Elementary Linear Algebra, Application Version, International Adaptation, Revised Edition

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

Network Analysis 98

Electrical Circuits 100

Balancing Chemical Equations 103

Polynomial Interpolation 105

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 R n 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 Si