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1.MATRICES AND SYSTEMS OF LINEAR EQUATIONS 1

1.1 Introduction to Matrices and Systems of Linear Equations 2

1.2 Echelon Form and Gauss-Jordan Elimination 14

1.3 Consistent Systems of Linear Equations 28

1.4 Applications（Optional） 39

1.5 Matrix Operations 46

1.6 Algebraic Properties of Matrix Operations 61

1.7 Linear Independence and Nonsingular Matrices 71

1.8 Data Fitting, Numerical Integration, and Numerical Differentiation （Optional） 80

1.9 Matrix Inverses and Their Properties 92

2.VECTORS IN 2-SPACE AND 3-SPACE 113

2.1 Vectors in the Plane 114

2.2 Vectors in Space 128

2.3 The Dot Product and the Cross Product 135

2.4 Lines and Planes in Space 148

3.THE VECTOR SPACE Rn 163

3.1 Introduction 164

3.2 Vector Space Properties of Rn 167

3.3 Examples of Subspaces 176

3.4 Bases for Subspaces 188

3.5 Dimension 202

3.6 Orthogonal Bases for Subspaces 214

3.7 Linear Transformations from Rn to Rm 225

3.8 Least-Squares Solutions to Inconsistent Systems,with Applications to Data Fitting 243

3.9 Theory and Practice of Least Squares 255

4.THE EIGENVALUE PROBLEM 275

4.1 The Eigenvalue Problem for （2×2） Matrices 276

4.2 Determinants and the Eigenvalue Problem 280

4.3 Elementary Operations and Determinants （Optional） 290

4.4 Eigenvalues and the Characteristic Polynomial 298

4.5 Eigenvectorsand Eigenspaces 307

4.6 Complex Eigenvalues and Eigenvectors 315

4.7 Similarity Transformations and Diagonalization 325

4.8 Difference Equations; Markov Chains; Systems of Differential Equations （Optional） 338

5.VECTOR SPACES AND LINEAR TRANSFORMATIONS 357

5.1 Introduction 358

5.2 Vector Spaces 360

5.3 Subspaces 368

5.4 Linear Independence, Bases, and Coordinates 375

5.5 Dimension 388

5.6 Inner-Product Spaces, Orthogonal Bases, and Projections （Optional） 392

5.7 Linear Transformations 403

5.8 Operations with Linear Transformations 411

5.9 Matrix Representations for Linear Transformations 419

5.10 Change of Basis and Diagonalization 431

6.DETERMINANTS 447

6.1 Introduction 448

6.2 Cofactor Expansions of Determinants 448

6.3 Elementa Operations and Determinants 455

6.4 Cramer's Rule 465

6.5 Applications of Determinants:Inverses and Wronksians 471

7.EIGENVALUES AND APPLICATIONS 483

7.1 Quadratic Forms 484

7.2 Systems of Differential Equations 493

7.3 Transformation to Hessenberg Form 502

7.4 Eigenvalues of Hessenberg Matrices 510

7.5 Householder Transformations 519

7.6 The QR Factorization and Least-Squares Solutions 531

7.7 Matrix Polynomials and the Cayley-Hamilton Theorem 540

7.8 Generalized Eigenvectors and Solutions of Systemsof Differential Equations 546