplication With Matrices - p. 37 21. Fundamental Properties of Addition and Scalar Multiplication - p. 37 22. The Transpose of a Matrix and Its Properties - p. 38 23. Symmetric and Skew-Symmetric Matrices - p. 39 24. New Vocabulary - p. 41 25. Highlights - p. 41 26. Exercises for Section 1.4 - p. 41 27. 1.5 Matrix Multiplication - p. 43 28. Exercises for Section 1.5 - p. 50 29. Review Exercises for Chapter 1 - p. 54 30. Chapter 2 : Systems of Linear Equations - p. 55 31. Abstract - p. 55 32. 2.1 Solving Linear Systems Using Gaussian Elimination - p. 57 33. Exercises for Section 2.1 - p. 69 34. 2.2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form - p. 70 35. Introduction to Gauss-Jordan Row Reduction - p. 70 36. Reduced Row Echelon Form - p. 72 37. Number of Solutions - p. 73 38. Homogeneous Systems - p. 73 39. Fundamental Solutions for a Homogeneous System - p. 74 40. Application: Balancing Chemical Equations - p. 75 41. Solving Several Systems Simultaneously - p. 76 42. New Vocabulary - p. 77 43. Highlights - p. 77 44. Exercises for Section 2.2 - p. 77 45. 2.3 Equivalent Systems, Rank, and Row Space - p. 80 46. Equivalent Systems and Row Equivalence of Matrices - p. 80 47. Rank of a Matrix - p. 82 48. Linear Combinations of Vectors - p. 82 49. The Row Space of a Matrix - p. 83 50. Row Equivalence Determines the Row Space - p. 85 51. New Vocabulary - p. 86 52. Highlights - p. 86 53. Exercises for Section Page 87 2.3 - p. 86 54. 2.4 Inverses of Matrices - p. 90 55. Multiplicative Inverse of a Matrix - p. 90 56. Properties of the Matrix Inverse - p. 91 57. Inverses for 2 × 2 Matrices - p. 92 58. Inverses of Larger Matrices - p. 93 59. Justification of the Inverse Method - p. 94 60. Solving a System Using the Inverse of the Coefficient Matrix - p. 95 61. New Vocabulary - p. 96 62. Highlights - p. 96 63. Exercises for Section 2.4 - p. 96 64. Review Exercises for Chapter 2 - p. 98 65. Chapter 3 : Determinants and Eigenvalues - p. 100 66. Abstract - p. 100 67. 3.1 Introduction to Determinants - p. 101 68. Exercises for Section 3.1 - p. 106 69. 3.2 Determinants and Row Reduction - p. 109 70. Exercises for Section 3.2 - p. 113 71. 3.3 Further Properties of the Determinant - p. 116 72. Exercises for Section 3.3 - p. 120 73. 3.4 Eigenvalues and Diagonalization - p. 123 74. Exercises for Section 3.4 - p. 132 75. Review Exercises for Chapter 3 - p. 136 76. Chapter 4 : Finite Dimensional Vector Spaces - p. 139 77. Abstract - p. 139 78. 4.1 Introduction to Vector Spaces - p. 141 79. Definition of a Vector Space - p. 141 80. Examples of Vector Spaces - p. 142 81. Two Unusual Vector Spaces - p. 144 82. Some Elementary Properties of Vector Spaces - p. 145 83. Failure of the Vector Space Conditions - p. 146 84. New Vocabulary - p. 147 85. Highlights - p. 147 86. Exercises for Section 4.1 - p. 147 87. 4.2 Subspaces - p. 148 88. Definition of a Subspace and Examples - p. 148 89. When Is a Subset a Subspace? - p. 149 90. Verifying Subspaces in M n n and R n - p. 150 91. Subsets That Are Not Subspaces - p. 151 92. Linear Combinations Remain in a Subspace - p. 152 93. An Eigenspace Is a Subspace - p. 153 94. New Vocabulary - p. 153 95. Highlights - p. 153 96. Exercises for Section 4.2 - p. 153 97. 4.3 Span - p. 156 98. Exercises for Section 4.3 - p. 161 99. 