Vectors, Pure and Applied: A General Introduction to Linear Algebra

Contents......Page 9
Introduction......Page 13
Part I Familiar vector spaces......Page 15
1.1 Two hundred years of algebra......Page 17
1.2 Computational matters......Page 22
1.3 Detached coefficients......Page 26
1.4 Another fifty years......Page 29
1.5 Further exercises......Page 32
2.1 Geometric vectors......Page 34
2.2 Higher dimensions......Page 38
2.3 Euclidean distance......Page 41
2.4 Geometry, plane and solid......Page 46
2.5 Further exercises......Page 50
3.1 The summation convention......Page 56
3.2 Multiplying matrices......Page 57
3.3 More algebra for square matrices......Page 59
3.4 Decomposition into elementary matrices......Page 63
3.5 Calculating the inverse......Page 68
3.6 Further exercises......Page 70
4.1 The area of a parallelogram......Page 74
4.2 Rescaling......Page 78
4.3 3 × 3 determinants......Page 80
4.4 Determinants of n × n matrices......Page 86
4.5 Calculating determinants......Page 89
4.6 Further exercises......Page 95
5.1 The space Cn......Page 101
5.2 Abstract vector spaces......Page 102
5.3 Linear maps......Page 105
5.4 Dimension......Page 109
5.5 Image and kernel......Page 117
5.6 Secret sharing......Page 125
5.7 Further exercises......Page 128
6.1 Linear maps, bases and matrices......Page 132
6.2 Eigenvectors and eigenvalues......Page 136
6.3 Diagonalisation and eigenvectors......Page 139
6.4 Linear maps from C2 to itself......Page 141
6.5 Diagonalising square matrices......Page 146
6.6 Iteration's artful aid......Page 150
6.7 LU factorisation......Page 155
6.8 Further exercises......Page 160
7.1 Orthonormal bases......Page 174
7.2 Orthogonal maps and matrices......Page 178
7.3 Rotations and reflections in R2 and R3......Page 183
7.4 Reflections in Rn......Page 188
7.5 QR factorisation......Page 191
7.6 Further exercises......Page 196
8.1 Symmetric maps......Page 206
8.2 Eigenvectors for symmetric linear maps......Page 209
8.3 Stationary points......Page 215
8.4 Complex inner product......Page 217
8.5 Further exercises......Page 221
9.1 Physical vectors......Page 225
9.2 General Cartesian tensors......Page 228
9.3 More examples......Page 230
9.4 The vector product......Page 234
9.5 Further exercises......Page 241
10.1 Some tensorial theorems......Page 247
10.2 A (very) little mechanics......Page 251
10.3 Left-hand, right-hand......Page 256
10.4 General tensors......Page 258
10.5 Further exercises......Page 261
Part II General vector spaces......Page 271
11.1 A look at L(U,V)......Page 273
11.2 A look at L(U,U)......Page 280
11.3 Duals (almost) without using bases......Page 283
11.4 Duals using bases......Page 290
11.5 Further exercises......Page 297
12.1 Direct sums......Page 305
12.2 The Cayley-Hamilton theorem......Page 310
12.3 Minimal polynomials......Page 315
12.4 The Jordan normal form......Page 321
12.5 Applications......Page 326
12.6 Further exercises......Page 330
13.2 Vector spaces over fields......Page 343
13.3 Error correcting codes......Page 348
13.4 Further exercises......Page 354
14.1 Orthogonal polynomials......Page 358
14.2 Inner products and dual spaces......Page 367
14.3 Complex inner product spaces......Page 373
14.4 Further exercises......Page 378
15.1 Distance on L(U,U)......Page 383
15.2 Inner products and triangularisation......Page 390
15.3 The spectral radius......Page 393
15.4 Normal maps......Page 397
15.5 Further exercises......Page 401
16.1 Bilinear forms......Page 413
16.2 Rank and signature......Page 421
16.3 Positive definiteness......Page 428
16.4 Antisymmetric bilinear forms......Page 435
16.5 Further exercises......Page 439
References......Page 452
Index......Page 454