Linear algebra : concepts and methods

Linear Algebra: Concepts and Methods......Page 3
Title\n......Page 5
Copyright\n......Page 6
Dedication\n......Page 7
Contents......Page 9
Preface......Page 15
 Sets and set notation......Page 17
 Numbers......Page 18
 Mathematical terminology......Page 19
 Powers......Page 20
 Quadratic equations......Page 21
 Polynomial equations......Page 22
 Trigonometry......Page 23
 A little bit of logic......Page 24
1.1 What is a matrix?......Page 26
1.2 Matrix addition and scalar multiplication......Page 27
1.3 Matrix multiplication......Page 28
1.4 Matrix algebra......Page 30
1.5.1 The inverse of a matrix......Page 32
1.5.2 Properties of the inverse......Page 35
1.7.1 The transpose of a matrix......Page 36
1.7.2 Properties of the transpose......Page 37
1.7.3 Symmetric matrices......Page 38
1.8.1 Vectors......Page 39
1.8.2 The inner product of two vectors......Page 40
1.8.3 Vectors and matrices......Page 42
1.9.1 Vectors in R2......Page 43
1.9.2 Inner product......Page 46
1.9.3 Vectors in R3......Page 48
1.10.1 Lines in R2......Page 49
1.10.2 Lines in R3......Page 52
1.11 Planes in R3......Page 55
1.12.1 Vectors and lines in Rn......Page 62
1.13 Learning outcomes......Page 63
1.14 Comments on activities......Page 64
1.15 Exercises......Page 69
1.16 Problems......Page 71
2.1 Systems of linear equations......Page 75
2.2 Row operations......Page 78
2.3.1 The algorithm: reduced row echelon form......Page 80
2.3.2 Consistent and inconsistent systems......Page 85
2.3.3 Linear systems with free variables......Page 87
2.3.4 Solution sets......Page 90
2.4.1 Homogeneous systems......Page 91
2.4.2 Null space......Page 94
2.5 Learning outcomes......Page 97
2.6 Comments on activities......Page 98
2.7 Exercises......Page 100
2.8 Problems......Page 102
3.1.1 Elementary matrices......Page 106
3.1.3 The main theorem......Page 110
3.1.4 Using row operations to find the inverse matrix......Page 112
3.1.5 Verifying an inverse......Page 113
3.2.1 Determinant using cofactors......Page 114
3.2.2 Determinant as a sum of elementary signed products......Page 117
3.3 Results on determinants......Page 120
3.3.1 Determinant using row operations......Page 122
3.3.2 The determinant of a product......Page 128
3.4.1 Using determinants to find an inverse......Page 129
3.4.2 Cramer\'s rule......Page 133
3.5 Leontief input–output analysis\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 135
3.6 Learning outcomes......Page 137
3.7 Comments on activities......Page 138
3.8 Exercises......Page 141
3.9 Problems......Page 144
4.1.1 The definition of rank......Page 147
4.2.1 General solution and rank......Page 149
4.2.2 General solution in vector notation......Page 154
4.3 Range......Page 155
4.5 Comments on activities......Page 158
4.6 Exercises......Page 160
4.7 Problems......Page 162
5.1.1 Definition of a vector space......Page 165
5.1.2 Examples......Page 167
5.1.3 Linear combinations......Page 169
5.2.1 Definition of a subspace......Page 170
5.2.2 Examples......Page 171
5.2.3 Deciding if a subset is a subspace......Page 173
5.2.4 Null space and range of a matrix......Page 174
5.3 Linear span......Page 176
5.3.1 Row space and column space of a matrix......Page 177
5.3.2 Lines and planes in R3......Page 178
5.5 Comments on activities......Page 180
5.6 Exercises......Page 184
5.7 Problems......Page 186
6.1 Linear independence......Page 188
6.1.2 Testing for linear independence in Rn......Page 191
6.1.3 Linear independence and span......Page 194
6.1.4 Linear independence and span in Rn......Page 196
6.2 Bases......Page 197
6.3 Coordinates......Page 201
6.4.1 Definition of dimension......Page 202
6.4.2 Dimension and bases of subspaces......Page 206
6.5.1 Row space, column space and null space......Page 207
6.5.2 The rank–nullity theorem\n......Page 211
6.7 Comments on activities......Page 215
6.8 Exercises......Page 218
6.9 Problems......Page 221
7.1 Linear transformations......Page 226
7.1.1 Examples......Page 227
7.1.2 Linear transformations and matrices......Page 228
7.1.3 Linear transformations on R2......Page 230
7.1.4 Identity and zero linear transformations......Page 232
7.1.5 Composition and combinations of linear transformations......Page 233
7.1.6 Inverse linear transformations......Page 234
7.1.7 Linear transformations from V to W......Page 235
7.2.1 Definitions of range and null space......Page 236
7.2.2 Rank–nullity theorem for linear transformations\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 237
7.3 Coordinate change......Page 239
7.3.1 Change of coordinates from standard to basis B......Page 240
7.3.2 Change of basis as a linear transformation......Page 242