4.4 Linear Independence - p. 163 100. Linear Independence and Dependence - p. 163 101. Linear Dependence and Independence With One- and Two-Element Sets - p. 164 102. Using Row Reduction to Test for Linear Independence - p. 165 103. Alternate Characterizations of Linear Independence - p. 167 104. Linear Independence of Infinite Sets - p. 169 105. Uniqueness of Expression of a Vector as a Linear Combination - p. 169 106. Summary of Results - p. 170 107. New Vocabulary - p. 171 108. Highlights - p. 171 109. Exercises for Section 4.4 - p. 171 110. 4.5 Basis and Dimension - p. 174 111. Definition of Basis - p. 174 112. A Technical Lemma - p. 175 113. Dimension - p. 176 114. Sizes of Spanning Sets and Linearly Independent Sets - p. 177 115. Dimension of a Subspace - p. 178 116. Diagonalization and Bases - p. 179 117. New Vocabulary - p. 179 118. Highlights - p. 179 119. Exercises for Section 4.5 - p. 180 120. 4.6 Constructing Special Bases - p. 182 121. Exercises for Section 4.6 - p. 186 122. 4.7 Coordinatization - p. 188 123. Coordinates With Respect to an Ordered Basis - p. 188 124. Using Row Reduction to Coordinatize a Vector - p. 190 125. Fundamental Properties of Coordinatization - p. 191 126. The Transition Matrix for Change of Coordinates - p. 192 127. Change of Coordinates Using the Transition Matrix - p. 194 128. Algebra of the Transition Matrix - p. 195 129. Diagonalization and the Transition Matrix - p. 196 130. New Vocabulary - p. 198 131. Highlights - p. 198 132. Exercises for Section 4.7 - p. 198 133. Review Exercises for Chapter 4 - p. 201 134. Chapter 5 : Linear Transformations - p. 204 135. Abstract - p. 204 136. 5.1 Introduction to Linear Transformations - p. 205 137. Exercises for Section 5.1 - p. 212 138. 5.2 The Matrix of a Linear Transformation - p. 215 139. Exercises for Section 5.2 - p. 223 140. 5.3 The Dimension Theorem - p. 227 141. Exercises for Section 5.3 - p. 232 142. 5.4 One-to-One and Onto Linear Transformations - p. 235 143. Exercises for Section 5.4 - p. 238 144. 5.5 Isomorphism - p. 240 145. Exercises for Section 5.5 - p. 246 146. 5.6 Diagonalization of Linear Operators - p. 248 147. Exercises for Section 5.6 - p. 257 148. Review Exercises for Chapter 5 - p. 260 149. Chapter 6 : Orthogonality - p. 264 150. Abstract - p. 264 151. 6.1 Orthogonal Bases and the Gram-Schmidt Process - p. 265 152. Exercises for Section 6.1 - p. 272 153. 6.2 Orthogonal Complements - p. 274 154. Orthogonal Complements - p. 274 155. Properties of Orthogonal Complements - p. 275 156. Orthogonal Projection Onto a Subspace - p. 277 157. Application: Orthogonal Projections and Reflections in R 3 - p. 279 158. Application: Distance From a Point to a Subspace - p. 281 159. New Vocabulary - p. 282 160. Highlights - p. 282 161. Exercises for Section 6.2 - p. 282 162. 6.3 Orthogonal Diagonalization - p. 285 163. Exercises for Section 6.3 - p. 292 164. Review Exercises for Chapter 6 - p. 295 165. Chapter 7 : Complex Vector Spaces and General Inner Products - p. 297 166. Abstract - p. 297 167. 7.1 Complex n -Vectors and Matrices - p. 299 168. Exercises for Section 7.1 - p. 303 169. 7.2 Complex Eigenvalues and Complex Eigenvectors - p. 305 170. Exercises for Section 7.2 - p. 308 171. 7.3 Complex Vector Spaces - p. 309 172. Exercises for Section 7.3 - p. 311 173. 