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Chapter 1 Systems of Linear Equations and Matrices |
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1 | (104) |
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1.1 Introduction to Systems of Linear Equations |
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2 | (9) |
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11 | (14) |
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1.3 Matrices and Matrix Operations |
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25 | (14) |
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1.4 Inverses; Algebraic Properties of Matrices |
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39 | (13) |
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1.5 Elementary Matrices and a Method for Finding A-1 |
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52 | (9) |
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1.6 More on Linear Systems and Invertible Matrices |
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61 | (6) |
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1.7 Diagonal, Triangular, and Symmetric Matrices |
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67 | (8) |
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1.8 Matrix Transformations |
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75 | (9) |
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1.9 Applications of Linear Systems |
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84 | (12) |
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Network Analysis (Traffic Flow) |
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84 | (2) |
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86 | (2) |
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Balancing Chemical Equations |
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88 | (3) |
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91 | (5) |
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1.10 Application: Leontief Input-Output Models |
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96 | (9) |
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105 | (26) |
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2.1 Determinants by Cofactor Expansion |
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105 | (8) |
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2.2 Evaluating Determinants by Row Reduction |
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113 | (5) |
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2.3 Properties of Determinants; Cramer's Rule |
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118 | (13) |
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Chapter 3 Euclidean Vector Spaces |
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131 | (52) |
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3.1 Vectors in 2-Space, 3-Space, and n-Space |
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131 | (11) |
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3.2 Norm, Dot Product, and Distance in Rn |
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142 | (13) |
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155 | (9) |
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3.4 The Geometry of Linear Systems |
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164 | (8) |
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172 | (11) |
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Chapter 4 General Vector Spaces |
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183 | (108) |
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183 | (8) |
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191 | (11) |
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202 | (10) |
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4.4 Coordinates and Basis |
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212 | (9) |
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221 | (8) |
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229 | (8) |
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4.7 Row Space, Column Space, and Null Space |
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237 | (11) |
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4.8 Rank, Nullity, and the Fundamental Matrix Spaces |
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248 | (11) |
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4.9 Basic Matrix Transformations in R2 and R3 |
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259 | (11) |
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4.10 Properties of Matrix Transformations |
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270 | (10) |
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4.11 Application: Geometry of Matrix Operators on R2 |
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280 | (11) |
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Chapter 5 Eigenvalues and Eigenvectors |
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291 | (54) |
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5.1 Eigenvalues and Eigenvectors |
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291 | (11) |
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302 | (11) |
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5.3 Complex Vector Spaces |
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313 | (13) |
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5.4 Application: Differential Equations |
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326 | (6) |
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5.5 Application: Dynamical Systems and Markov Chains |
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332 | (13) |
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Chapter 6 Inner Product Spaces |
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345 | (56) |
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345 | (10) |
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6.2 Angle and Orthogonality in Inner Product Spaces |
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355 | (9) |
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6.3 Gram--Schmidt Process; QR-Decomposition |
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364 | (14) |
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6.4 Best Approximation; Least Squares |
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378 | (9) |
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6.5 Application: Mathematical Modeling Using Least Squares |
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387 | (7) |
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6.6 Application: Function Approximation; Fourier Series |
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394 | (7) |
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Chapter 7 Diagonalization and Quadratic Forms |
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401 | (46) |
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401 | (8) |
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7.2 Orthogonal Diagonalization |
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409 | (8) |
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417 | (12) |
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7.4 Optimization Using Quadratic Forms |
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429 | (8) |
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7.5 Hermitian, Unitary, and Normal Matrices |
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437 | (10) |
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Chapter 8 General Linear Transformations |
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447 | (44) |
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8.1 General Linear Transformations |
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447 | (11) |
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8.2 Compositions and Inverse Transformations |
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458 | (8) |
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466 | (6) |
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8.4 Matrices for General Linear Transformations |
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472 | (9) |
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481 | (10) |
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Chapter 9 Numerical Methods |
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491 | |
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491 | (10) |
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501 | (8) |
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9.3 Comparison of Procedures for Solving Linear Systems |
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509 | (5) |
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9.4 Singular Value Decomposition |
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514 | (7) |
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9.5 Application: Data Compression Using Singular Value Decomposition |
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521 | |
Appendix A Working with Proofs |
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1 | (4) |
Appendix B Complex Numbers |
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5 | (8) |
Answers to Exercises |
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13 | |
Index |
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1 | |