Preface |
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vii | |
To the Instructor |
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xvii | |
To the Student |
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xxiii | |
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1 | (57) |
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Introduction: The Racetrack Game |
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1 | (2) |
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The Geometry and Algebra of Vectors |
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3 | (12) |
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Length and Angle: The Dot Product |
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15 | (16) |
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Exploration: Vectors and Geometry |
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29 | (2) |
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31 | (16) |
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Exploration: The Cross Product |
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45 | (2) |
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Code Vectors and Modular Arithmetic |
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47 | (11) |
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Vignette: The Codabar System |
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55 | (1) |
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56 | (2) |
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Systems of Linear Equations |
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58 | (76) |
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58 | (1) |
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Introduction to Systems of Linear Equations |
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59 | (9) |
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Exploration: Lies My Computer Told Me |
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66 | (2) |
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Direct Methods for Solving Linear Systems |
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68 | (22) |
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Exploration: Partial Pivoting |
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86 | (1) |
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Exploration: Counting Operations: An Introduction to the Analysis of Algorithms |
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87 | (3) |
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Spanning Sets and Linear Independence |
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90 | (11) |
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101 | (21) |
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101 | (2) |
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Balancing Chemical Equations |
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103 | (1) |
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104 | (2) |
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106 | (3) |
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109 | (10) |
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Vignette: The Global Positioning System |
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119 | (3) |
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Iterative Methods for Solving Linear Systems |
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122 | (12) |
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132 | (2) |
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134 | (118) |
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Introduction: Matrices in Action |
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134 | (2) |
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136 | (16) |
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152 | (9) |
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161 | (17) |
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178 | (11) |
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Subspaces, Basis, Dimension, and Rank |
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189 | (20) |
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Introduction to Linear Transformations |
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209 | (19) |
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224 | (4) |
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228 | (24) |
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228 | (5) |
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233 | (2) |
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235 | (5) |
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240 | (10) |
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250 | (2) |
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Eigenvalues and Eigenvectors |
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252 | (111) |
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Introduction: A Dynamical System on Graphs |
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252 | (1) |
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Introduction to Eigenvalues and Eigenvectors |
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253 | (9) |
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262 | (27) |
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Exploration: Geometric Applications of Determinants |
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283 | (6) |
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Eigenvalues and Eigenvectors of nxn Matrices |
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289 | (9) |
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Similarity and Diagonalization |
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298 | (10) |
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Iterative Methods for Computing Eigenvalues |
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308 | (14) |
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Applications and the Perron-Frobenius Theorem |
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322 | (41) |
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322 | (5) |
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327 | (2) |
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The Perron-Frobenius Theorem |
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329 | (3) |
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Linear Recurrence Relations |
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332 | (5) |
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Systems of Linear Differential Equations |
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337 | (8) |
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Discrete Linear Dynamical Systems |
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345 | (8) |
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Vignette: Ranking Sports Teams and Searching the Internet |
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353 | (8) |
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361 | (2) |
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363 | (68) |
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Introduction: Shadows on a Wall |
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363 | (2) |
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365 | (10) |
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Orthogonal Complements and Orthogonal Projections |
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375 | (10) |
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The Gram-Schmidt Process and the QR Factorization |
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385 | (12) |
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Exploration: The Modified QR Factorization |
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393 | (2) |
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Exploration: Approximating Eigenvalues with the QR Algorithm |
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395 | (2) |
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Orthogonal Diagonalization of Symmetric Matrices |
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397 | (8) |
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405 | (26) |
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405 | (6) |
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411 | (7) |
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Graphing Quadratic Equations |
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418 | (11) |
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429 | (2) |
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431 | (107) |
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Introduction: Fibonacci in (Vector) Space |
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431 | (2) |
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Vector Spaces and Subspaces |
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433 | (14) |
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Linear Independence, Basis, and Dimension |
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447 | (20) |
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Exploration: Magic Squares |
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464 | (3) |
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467 | (9) |
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476 | (9) |
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The Kernel and Range of a Linear Transformation |
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485 | (16) |
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The Matrix of a Linear Transformation |
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501 | (21) |
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Exploration: Tilings, Lattices, and the Crystallographic Restriction |
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519 | (3) |
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522 | (16) |
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Homogeneous Linear Differential Equations |
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522 | (7) |
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529 | (7) |
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536 | (2) |
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Distance and Approximation |
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538 | (96) |
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Introduction: Taxicab Geometry |
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538 | (2) |
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540 | (21) |
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Exploration: Vectors and Matrices with Complex Entries |
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552 | (4) |
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Exploration: Geometric Inequalities and Optimization Problems |
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556 | (5) |
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Norms and Distance Functions |
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561 | (16) |
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Least Squares Approximation |
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577 | (22) |
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The Singular Value Decomposition |
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599 | (20) |
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Vignette: Digital Image Compression |
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616 | (3) |
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619 | (15) |
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Approximation of Functions |
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619 | (7) |
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626 | (5) |
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631 | (3) |
Appendix A Mathematical Notation and Methods of Proof |
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634 | (9) |
Appendix B Mathematical Induction |
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643 | (7) |
Appendix C Complex Numbers |
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650 | (11) |
Appendix D Polynomials |
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661 | (10) |
Answers to Selected Odd-Numbered Exercises |
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671 | (35) |
Index |
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706 | |