7.4 Orthogonality in C n - p. 311 174. Prerequisite: Section 6.3 , Orthogonal Diagonalization - p. 311 175. Orthogonal Bases and the Gram-Schmidt Process - p. 311 176. Unitary Matrices - p. 312 177. Unitarily Diagonalizable Matrices - p. 313 178. Self-Adjoint Operators and Hermitian Matrices - p. 314 179. New Vocabulary - p. 315 180. Highlights - p. 315 181. Exercises for Section 7.4 - p. 315 182. 7.5 Inner Product Spaces - p. 316 183. Prerequisite: Section 6.3 , Orthogonal Diagonalization - p. 316 184. Inner Products - p. 317 185. Length, Distance, and Angles in Inner Product Spaces - p. 318 186. Orthogonality in Inner Product Spaces - p. 319 187. The Generalized Gram-Schmidt Process - p. 320 188. Orthogonal Complements and Orthogonal Projections in Inner Product Spaces - p. 322 189. New Vocabulary - p. 323 190. Highlights - p. 324 191. Exercises for Section 7.5 - p. 324 192. Review Exercises for Chapter 7 - p. 327 193. Chapter 8 : Additional Applications - p. 329 194. Abstract - p. 329 195. 8.1 Graph Theory - p. 331 196. Exercises for Section 8.1 - p. 336 197. 8.2 Ohm's Law - p. 340 198. Exercises for Section 8.2 - p. 342 199. 8.3 Least-Squares Polynomials - p. 342 200. Exercises for Section 8.3 - p. 347 201. 8.4 Markov Chains - p. 348 202. Exercises for Section 8.4 - p. 354 203. 8.5 Hill Substitution: An Introduction to Coding Theory - p. 356 204. Exercises for Section 8.5 - p. 358 205. 8.6 Linear Recurrence Relations and the Fibonacci Sequence - p. 359 206. Exercises for Section 8.6 - p. 366 207. 8.7 Rotation of Axes for Conic Sections - p. 369 208. Exercises for Section 8.7 - p. 373 209. 8.8 Computer Graphics - p. 373 210. Exercises for Section 8.8 - p. 381 211. 8.9 Differential Equations - p. 385 212. Exercises for Section 8.9 - p. 390 213. 8.10 Least-Squares Solutions for Inconsistent Systems - p. 391 214. Prerequisite: Section 6.2 , Orthogonal Complements - p. 391 215. Finding Approximate Solutions - p. 391 216. Non-Unique Least-Squares Solutions - p. 392 217. Approximate Eigenvalues and Eigenvectors - p. 393 218. Least-Squares Polynomials - p. 394 219. New Vocabulary - p. 394 220. Highlights - p. 395 221. Exercises for Section 8.10 - p. 395 222. 8.11 Quadratic Forms - p. 396 223. Exercises for Section 8.11 - p. 400 224. Chapter 9 : Numerical Techniques - p. 400 225. Abstract - p. 400 226. 9.1 Numerical Techniques for Solving Systems - p. 401 227. Exercises for Section 9.1 - p. 408 228. 9.2 LDU Decomposition - p. 410 229. 9.3 The Power Method for Finding Eigenvalues - p. 415 230. Exercises for Section 9.3 - p. 419 231. 9.4 QR Factorization - p. 420 232. Prerequisite: Section 6.1 , Orthogonal Bases and the Gram-Schmidt Process - p. 420 233. QR Factorization Theorem - p. 420 234. QR Factorization and Least Squares - p. 423 235. A More General QR Factorization - p. 424 236. New Vocabulary - p. 424 237. Highlights - p. 424 238. Exercises for Section 9.4 - p. 425 239. 9.5 Singular Value Decomposition - p. 426 240. Prerequisite: Section 6.3 : Orthogonal Diagonalization - p. 426 241. Singular Values and Right Singular Vectors - p. 426 242. Singular Values and Left Singular Vectors - p. 427 243. Orthonormal Bases Derived From the Left and Right Singular Vectors - p. 428 244. Singular Value Decomposition - p. 429 245. A Geometric Interpretation - p. 431 246. The Outer Product Form for Singular Value Decomposition - p. 432 247. Digital Images - p. 433 248. The Pseudoinverse - p. 433 249. New Vocabulary - p. 436 250. Highlights - p. 436 251. Exercises for Section 9.5 - p. 437 252. Appendix A : Miscellaneous Proofs - p. 439 253. Appendix B : Functions - p. 445 254. Appendix C : Complex Numbers - p. 453 255. Appendix D : Elementary Matrices - p. 457 256. Appendix E : Answers to Selected Exercises - p. 463 257. Section 1.1 (p. 10–12) - p. 465 258. Section 1.2 (p. 20–22) - p. 466 259. Section 1.3 (p. 32–34) - p. 467 260. Section 1.4 (p. 41–43) - p. 468 261. Section 1.5 (p. 50–54) - p. 469 262. Chapter 1 Review Exercises (p. 54–55) - p. 470 263. Section 2.1 (p. 69–70) - p. 470 264. Section 2.2 (p. 77–80) - p. 470 265. Section 2.3 (p. 86–89) - p. 471 266. Section 2.4 (p. 96–98) - p. 473 267. Chapter 2 Review Exercises (p. 98–100) - p. 475 268. Section 3.1 (p. 106–109) - p. 475 269. Section 3.2 (p. 113–116) - p. 475 270. Section 3.3 (p. 121–123) - p. 476 271. Section 3.4 (p. 132–136) - p. 476 272. Chapter 3 Review Exercises (p. 136–138) - p. 477 273. Section 4.1 (p. 147–148) - p. 478 274. Section 4.2 (p. 153–155) - p. 478 275. Section 4.3 (p. 161–163) - p. 478 276. Section 4.4 (p. 172–174) - p. 479 277. Section 4.5 (p. 180–182) - p. 479 278. Section 4.6 (p. 186–188) - p. 480 279. Section 4.7 (p. 198–201) - p. 480 280. Chapter 4 Review Exercises (p. 201–204) - p. 482 281. Section 5.1 (p. 212–215) - p. 483 282. Section 5.2 (p. 223–227) - p. 483 283. Section 5.3 (p. 232–235) - p. 484 284. Section 5.4 (p. 238–240) - p. 484 285. Section 5.5 (p. 246–248) - p. 485 286. Section 5.6 (p. 257–260) - p. 486 287. Chapter 5 Review Exercises (p. 260–264) - p. 487 288. Section 6.1 (p. 272–274) - p. 488 289. Section 6.2 (p. 282–284) - p. 488 290. Section 6.3 (p. 292–295) - p. 488 291. Chapter 6 Review Exercises (p. 295–297) - p. 490 292. Section 7.1 (p. 304–305) - p. 490 293. Section 7.2 (p. 308–309) - p. 490 294. Section 7.3 (p. 311) - p. 491 295. Section 7.4 (p. 315–316) - p. 491 296. Section 7.5 (p. 324–327) - p. 491 297. Chapter 7 Review Exercises (p. 327–329) - p. 492 298. Section 8.1 (p. 336–340) - p. 492 299. Section 8.2 (p. 342) - p. 494 300. Section 8.3 (p. 347–348) - p. 495 301. Section 8.4 (p. 354–356) - p. 495 302. Section 8.5 (p. 358–359) - p. 496 303. Section 8.6 (p. 366–369) - p. 496 304. Section 8.7 (p. 373) - p. 496 305. Section 8.8 (p. 381–385) - p. 497 306. Section 8.9 (p. 390–391) - p. 498 307. Section 8.10 (p. 395–396) - p. 498 308. Section 8.11 (p. 400) - p. 499 309. Section 9.1 (p. 408–409) - p. 499 310. Section 9.2 (p. 415) - p. 500 311. Section 9.3 (p. 419–420) - p. 501 312. Section 9.4 (p. 425) - p. 501 313. Section 9.5 (p. 437–439) - p. 501 314. Appendix B (p. 452–453) - p. 503 315. Appendix C (p. 457) - p. 503 316. Appendix D (p. 462–463) - p. 503 317. Index - p. 504 318. Endpapers - p. 513
Elementary Linear Algebra, Sixth Edition provides a solid introduction to both the computational and theoretical aspects of linear algebra, covering many important real-world applications, including graph theory, circuit theory, Markov chains, elementary coding theory, least-squares polynomials and least-squares solutions for inconsistent systems, differential equations, computer graphics and quadratic forms. In addition, many computational techniques in linear algebra are presented, including iterative methods for solving linear systems, LDU Decomposition, the Power Method for finding eigenvalues, QR Decomposition, and Singular Value Decomposition and its usefulness in digital imaging.
