Elementary Linear Algebra: Applications Version, 12th Edition, gives an elementary treatment of linear algebra that is suitable for a first course for undergraduate students. The classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation. In this edition, a new section has been added to describe the applications of linear algebra in emerging fields such as data science, machine learning, climate science, geomatics, and biological modeling. New exercises have been added with special attention to the expanded early introduction
to linear transformations and new examples have been added, where needed, to support the exercise sets. Calculus is not a prerequisite, but there are clearly labeled exercises and examples (which can be omitted without loss of continuity) for students who have studied calculus.
1 Systems of Linear Equations and Matrices 1
1.1 Introduction to Systems of Linear Equations 2
1.2 Gaussian Elimination 11
1.3 Matrices and Matrix Operations 25
1.4 Inverses; Algebraic Properties of Matrices 40
1.5 Elementary Matrices and a Method for Finding A-1 53
1.6 More on Linear Systems and Invertible Matrices 62
1.7 Diagonal, Triangular, and Symmetric Matrices 69
1.8 Introduction to Linear Transformations 76
1.9 Compositions of Matrix Transformations 90
1.10 Applications of Linear Systems 98
1.11 Leontief Input-Output Models 110
2 Determinants 119
2.1 Determinants by Cofactor Expansion 119
2.2 Evaluating Determinants by Row Reduction 127
2.3 Properties of Determinants; Cramer's Rule 134
3 Euclidean Vector Spaces 149
3.1 Vectors in 2-Space, 3-Space, and n-Space 149
3.2 Norm, Dot Product, and Distance in Rn 161
3.3 Orthogonality 175
3.4 The Geometry of Linear Systems 186
3.5 Cross Product 193
4 General Vector Spaces 205
4.1 Real Vector Spaces 205
4.2 Subspaces 214
4.3 Spanning Sets 223
4.4 Linear Independence 231
4.5 Coordinates and Basis 241
4.6 Dimension 251
4.7 Change of Basis 259
4.8 Row Space, Column Space, and Null Space 266
4.9 Rank, Nullity, and the Fundamental Matrix Spaces 279
5 Eigenvalues and Eigenvectors 295
5.1 Eigenvalues and Eigenvectors 295
5.2 Diagonalization 305
5.3 Complex Vector Spaces 315
5.4 Differential Equations 327
5.5 Dynamical Systems and Markov Chains 333
6 Inner Product Spaces 347
6.1 Inner Products 347
6.2 Angle and Orthogonality in Inner Product Spaces 358
6.3 Gram-Schmidt Process; QR-Decomposition 367
6.4 Best Approximation; Least Squares 382
6.5 Mathematical Modeling Using Least Squares 391
6.6 Function Approximation; Fourier Series 398
7 Diagonalization and Quadratic Forms 407
7.1 Orthogonal Matrices 407
7.2 Orthogonal Diagonalization 416
7.3 Quadratic Forms 424
7.4 Optimization Using Quadratic Forms 437
7.5 Hermitian, Unitary, and Normal Matrices 444
8 General Linear Transformations 455
8.1 General Linear Transformations 455
8.2 Compositions and Inverse Transformations 468
8.3 Isomorphism 480
8.4 Matrices for General Linear Transformations 486
8.5 Similarity 496
8.6 Geometry of Matrix Operators 502
9 Numerical Methods 519
9.1 LU-Decompositions 519
9.2 The Power Method 529
9.3 Comparison of Procedures for Solving Linear Systems 538
9.4 Singular Value Decomposition 542
9.5 Data Compression Using Singular Value Decomposition 550
10 Applications of Linear Algebra 555
10.1 Constructing Curves and Surfaces Through Specified Points 555
10.2 The Earliest Applications of Linear Algebra 561
10.3 Cubic Spline Interpolation 568
10.4 Markov Chains 578
10.5 Graph Theory 587
10.6 Games of Strategy 597
10.7 Forest Management 605
10.8 Computer Graphics 612
10.9 Equilibrium Temperature Distributions 620
10.10 Computed Tomography 629
10.11 Fractals 639
10.12 Chaos 655
10.13 Cryptography 668
10.14 Genetics 679
10.15 Age-Specific Population Growth 688
10.16 Harvesting of Animal Populations 697
10.17 A Least Squares Model for Human Hearing 705
10.18 Warps and Morphs 711
10.19 Internet Search Engines 720
10.20 Facial Recognition 726
APPENDIX A Working with Proofs A1
APPENDIX B Complex Numbers A7
ANSWERS TO EXERCISES A15
INDEX I1
SUPPLEMENTAL ONLINE TOPICS
LINEAR PROGRAMMING: A GEOMETRIC APPROACH
LINEAR PROGRAMMING: BASIC CONCEPTS
LINEAR PROGRAMMING: THE SIMPLEX METHOD
VECTORS IN PLANE GEOMETRY
EQUILIBRIUM OF RIGID BODIES
THE ASSIGNMENT PROBLEM
THE DETERMINANT FUNCTION
EVALUATING DETERMINANTS BY ROW REDUCTION
LEONTIEF ECONOMIC MODELS
New to this Edition
New Application Section ' A new section on the mathematics of facial recognition has been added to Chapter 10.
Earlier Linear Transformations ' Selected material on linear transformations that was covered later in
the previous edition has been moved to Chapter 1 to provide a more complete early introduction to the topic. Specifically, some of the material in Sections 4.10 and 4.11 of the previous edition was extracted to form the new Section 1.9, and the remaining material is now in Section 8.6.
New Section 4.3 Devoted to Spanning Sets ' Section 4.2 of the previous edition dealt with both subspaces and spanning sets. Classroom experience has suggested that too many concepts were being introduced at once, so we have slowed down the pace and split off the material on spanning sets to create a new Section 4.3.
New Box Element ' A new section has been added to describe the applications of linear algebra in emerging fields such as data science, machine learning, climate science, geomatics, and biological modeling.
New Examples ' New examples have been added, where needed, to support the exercise sets.
New Exercises ' New exercises have been added with special attention to the expanded early introduction to linear transformations.
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