A Brief Course in Linear Algebra

Chapter I. Linear Algebra, Basic Notions 1
1.1 Introduction 1
1.2 Matrix Algebra 4
1.3 Formal Rules 12
1.4 Linear Systems of Algebraic Equations 15
1.5 Singularity, Pivots, and Invertible Matrices 24
1.6 Gauss-Jordan Reduction in the General Case 36
1.7 Homogeneous Systems and Vector Subspaces 46
1.8 Linear Independence, Bases, and Dimension 51
1.9 Calculations in R n 62
1.10 Review Problems 67
Chapter II. Determinants and Eigenvalues 71
2.1 Introduction 71
2.2 Definition of the Determinant 74
2.3 Some Important Properties of Determinants 82
2.4 Eigenvalues and Eigenvectors 89
2.5 Diagonalization 100
2.6 The Exponential of a Matrix 105
2.7 Review 108
Chapter III. Applications 111
3.1 Real Symmetric Matrices 111
3.2 Repeated Eigenvalues, The Gram–Schmidt Process 113
3.3 Change of Coordinates 118
3.4 Classification of Conics and Quadrics 125
3.5 Conics and the Method of Lagrange Multipliers 133
3.6 Normal Modes 139
3.7 Review 147
Chapter IV. Index 149