1. 1.3 An Introduction to Proof Techniques - p. 22 2. Proof Technique: Direct Proof - p. 22 3. “If A Then B ” Proofs - p. 23 4. Working “Backward” to Discover a Proof - p. 23 5. “ A If and Only If B ” Proofs - p. 24 6. “If A Then ( B or C )” Proofs - p. 25 7. Proof Technique: Proof by Contrapositive - p. 26 8. Converse and Inverse - p. 26 9. Proof Technique: Proof by Contradiction - p. 27 10. Proof Technique: Proof by Induction - p. 28 11. Proof Technique: Reducing to a Previous Result - p. 29 12. Negating Statements With Quantifiers and Connectives - p. 30 13. Disproving Statements - p. 30 14. New Vocabulary - p. 32 15. Highlights - p. 32 16. Exercises for Section 1.3 - p. 32 17. 1.4 Fundamental Operations With Matrices - p. 35 18. Definition of a Matrix - p. 35 19. Special Types of Matrices - p. 36 20. Addition and Scalar Multiplication With Matrices - p. 37 21. Fundamental Properties of Addition and Scalar Multiplication - p. 37 22. The Transpose of a Matrix and Its Properties - p. 38 23. Symmetric and Skew-Symmetric Matrices - p. 39 24. New Vocabulary - p. 41 25. Highlights - p. 41 26. Exercises for Section 1.4 - p. 41 27. 1.5 Matrix Multiplication - p. 43 28. Exercises for Section 1.5 - p. 50 29. Review Exercises for Chapter 1 - p. 54 30. Chapter 2 : Systems of Linear Equations - p. 55 31. Abstract - p. 55 32. 2.1 Solving Linear Systems Using Gaussian Elimination - p. 57 33. Exercises for Section 2.1 - p. 69 34. 2.2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form - p. 70 35. Introduction to Gauss-Jordan Row Reduction - p. 70 36. Reduced Row Echelon Form - p. 72 37. Number of Solutions - p. 73 38. Homogeneous Systems - p. 73 39. Fundamental Solutions for a Homogeneous System - p. 74 40. Application: Balancing Chemical Equations - p. 75 41. Solving Several Systems Simultaneously - p. 76 42. New Vocabulary - p. 77 43. Highlights - p. 77 44. Exercises for Section 2.2 - p. 77 45. 2.3 Equivalent Systems, Rank, and Row Space - p. 80 46. Equivalent Systems and Row Equivalence of Matrices - p. 80 47. Rank of a Matrix - p. 82 48. Linear Combinations of Vectors - p. 82 49. The Row Space of a Matrix - p. 83 50. Row Equivalence Determines the Row Space - p. 85 51. New Vocabulary - p. 86 52. Highlights - p. 86 53. Exercises for Section Page 87 2.3 - p. 86 54. 2.4 Inverses of Matrices - p. 90 55. Multiplicative Inverse of a Matrix - p. 90 56. Properties of the Matrix Inverse - p. 91 57. Inverses for 2 × 2 Matrices - p. 92 58. Inverses of Larger Matrices - p. 93 59. Justification of the Inverse Method - p. 94 60. Solving a System Using the Inverse of the Coefficient Matrix - p. 95 61. New Vocabulary - p. 96 62. Highlights - p. 96 63. Exercises for Section 2.4 - p. 96 64. Review Exercises for Chapter 2 - p. 98 65. Chapter 3 : Determinants and Eigenvalues - p. 100 66. Abstract - p. 100 67. 3.1 Introduction to Determinants - p. 101 68. Exercises for Section 3.1 - p. 106 69. 3.2 Determinants and Row Reduction - p. 109 70. Exercises for Section 3.2 - p. 113 71. 3.3 Further Properties of the Determinant - p. 116 72. Exercises for Section 3.3 - p. 120 73. 3.4 Eigenvalues and Diagonalization - p. 123 74. Exercises for Section 3.4 - p. 132 75. Review Exercises for Chapter 3 - p. 136 76. Chapter 4 : Finite Dimensional Vector Spaces - p. 139 77. Abstract - p. 139 78. 