Linear Algebra : Concepts and Methods

Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarize the material to help students consolidate their learning as they progress through the book. Linear Algebra: Concepts and Methods......Page 3 Title ......Page 5 Copyright ......Page 6 Dedication ......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......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 ......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......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......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......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 "Any student studying linear algebra will welcome this textbook, which provides a thorough, yet concise, treatment of key topics in university linear algebra courses. Blending practice and theory, the book enables students to practice and master the standard methods as well as understand how they actually work. At every stage the authors take care to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses only on the fundamental topics. Hundreds of examples and exercises, including solutions, give students plenty of hands-on practice End-of-chapter sections summarize material to help students consolidate their learning Ideal as a course text and for self-study Instructors can use the many examples and exercises to supplement their own assignments Both authors have extensive experience of undergraduate teaching and of preparation of distance learning materials"-- Provided by publisher "Any student studying linear algebra will welcome this textbook, which provides a thorough, yet concise, treatment of key topics in university linear algebra courses. Blending practice and theory, the book enables students to practice and master the standard methods as well as understand how they actually work. At every stage the authors take care to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses only on the fundamental topics. Hundreds of examples and exercises, including solutions, give students plenty of hands-on practice End-of-chapter sections summarise material to help students consolidate their learning Ideal as a course text and for self-study Instructors can use the many examples and exercises to supplement their own assignments Both authors have extensive experience of undergraduate teaching and of preparation of distance learning materials"-- Provided by publisher Any student studying linear algebra will welcome this textbook, which provides a thorough, yet concise, treatment of key topics in university linear algebra courses. The authors, who have extensive teaching experience, provide hundreds of examples and exercises with a complete list of solutions, to enable students to practise and master the standard methods. Crucially, the authors also give clear explanations of how the methods really work, so that readers can gain a sound understanding of the underlying theory. End-of-chapter sections summarise the material to help students consolidate their learning as they progress through the book. At every stage the authors take care to ensure that the discussion is no more complicated or abstract than it needs to be and focuses only on the fundamental topics. Instructors can draw on the many examples and exercises to supplement their own assignments. Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarise the material to help students consolidate their learning as they progress through the book. Machine generated contents note: Preface; Preliminaries: before we begin; 1. Matrices and vectors; 2. Systems of linear equations; 3. Matrix inversion and determinants; 4. Rank, range and linear equations; 5. Vector spaces; 6. Linear independence, bases and dimension; 7. Linear transformations and change of basis; 8. Diagonalisation; 9. Applications of diagonalisation; 10. Inner products and orthogonality; 11. Orthogonal diagonalisation and its applications; 12. Direct sums and projections; 13. Complex matrices and vector spaces; 14. Comments on exercises; Index. This thorough, yet concise, textbook covers key topics in first- and second-year university courses and includes many examples and exercises with solutions to help students practise and master the relevant methods. Crucially, it fully develops the underlying theory so that students can understand how these methods really work.