Elementary linear algebra with applications

cover......Page 1
Preface......Page 2
Acknowledgements......Page 6
1.0. Systems of Linear Equations and Matrices......Page 8
1.1. Introduction to Systems of Linear Equations......Page 9
1.2. Gaussian Elimination......Page 18
1.3. Matrices and Matrix Operations......Page 40
1.4. Inverses; Rules of Matrix Arithmetic......Page 63
1.5. Elementary Matrices and a Method for Finding A-1......Page 82
1.6. Further Results on Systems of Equations and Invertibility......Page 96
1.7. Diagonal, Triangular, and Symmetric Matrices......Page 107
Supplementary Exercises......Page 119
Technology Exercises......Page 124
2.0. Determinants......Page 127
2.1. Determinants by Cofactor Expansion......Page 128
2.2. Evaluating Determinants by Row Reduction......Page 145
2.3. Properties of the Determinant Function......Page 155
2.4. A Combinatorial Approach to Determinants......Page 167
Supplementary Exercises......Page 177
Technology Exercises......Page 181
3.0. Vectors in 2-Space and 3-Space......Page 183
3.1. Introduction to Vectors (Geometric)......Page 184
3.2. Norm of a Vector; Vector Arithmetic......Page 199
3.3. Dot Product; Projections......Page 207
3.4. Cross Product......Page 221
3.5. Lines and Planes in 3-Space......Page 238
Abbreviations......Page 254
Technology Exercises......Page 255
4.0. Euclidean Vector Spaces......Page 257
4.1. Euclidean n-Space......Page 258
4.2. Linear Transformations from Rn to Rm......Page 279
4.3. Properties of Linear Transformations from Rn to Rm......Page 304
4.4. Linear Transformations and Polynomials......Page 324
Technology Exercises......Page 338
5.0. General Vector Spaces......Page 339
5.1. Real Vector Spaces......Page 340
5.2. Subspaces......Page 351
5.3. Linear Independence......Page 370
5.4. Basis and Dimension......Page 383
5.5. Row Space, Column Space, and Nullspace......Page 404
5.6. Rank and Nullity......Page 421
Abbreviations......Page 436
Supplementary Exercises......Page 437
Technology Exercises......Page 441
6.0. Inner Product Spaces......Page 443
6.1. Inner Products......Page 444
6.2. Angle and Orthogonality in Inner Product Spaces......Page 460
6.3. Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition......Page 478
6.4. Best Approximation; Least Squares......Page 498
6.5. Change of Basis......Page 512
6.6. Orthogonal Matrices......Page 521
Supplementary Exercises......Page 532
Technology Exercises......Page 535
7.0. Eigenvalues, Eigenvectors......Page 537
7.1. Eigenvalues and Eigenvectors......Page 538
7.2. Diagonalization......Page 552
7.3. Orthogonal Diagonalization......Page 566
Supplementary Exercises......Page 573
Technology Exercises......Page 576
8.0. Linear Transformations......Page 578
8.1. General Linear Transformations......Page 579
8.2. Kernel and Range......Page 594
8.3. Inverse Linear Transformations......Page 605
8.4. Matrices of General Linear Transformations......Page 619
8.5. Similarity......Page 636
8.6. Isomorphism......Page 651
Supplementary Exercises......Page 657
Technology Exercises......Page 663
9.0. Additional Topics......Page 664
9.1. Application to Differential Equations......Page 665
9.2. Geometry of Linear Operators on R2......Page 673
9.3. Least Squares Fitting to Data......Page 690
9.4. Approximation Problems; Fourier Series......Page 699
9.5. Quadratic Forms......Page 706
9.6. Diagonalizing Quadratic Forms; Conic Sections......Page 719
9.7. Quadric Surfaces......Page 733
9.8. Comparison of Procedures for Solving Linear Systems......Page 740
9.9. LU-Decompositions......Page 751
Technology Exercises......Page 761
10.0. Complex Vector Spaces......Page 763
10.1. Complex Numbers......Page 764
10.2. Division of Complex Numbers......Page 776
10.3. Polar Form of a Complex Number......Page 789
10.4. Complex Vector Spaces......Page 801
10.5. Complex Inner Product Spaces......Page 813
10.6. Unitary, Normal, and Hermitian Matrices......Page 825
Supplementary Exercises......Page 837
Technology Exercises......Page 840
11.0. Applications of Linear Algebra......Page 841
11.1. Constructing Curves and Surfaces through Specified Points......Page 842
11.2. Electrical Networks......Page 851
11.3. Geometric Linear Programming......Page 857
11.4. The Earliest Applications of Linear Algebra......Page 870
11.5. Cubic Spline Interpolation......Page 880
11.6. Markov Chains......Page 893
11.7. Graph Theory......Page 906
11.8. Games of Strategy......Page 923
11.9. Leontief Economic Models......Page 935
11.10. Forest Management......Page 946
11.11. Computer Graphics......Page 955
11.12. Equilibrium Temperature Distributions......Page 968
11.13. Computed Tomography......Page 981
11.14. Fractals......Page 997
11.15. Chaos......Page 1020
11.16. Cryptography......Page 1035
11.17. Genetics......Page 1049
11.18. Age-Specific Population Growth......Page 1062
11.19. Harvesting of Animal Populations......Page 1075
11.20. A Least Squares Model for Human Hearing......Page 1083
11.21. Warps and Morphs......Page 1091
Abbreviations......Page 1104
Answers to Exercises......Page 1105