4.1 Introduction to Vector Spaces - p. 141 79. Definition of a Vector Space - p. 141 80. Examples of Vector Spaces - p. 142 81. Two Unusual Vector Spaces - p. 144 82. Some Elementary Properties of Vector Spaces - p. 145 83. Failure of the Vector Space Conditions - p. 146 84. New Vocabulary - p. 147 85. Highlights - p. 147 86. Exercises for Section 4.1 - p. 147 87. 4.2 Subspaces - p. 148 88. Definition of a Subspace and Examples - p. 148 89. When Is a Subset a Subspace? - p. 149 90. Verifying Subspaces in M n n and R n - p. 150 91. Subsets That Are Not Subspaces - p. 151 92. Linear Combinations Remain in a Subspace - p. 152 93. An Eigenspace Is a Subspace - p. 153 94. New Vocabulary - p. 153 95. Highlights - p. 153 96. Exercises for Section 4.2 - p. 153 97. 4.3 Span - p. 156 98. Exercises for Section 4.3 - p. 161 99. 4.4 Linear Independence - p. 163 100. Linear Independence and Dependence - p. 163 101. Linear Dependence and Independence With One- and Two-Element Sets - p. 164 102. Using Row Reduction to Test for Linear Independence - p. 165 103. Alternate Characterizations of Linear Independence - p. 167 104. Linear Independence of Infinite Sets - p. 169 105. Uniqueness of Expression of a Vector as a Linear Combination - p. 169 106. Summary of Results - p. 170 107. New Vocabulary - p. 171 108. Highlights - p. 171 109. Exercises for Section 4.4 - p. 171 110. 4.5 Basis and Dimension - p. 174 111. Definition of Basis - p. 174 112. A Technical Lemma - p. 175 113. Dimension - p. 176 114. Sizes of Spanning Sets and Linearly Independent Sets - p. 177 115. Dimension of a Subspace - p. 178 116. Diagonalization and Bases - p. 179 117. New Vocabulary - p. 179 118. Highlights - p. 179 119. Exercises for Section 4.5 - p. 180 120. 4.6 Constructing Special Bases - p. 182 121. Exercises for Section 4.6 - p. 186 122. 4.7 Coordinatization - p. 188 123. Coordinates With Respect to an Ordered Basis - p. 188 124. Using Row Reduction to Coordinatize a Vector - p. 190 125. Fundamental Properties of Coordinatization - p. 191 126. The Transition Matrix for Change of Coordinates - p. 192 127. Change of Coordinates Using the Transition Matrix - p. 194 128. Algebra of the Transition Matrix - p. 195 129. Diagonalization and the Transition Matrix - p. 196 130. New Vocabulary - p. 198 131. Highlights - p. 198 132. Exercises for Section 4.7 - p. 198 133. Review Exercises for Chapter 4 - p. 201 134. Chapter 5 : Linear Transformations - p. 204 135. Abstract - p. 204 136. 5.1 Introduction to Linear Transformations - p. 205 137. Exercises for Section 5.1 - p. 212 138. 5.2 The Matrix of a Linear Transformation - p. 215 139. Exercises for Section 5.2 - p. 223 140. 5.3 The Dimension Theorem - p. 227 141. Exercises for Section 5.3 - p. 232 142. 5.4 One-to-One and Onto Linear Transformations - p. 235 143. Exercises for Section 5.4 - p. 238 144. 5.5 Isomorphism - p. 240 145. Exercises for Section 5.5 - p. 246 146. 5.6 Diagonalization of Linear Operators - p. 248 147. Exercises for Section 5.6 - p. 257 148. Review Exercises for Chapter 5 - p. 260 149. Chapter 6 : Orthogonality - p. 264 150. Abstract - p. 264 151. 6.1 Orthogonal Bases and the Gram-Schmidt Process - p. 265 152. Exercises for Section 6.1 - p. 272 153. 6.2 Orthogonal Complements - p. 274 154. Orthogonal Complements - p. 274 155. Properties of Orthogonal Complements - p. 275 156. Orthogonal Projection Onto a Subspace - p. 277 157. Application: Orthogonal Projections and Reflections in R 3 - p. 279 158. Application: Distance From a Point to a Subspace - p. 281 159. New Vocabulary - p. 282 160. Highlights - p. 282 161. Exercises for Section 6.2 - p. 282 162. 