Linear Algebra and Analytic Geometry for Physical Sciences

Contents......Page 7
Introduction......Page 10
 1.1 Applied Vectors......Page 12
 1.2 Coordinate Systems......Page 16
 1.3 More Vector Operations......Page 20
 1.4 Divergence, Rotor, Gradient and Laplacian......Page 26
 2.1 Definition and Basic Properties......Page 28
 2.2 Vector Subspaces......Page 32
 2.3 Linear Combinations......Page 35
 2.4 Bases of a Vector Space......Page 39
 2.5 The Dimension of a Vector Space......Page 44
 3.1 Scalar Product, Norm......Page 46
 3.2 Orthogonality......Page 50
 3.3 Orthonormal Basis......Page 52
 3.4 Hermitian Products......Page 56
 4.1 Basic Notions......Page 57
 4.2 The Rank of a Matrix......Page 63
 4.3 Reduced Matrices......Page 68
 4.4 Reduction of Matrices......Page 70
 4.5 The Trace of a Matrix......Page 76
 5.1 A Multilinear Alternating Mapping......Page 78
 5.2 Computing Determinants via a Reduction Procedure......Page 83
 5.3 Invertible Matrices......Page 86
 6.1 Basic Notions......Page 88
 6.2 The Space of Solutions for Reduced Systems......Page 90
 6.3 The Space of Solutions for a General Linear System......Page 93
 6.4 Homogeneous Linear Systems......Page 103
 7.1 Linear Transformations and Matrices......Page 105
 7.3 Kernel and Image of a Linear Map......Page 112
 7.4 Isomorphisms......Page 115
 7.5 Computing the Kernel of a Linear Map......Page 116
 7.6 Computing the Image of a Linear Map......Page 119
 7.7 Injectivity and Surjectivity Criteria......Page 122
 7.8 Composition of Linear Maps......Page 124
 7.9 Change of Basis in a Vector Space......Page 126
 8.1 The Dual of a Vector Space......Page 133
 8.2 The Dirac's Bra-Ket Formalism......Page 136
 9.1 Endomorphisms......Page 139
 9.2 Eigenvalues and Eigenvectors......Page 141
 9.3 The Characteristic Polynomial of an Endomorphism......Page 146
 9.4 Diagonalisation of an Endomorphism......Page 151
 9.5 The Jordan Normal Form......Page 155
 10.1 Orthogonal Matrices and Isometries......Page 159
 10.2 Self-adjoint Endomorphisms......Page 164
 10.3 Orthogonal Projections......Page 166
 10.4 The Diagonalization of Self-adjoint Endomorphisms......Page 171
 10.5 The Diagonalization of Symmetric Matrices......Page 175
 11.1 Skew-Adjoint Endomorphisms......Page 181
 11.2 The Exponential of a Matrix......Page 186
 11.3 Rotations in Two Dimensions......Page 188
 11.4 Rotations in Three Dimensions......Page 190
 11.5 The Lie Algebra mathfrakso(3)......Page 196
 11.6 The Angular Velocity......Page 199
 11.7 Rigid Bodies and Inertia Matrix......Page 202
 12.1 The Adjoint Endomorphism......Page 205
 12.2 Spectral Theory for Normal Endomorphisms......Page 211
 12.3 The Unitary Group......Page 215
 13.1 Quadratic Forms on Real Vector Spaces......Page 220
 13.2 Quadratic Forms on Complex Vector Spaces......Page 229
 13.3 The Minkowski Spacetime......Page 231
 13.4 Electro-Magnetism......Page 236
 14.1 Affine Spaces......Page 241
 14.2 Lines and Planes......Page 245
 14.3 General Linear Affine Varieties and Parallelism......Page 251
 14.4 The Cartesian Form of Linear Affine Varieties......Page 255
 14.5 Intersection of Linear Affine Varieties......Page 264
 15.1 Euclidean Affine Spaces......Page 274
 15.2 Orthogonality Between Linear Affine Varieties......Page 277
 15.3 The Distance Between Linear Affine Varieties......Page 281
 15.4 Bundles of Lines and of Planes......Page 288
 15.5 Symmetries......Page 292
 16.1 Conic Sections as Geometric Loci......Page 298
 16.2 The Equation of a Conic in Matrix Form......Page 303
 16.3 Reduction to Canonical Form of a Conic: Translations......Page 306
 16.4 Eccentricity: Part 1......Page 312
 16.5 Conic Sections and Kepler Motions......Page 314
 16.6 Reduction to Canonical Form of a Conic: Rotations......Page 315
 16.7 Eccentricity: Part 2......Page 323
 16.8 Why Conic Sections......Page 328
 A.1 A Few Notions of Set Theory......Page 333
 A.2 Groups......Page 335
 A.3 Rings and Fields......Page 338
 A.4 Maps Preserving Algebraic Structures......Page 341
 A.5 Complex Numbers......Page 343
 A.6 Integers Modulo A Prime Number......Page 344
Index......Page 346