7.3.3 Change of coordinates from basis B to basis B\'......Page 243
7.4.1 Change of basis and linear transformations......Page 245
7.4.2 Similarity......Page 247
7.6 Comments on activities......Page 251
7.7 Exercises......Page 255
7.8 Problems......Page 258
8.1.2 Finding eigenvalues and eigenvectors......Page 263
8.1.3 Eigenspaces......Page 268
8.1.4 Eigenvalues and the matrix......Page 269
8.2.1 Diagonalisation......Page 272
8.2.2 General method......Page 273
8.2.3 Geometrical interpretation......Page 276
8.2.4 Similar matrices......Page 278
8.3 When is diagonalisation possible?......Page 279
8.3.1 Examples of non-diagonalisable matrices......Page 280
8.3.2 Matrices with distinct eigenvalues......Page 281
8.3.3 The general case......Page 282
8.3.4 Algebraic and geometric multiplicity......Page 285
8.4 Learning outcomes......Page 288
8.5 Comments on activities......Page 289
8.6 Exercises......Page 290
8.7 Problems......Page 292
9.1 Powers of matrices......Page 295
9.2.2 Systems of difference equations......Page 298
9.2.3 Solving using matrix powers......Page 300
9.2.4 Solving by change of variable......Page 302
9.2.5 Another example......Page 304
9.2.6 Markov Chains......Page 306
9.3 Linear systems of differential equations......Page 312
9.5 Comments on activities......Page 319
9.6 Exercises......Page 321
9.7 Problems......Page 324
10.1.1 The inner product of real n-vectors......Page 328
10.1.2 Inner products more generally......Page 329
10.1.4 The Cauchy–Schwarz inequality\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 331
10.2.1 Orthogonal vectors......Page 332
10.2.2 A generalised Pythagoras theorem......Page 333
10.2.3 Orthogonality and linear independence......Page 334
10.3.1 Definition of orthogonal matrix......Page 335
10.3.2 Orthonormal sets......Page 336
10.4 Gram–Schmidt orthonormalisation process\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 337
10.5 Learning outcomes......Page 339
10.6 Comments on activities......Page 340
10.7 Exercises......Page 341
10.8 Problems......Page 342
11.1.1 Orthogonal diagonalisation......Page 345
11.1.2 When is orthogonal diagonalisation possible?......Page 347
11.1.3 The case of distinct eigenvalues......Page 348
11.1.4 When eigenvalues are not distinct......Page 350
11.1.5 The general case......Page 353
11.2 Quadratic forms......Page 355
11.2.1 Quadratic forms......Page 356
11.2.2 Definiteness of quadratic forms......Page 357
11.2.3 The characterisation of positive-definiteness......Page 362
11.2.4 Quadratic forms in R2: conic sections......Page 367
11.3 Learning outcomes......Page 371
11.4 Comments on activities......Page 372
11.5 Exercises......Page 374
11.6 Problems......Page 376
12.1.1 The sum of two subspaces......Page 380
12.1.2 Direct sums......Page 381
12.2.1 The orthogonal complement of a subspace......Page 383
12.2.2 Orthogonal complements of null spaces and ranges......Page 385
12.3.1 The definition of a projection......Page 388
12.3.2 An example......Page 389
12.4.1 Projections are idempotents......Page 390
12.5 Orthogonal projection onto the range of a matrix......Page 392
12.6 Minimising the distance to a subspace......Page 395
12.7.2 A linear algebra view......Page 396
12.7.3 Examples......Page 397
12.8 Learning outcomes......Page 399
12.9 Comments on activities......Page 400
12.10 Exercises......Page 401
12.11 Problems......Page 402
13.1 Complex numbers......Page 405
13.1.2 Algebra of complex numbers......Page 406
13.1.3 Roots of polynomials......Page 407
13.1.5 Polar form......Page 409
13.1.6 Exponential form and Euler\'s formula......Page 411
13.2 Complex vector spaces......Page 414
13.3 Complex matrices......Page 415
13.4.1 The inner product on Cn......Page 417
13.4.2 Complex inner product in general......Page 418
13.4.3 Orthogonal vectors......Page 420
13.5.1 The Hermitian conjugate......Page 423
13.5.2 Hermitian matrices......Page 424
13.5.3 Unitary matrices......Page 426
13.6 Unitary diagonalisation and normal matrices......Page 428
13.7 Spectral decomposition......Page 431
13.8 Learning outcomes......Page 436
13.9 Comments on activities......Page 437
13.10 Exercises......Page 440
13.11 Problems......Page 442
 Chapter 1 exercises......Page 447
 Chapter 2 exercises......Page 452
 Chapter 3 exercises......Page 457
 Chapter 4 exercises......Page 465
 Chapter 5 exercises......Page 472
 Chapter 6 exercises......Page 476
 Chapter 7 exercises......Page 484
 Chapter 8 exercises......Page 491
 Chapter 9 exercises......Page 497
 Chapter 10 exercises......Page 508
 Chapter 11 exercises......Page 512
 Chapter 12 exercises......Page 520
 Chapter 13 exercises......Page 523
Index......Page 529