6.3 Orthogonal Diagonalization - p. 285 163. Exercises for Section 6.3 - p. 292 164. Review Exercises for Chapter 6 - p. 295 165. Chapter 7 : Complex Vector Spaces and General Inner Products - p. 297 166. Abstract - p. 297 167. 7.1 Complex n -Vectors and Matrices - p. 299 168. Exercises for Section 7.1 - p. 303 169. 7.2 Complex Eigenvalues and Complex Eigenvectors - p. 305 170. Exercises for Section 7.2 - p. 308 171. 7.3 Complex Vector Spaces - p. 309 172. Exercises for Section 7.3 - p. 311 173. 7.4 Orthogonality in C n - p. 311 174. Prerequisite: Section 6.3 , Orthogonal Diagonalization - p. 311 175. Orthogonal Bases and the Gram-Schmidt Process - p. 311 176. Unitary Matrices - p. 312 177. Unitarily Diagonalizable Matrices - p. 313 178. Self-Adjoint Operators and Hermitian Matrices - p. 314 179. New Vocabulary - p. 315 180. Highlights - p. 315 181. Exercises for Section 7.4 - p. 315 182. 7.5 Inner Product Spaces - p. 316 183. Prerequisite: Section 6.3 , Orthogonal Diagonalization - p. 316 184. Inner Products - p. 317 185. Length, Distance, and Angles in Inner Product Spaces - p. 318 186. Orthogonality in Inner Product Spaces - p. 319 187. The Generalized Gram-Schmidt Process - p. 320 188. Orthogonal Complements and Orthogonal Projections in Inner Product Spaces - p. 322 189. New Vocabulary - p. 323 190. Highlights - p. 324 191. Exercises for Section 7.5 - p. 324 192. Review Exercises for Chapter 7 - p. 327 193. Chapter 8 : Additional Applications - p. 329 194. Abstract - p. 329 195. 8.1 Graph Theory - p. 331 196. Exercises for Section 8.1 - p. 336 197. 8.2 Ohm's Law - p. 340 198. Exercises for Section 8.2 - p. 342 199. 8.3 Least-Squares Polynomials - p. 342 200. Exercises for Section 8.3 - p. 347 201. 8.4 Markov Chains - p. 348 202. Exercises for Section 8.4 - p. 354 203. 8.5 Hill Substitution: An Introduction to Coding Theory - p. 356 204. Exercises for Section 8.5 - p. 358 205. 8.6 Linear Recurrence Relations and the Fibonacci Sequence - p. 359 206. Exercises for Section 8.6 - p. 366 207. 8.7 Rotation of Axes for Conic Sections - p. 369 208. Exercises for Section 8.7 - p. 373 209. 8.8 Computer Graphics - p. 373 210. Exercises for Section 8.8 - p. 381 211. 8.9 Differential Equations - p. 385 212. Exercises for Section 8.9 - p. 390 213. 8.10 Least-Squares Solutions for Inconsistent Systems - p. 391 214. Prerequisite: Section 6.2 , Orthogonal Complements - p. 391 215. Finding Approximate Solutions - p. 391 216. Non-Unique Least-Squares Solutions - p. 392 217. Approximate Eigenvalues and Eigenvectors - p. 393 218. Least-Squares Polynomials - p. 394 219. New Vocabulary - p. 394 220. Highlights - p. 395 221. Exercises for Section 8.10 - p. 395 222. 8.11 Quadratic Forms - p. 396 223. Exercises for Section 8.11 - p. 400 224. Chapter 9 : Numerical Techniques - p. 400 225. Abstract - p. 400 226. 9.1 Numerical Techniques for Solving Systems - p. 401 227. Exercises for Section 9.1 - p. 408 228. 9.2 LDU Decomposition - p. 410 229. 9.3 The Power Method for Finding Eigenvalues - p. 415 230. Exercises for Section 9.3 - p. 419 231. 9.4 QR Factorization - p. 420 232. Prerequisite: Section 6.1 , Orthogonal Bases and the Gram-Schmidt Process - p. 420 233. QR Factorization Theorem - p. 420 234. QR Factorization and Least Squares - p. 423 235. A More General QR Factorization - p. 424 236. New Vocabulary - p. 424 237. Highlights - p. 424 238. Exercises for Section 9.4 - p. 425 239. 9.5 Singular Value Decomposition - p. 426 240. Prerequisite: Section 6.3 : Orthogonal Diagonalization - p. 426 241. Singular Values and Right Singular Vectors - p. 426 242. Singular Values and Left Singular Vectors - p. 427 243. Orthonormal Bases Derived From the Left and Right Singular Vectors - p. 428 244. Singular Value Decomposition - p. 429 245. A Geometric Interpretation - p. 431 246. The Outer Product Form for Singular Value Decomposition - p. 432 247. Digital Images - p. 433 248. The Pseudoinverse - p. 433 249. New Vocabulary - p. 436 250. Highlights - p. 436 251. Exercises for Section 9.5 - p. 437 252. Appendix A : Miscellaneous Proofs - p. 439 253. Appendix B : Functions - p. 445 254. Appendix C : Complex Numbers - p. 453 255. Appendix D : Elementary Matrices - p. 457 256. Appendix E : Answers to Selected Exercises - p. 463 257. Section 1.1 (p. 10–12) - p. 465 258. Section 1.2 (p. 20–22) - p. 466 259. Section 1.3 (p. 32–34) - p. 467 260. Section 1.4 (p. 41–43) - p. 468 261. Section 1.5 (p. 50–54) - p. 469 262. Chapter 1 Review Exercises (p. 54–55) - p. 470 263. Section 2.1 (p. 69–70) - p. 470 264. Section 2.2 (p. 77–80) - p. 470 265. Section 2.3 (p. 86–89) - p. 471 266. Section 2.4 (p. 96–98) - p. 473 267. Chapter 2 Review Exercises (p. 98–100) - p. 475 268. Section 3.1 (p. 106–109) - p. 475 269. Section 3.2 (p. 113–116) - p. 475 270. Section 3.3 (p. 121–123) - p. 476 271. Section 3.4 (p. 132–136) - p. 476 272. Chapter 3 Review Exercises (p. 136–138) - p. 477 273. Section 4.1 (p. 147–148) - p. 478 274. Section 4.2 (p. 153–155) - p. 478 275. Section 4.3 (p. 161–163) - p. 478 276. Section 4.4 (p. 172–174) - p. 479 277. Section 4.5 (p. 180–182) - p. 479 278. Section 4.6 (p. 186–188) - p. 480 279. Section 4.7 (p. 198–201) - p. 480 280. Chapter 4 Review Exercises (p. 201–204) - p. 482 281. Section 5.1 (p. 212–215) - p. 483 282. Section 5.2 (p. 223–227) - p. 483 283. Section 5.3 (p. 232–235) - p. 484 284. Section 5.4 (p. 238–240) - p. 484 285. Section 5.5 (p. 246–248) - p. 485 286. Section 5.6 (p. 257–260) - p. 486 287. Chapter 5 Review Exercises (p. 260–264) - p. 487 288. Section 6.1 (p. 272–274) - p. 488 289. Section 6.2 (p. 282–284) - p. 488 290. Section 6.3 (p. 292–295) - p. 488 291. Chapter 6 Review Exercises (p. 295–297) - p. 490 292. Section 7.1 (p. 304–305) - p. 490 293. Section 7.2 (p. 308–309) - p. 490 294. Section 7.3 (p. 311) - p. 491 295. Section 7.4 (p. 315–316) - p. 491 296. Section 7.5 (p. 324–327) - p. 491 297. Chapter 7 Review Exercises (p. 327–329) - p. 492 298. Section 8.1 (p. 336–340) - p. 492 299. Section 8.2 (p. 342) - p. 494 300. Section 8.3 (p. 347–348) - p. 495 301. Section 8.4 (p. 354–356) - p. 495 302. Section 8.5 (p. 358–359) - p. 496 303. Section 8.6 (p. 366–369) - p. 496 304. Section 8.7 (p. 373) - p. 496 305. Section 8.8 (p. 381–385) - p. 497 306. Section 8.9 (p. 390–391) - p. 498 307. Section 8.10 (p. 395–396) - p. 498 308. Section 8.11 (p. 400) - p. 499 309. Section 9.1 (p. 408–409) - p. 499 310. Section 9.2 (p. 415) - p. 500 311. Section 9.3 (p. 419–420) - p. 501 312. Section 9.4 (p. 425) - p. 501 313. Section 9.5 (p. 437–439) - p. 501 314. Appendix B (p. 452–453) - p. 503 315. Appendix C (p. 457) - p. 503 316. Appendix D (p. 462–463) - p. 503 317. Index - p. 504 318. Endpapers - p. 513
